+
    ‰°³j:Ž  ã                   ó°  € ^ RI Ht ^RIHt ^RIHtHtHt ]RŒR l4       t]R�R l4       t	]R 4       t
]R 4       t. RNR	NR
NRNRNRNRNRNRNRNRNRNRNRNRNRNRNRNRNRNRNRNRNRNR NR!NR"NR#NR$NR%NR&NR'NR(NR)NR*NR+NR,NR-NR.NR/NR0NR1NR2NR3NR4NR5NR6NR7NR8NR9NR:NR;NR<NR=NR>NR?NR@NRANRBNRCNRDNRENRFNRGNRHNRINRJNRKNRLNRMNRNNRONRPNRQNRRNRSNRTNRUNRVNRWNRXNRYNRZNR[NR\NR]NR^NR_NR`NRaNRbNRcNRdNReNRfNRgNRhNRiNRjNRkNtRl t]RŽRm l4       t]Rn 4       t]Ro 4       t]Rp 4       t]Rq 4       t]Rr 4       t]Rs 4       t]R�Ru l4       tRv tRw tRx tRy t]Rz 4       t]R�R{ l4       t]R�R| l4       t]R} 4       t]R~ 4       t]R�R€ l4       t]R‘R� l4       t R‚ t!Rƒ t"]^ .Rt3R„ l4       t#]^.^ 3R… l4       t$R† t%R‡ t&Rˆ t'R‰ t(]R’RŠ l4       t)]R‹ 4       t*R# )“é    )Úprint_function)Úxrange)ÚdefunÚdefun_wrappedÚdefun_staticc                 óØ  a aaa	€ S P                  S4      oS P                  S4      oS^ 8  d   S P                  R4      # \        S R4      '       d   S P                  pM	/ ;pS n        S^8X  d;   S^ 8X  d   S P                  5# SV9   d    VS,          w  rEVS P
                  8¼  d   V5# ^o	VV V	V3R lpS P
                  p S^28”  d(   ^S n        S P                  V^ S P                  .^R7      o	V^
,           \        SR,          4      ,           S n        S P                  V^ S P                  .^R7      pS P                  S4      S,          ^S,          ,          S P                  S4      S^,           ,          S^,           ,          ,
          ^V,          S,          S	,          ,           pVS n        S^8X  d)   S P                  S4      '       d   S P
                  V3VS&   V5#   TS n        i ; i)r   z&Stieltjes constants defined for n >= 0Ústieltjes_cachec                 ój  <€ V S,          pVSP                   ,
          SP                  SSP                   V ,          ,
          4      S,          ,          ^V^,          ,           ,          SP                  ^SP                  ,          V ,          4      ^,
          ,          pSP	                  V4      S,          # ©é   )ÚjÚlnÚexpÚpiÚ_re)ÚxÚxaÚvÚaÚctxÚmagÚns   &  €€€€ÚV/home/luca/screen-checking/.venv/lib/python3.14/site-packages/mpmath/functions/zeta.pyÚfÚstieltjes.<locals>.f   sw   ø€ Øˆq�SˆØ�—‘�X�s—v‘v˜a §¡ a¥�iÓ(¨!Õ+Õ+¨Q¨r°1­u­WÕ5°s·w±w¸qÀÇÁ½xÈ½zÓ7JÈ1Õ7LÕMˆØ�w‰w�q‹z˜CÕÐó    )Ú	maxdegreeç      à?)ÚconvertÚ
bad_domainÚhasattrr	   ÚeulerÚprecÚquadÚinfÚintr   Úisint)
r   r   r   r	   r#   Úsr   Úorigr   r   s
   fff      @r   Ú	stieltjesr*      s¢  û€ à�‰�A‹€AØ�‰�A‹€AØˆ1„uØ�~‰~ÐFÓGÐGÜˆsÐ%×&Ò&Ø×-Ñ-‰à02Ð2ˆ˜#Ô-ØˆA„vØ�Œ6Ø—I‘I�:ÐØ�ÔØ% aÕ(‰GˆDØ�s—x‘xÔØ�r�	Ø
€C÷ ð  ð �8‰8€Dð
ð ˆrŒ6ØˆCŒHØ—(‘(˜1˜q §¡˜k°Q�(Ó7ˆCØ˜"•9œs 1 c¥6›{Õ*ˆŒØ�H‰H�Q˜˜3Ÿ7™7˜¨rˆHÓ2ˆØ�F‰F�1‹I�q�L˜!˜A�#Õ §¡¨£¨Q¨q­SÕ!1°1°Qµ3Õ!7Õ7¸!¸A½#¸a½%À½)ÕCˆàˆŒØˆA„v�#—)‘)˜A—,’,Ø!Ÿh™h¨˜]ˆ˜ÑØˆ2€Iøð ˆ�ús   Ã C'G  Ç 	G)c                 óš  € \        V4      pWP                  8X  g   WP                  8X  dC   V^8  d0   WP                  8X  d   V^ 8X  d   V P                  # V P                  # V P                  # V^ 8X  Ed	   V P	                  V4      '       d}   V P                  RRV,          ,           4      pV P                  RRV,          ,
          4      pV P                  V P                  4      ) ^,          V,          RWE,
          ,          ,
          # V P                  V4      '       d   V# V P	                  V P                  RRV,          ,           4      4      V P                  V P                  4      ^,          V,          ,
          # V^ 8”  EdF   RV^,
          ,          V P                  V^,
          RRV,          ,
          4      ,          pRV^,
          ,          V P                  V^,
          RRV,          ,           4      ,          pV P	                  V4      '       dM   V^8X  d7   RV P                  V P                  4      ,          RWE,           ,          ,           # RWE,           ,          # V^8X  dF   V P                  RV P                  V P                  4      ,          RWE,           ,          ,           4      # V P                  RWE,           ,          4      # R# )é   ç      Ð?y              à?Nù       €      à¿ç      à¿)r&   r%   ÚninfÚzeroÚ_imÚloggammar   r   ÚisinfÚ	polygammaÚlogr   )r   ÚtÚ
derivativeÚdr   Úbs   &&&   r   Úsiegelthetar;   ,   sê  € äˆJ‹€AØ	
�g‰gŒ˜Ÿh™hœØˆqŒ5Ø—H‘HŒ}  a¤Ø—x‘x�Ø—7‘7ˆNà—8‘8ˆOØˆA…vØ�7‰7�1�:Š:à—‘˜T $ q¥&�[Ó)ˆAØ—‘˜T $ q¥&�[Ó)ˆAØ—F‘F˜3Ÿ6™6“N�? 1Õ$ QÕ&¨¨q­s­Õ3Ð3à�y‰y˜�|Š|Ø�Ø—7‘7˜3Ÿ<™<¨¨T°!­V­Ó4Ó5¸¿¹¸s¿v¹v»ÀqÕ8HÈÕ8JÕJÐJØˆ1…uØ�a˜•c�N˜3Ÿ=™=¨¨1­¨d°4¸µ6­kÓ:Õ:ˆØ�Q�q•S�M˜#Ÿ-™-¨¨!­¨T°$°qµ&­[Ó9Õ9ˆØ�7‰7�1�:Š:Ø�AŒvØ˜CŸG™G C§F¡F›OÕ+¨D°!µ#­JÕ6Ð6à˜Q�S•zÐ!à�AŒvØ—w‘w˜t C§G¡G¨C¯F©F£OÕ3°D¸!½#µJÕ>Ó?Ð?à—w‘w˜t Q¥S�zÓ*Ð*ñ r   c           	      óþ   a a€ ^S P                   ,          S P                  ^S P                  ^S,          ^,           ^S P                  ,          ,          4      ,           4      ,          pS P	                  V V3R lV4      # )r,   c                 óV   <€ SP                  V 4      SP                  S,          ,
          # ©N)r;   r   )r7   r   r   s   &€€r   Ú<lambda>Úgrampoint.<locals>.<lambda>S   s   ø€  #§/¡/°!Ó"4°S·V±V¸AµXÖ"=r   )r   r   ÚlambertwÚeÚfindroot)r   r   Úgs   ff r   Ú	grampointrE   N   sU   ù€ ð 	
ˆ#�&‰&��—‘˜˜3Ÿ<™<¨¨1­¨Q­°°3·5±5µÕ(9Ó:Õ:Ó;Õ;€AØ�<‰<Õ=¸qÓAÐAr   c           
      ó€	  a aaaaaaaaaaaa€ \        VP                  R ^ 4      4      pS P                  V4      pS P                  V4      pS P	                  V4      pS P
                  p \        V4      RV,          8”  dJ   V^,          V8  d<   S P                  W4      pS P                  V4      '       d   S P                  V4      # V# S ;P
                  ^,          un        S P                  S P                  T4      4      pS P                  RS P                  T,          ,           4      oT^ 8X  d<   TS,          pTS n        S P                  T4      '       d   S P                  T4      # T5# S P                  RS P                  T,          ,           ^R7      oS P                  T^R7      oT^8X  d[   S P                  T,          SSS,          ,           ,          pTS n        S P                  T4      '       d   S P                  T4      # T5# S P                  RS P                  T,          ,           ^R7      oS P                  T^R7      oS^,          S P                  S,          ,
          oT^8X  dY   TTTTT3R lp	S P                  T	^4      pT) T,          pTS n        S P                  T4      '       d   S P                  T4      # T5# S ;P
                  ^
,          un        S P                  RS P                  T,          ,           ^R7      oS P                  T^R7      oS^,          ^S P                  ,          S,          S,          ,
          S,
          oT^8X  dl   TTTTTTT3R lp	S P                  T	^4      pS P                  ) T,          T,          pTS n        S P                  T4      '       d   S P                  T4      # T5# S P                  RS P                  T,          ,           ^R7      oS P                  T^R7      oT TTTT3R lp	S P                  T	^4      oT^8X  d\   TTT TTTTTTT3
R lp	S P                  T	^4      pY‡,          pTS n        S P                  T4      '       d   S P                  T4      # T5# T^8”  d!   T 3R lp
S P                  Y¡T^,
          R	7      # R
#   \         d     ELói ; i)r8   éô  r   ©r8   c                  ó8   <€ ^S,          S,          SSS ,          .# ©r,   © )Úcomb1Útheta1ÚzÚz1Úz2s   €€€€€r   ÚtermsÚsiegelz.<locals>.termsz   s   ø€ Ø�b•D˜•K  Q u¥WÐ-Ð-r   c                  ób   <€ ^S,          S,          ^S,          S ,          SSS,          ,           .# )é   rK   )rL   Úcomb2rM   rN   rO   rP   Úz3s   €€€€€€€r   rQ   rR   ‡   s(   ø€ Ø�v•X˜b•[ ! B¥$ u¥*¨b°°5µ­jÐ9Ð9r   c                  óÆ   <€ S^,          RS P                   ,          S^,          ,          S,          RS^,          ,          RS,          S,          S P                   S,          .# )é   éúÿÿÿéýÿÿÿéüÿÿÿ©r   )r   rM   Útheta2Útheta3Útheta4s   €€€€€r   rQ   rR   ‘   sL   ø€ Ø˜•	˜2˜cŸe™e�8 F¨A¥IÕ-¨fÕ4°b¸À½µlØˆv�I�fÕ˜cŸe™e F�lð,ð 	,r   c                  óÂ   <
€ ^S^,          ,          S,          RSP                   ,          S,          S,          ^S,          S,          ^S,          S ,          S	SS,          .# )é   rY   r\   )
rU   Úcomb3r   rM   r]   rN   rO   rP   rV   Úz4s
   €€€€€€€€€€r   rQ   rR   –   sL   ø€ Ø�v˜q•y•[ •^ R¨¯©¥X¨b¥[°Õ%7¸¸6½À"½Ø�2•�e•˜R  5¥ð*ð *r   c                 ó*   <€ SP                  V ^R7      # )rX   rH   )Úsiegelz)r   r   s   &€r   r?   Úsiegelz.<locals>.<lambda>    s   ø€ �c—k‘k !°�kÔ2r   )r   N)r&   Úgetr   r   r2   r#   ÚabsÚrs_zÚ_is_real_typeÚNotImplementedErrorÚexpjr;   Úzetar   Úsum_accuratelyÚdiff)r   r7   Úkwargsr9   Út1Út2r#   r   Úe1rQ   ÚhrL   rU   rb   rM   r]   r^   r_   rN   rO   rP   rV   rc   s   f&,        @@@@@@@@@@@@r   re   re   V   sÝ  ÿü€ äˆF�J‰J�| QÓ'Ó(€AØ�‰�A‹€AØ	�‰�‹€BØ	�‰�‹€BØ�8‰8€DðÜˆr‹7�S˜•XÔ " a¥%¨"¤*Ø—‘˜“ˆAØ× Ñ  ×#Ò#Ø—w‘w˜q“zÐ!ØˆHð ‡H‚H�…N…HØ	�‰�#—/‘/ !Ó$Ó	%€BØ�‰��S—U‘U˜1•W•Ó€AØˆA„vØˆq�DˆØˆŒØ×Ñ˜Q×ÒØ—7‘7˜1“:ÐØˆrˆ	Ø	�‰�#�c—e‘e˜A•g•+¨!ˆÓ	,€BØ�_‰_˜Q¨1ˆ_Ó-€FØˆA„vØ�U‰U�2�X�r˜!˜F�(•{Õ#ˆØˆŒØ×Ñ˜Q×ÒØ—7‘7˜1“:ÐØˆrˆ	Ø	�‰�#�c—e‘e˜A•g•+¨!ˆÓ	,€BØ�_‰_˜Q¨1ˆ_Ó-€FØ�A�I�c—e‘e˜F•lÕ"€EØˆA„v÷	.ñ 	.à×Ñ˜u aÓ(ˆØˆS��UˆØˆŒØ×Ñ˜Q×ÒØ—7‘7˜1“:ÐØˆrˆ	Ø‡H‚H�…N…HØ	�‰�#�c—e‘e˜A•g•+¨!ˆÓ	,€BØ�_‰_˜Q¨1ˆ_Ó-€FØ�A�I�a˜Ÿ™•g˜f•n VÕ+Õ+¨FÕ2€EØˆA„v÷	:ó 	:à×Ñ˜u aÓ(ˆØ�e‰eˆV�B�Y�q�[ˆØˆŒØ×Ñ˜Q×ÒØ—7‘7˜1“:ÐØˆrˆ	Ø	�‰�#�c—e‘e˜A•g•+¨!ˆÓ	,€BØ�_‰_˜Q¨1ˆ_Ó-€F÷,ñ ,ð ×Ñ˜u aÓ(€EØˆA„v÷	*ö 	*ð ×Ñ˜u aÓ(ˆØ�TˆØˆŒØ×Ñ˜Q×ÒØ—7‘7˜1“:ÐØˆrˆ	Øˆ1„uÜ2ˆØ�x‰x˜  !¥ˆxÓ$Ð$ñ øôy ô Úðús   Á)R. Â AR. ÃR. Ò.R=Ò<R=gŒ­�±úD,@g|¸Èc¤5@g‚÷�Ç9@glŠ®Äl>@gY°w@@gjDËâËB@gp`§•˜uD@g,¥Š‰Ý©E@ge¸¸È¨ H@gÜËPñãH@gæNƒ~3|J@gáC ¥9L@g7”ðk¬M@gï1ý·wjN@g?aóë3GP@g�ûž�ÅP@gMÎá>øbQ@gY`³OLR@g€§íR@gXàGEIS@gÊXe�—ÕS@giµ®CºT@gHõ&Q/U@gÜ©³7ÛU@gùK`zÈ3V@gÂwG{W@g‡¢êž¯©W@gr x¸÷W@gË»;I2µX@gSñ¾«WTY@gts7oîY@g´~Úx•\Z@gEA�†ÊÊZ@gKÐéãÁ[@g|jú÷[@gäZÝ~”\@gàU-î�]@gBí¼/œ²]@ggÁÆ °W^@gü'åÙ˜¼^@gè›·o_@g¾CYá_@gÚ¡«¾„2`@g#œ-XÎb`@gg’ví¯`@g®ñS5Ø`@gñåÓ�¶Ca@g™{�wa@gÎ:iõ£a@gÀñ¨=”ãa@g±l@b@g'B/K‡mb@gŸ;up¶Áb@gHýÙµ›Ýb@gÕ¬JÊ c@gŸ¤óóœƒc@gZUàx³c@gÒ�d3Ûc@gu…þ&d@g¼ªá’û`d@gÇÄ¸«/±d@g¸2«îæåd@g–1E#e@g× Lé.=e@gOÐ¢N+­e@g“ÆBV"Øe@gÄ'Ò: f@g£åx¹Lf@g�¦KÖS}f@g‡�c Æf@gÿ£ûg@goLc<)3g@gstUSgg@gô.õ+Q­g@g6×p^Ú h@g£™Î�"h@g%¢-!~hh@g@T¥#œh@go··j}Àh@g
¡Ùx(i@g®ã¸†ËOi@gxÛ™Ê†i@g5Í\¡¬i@gkµ£ ýi@gëŠ�Är2j@gÀqß‹vj@gÈ'"«j@gÖ�d�Ñj@g¹hÜlk@g‰Ñ¿)bk@g”y�à–k@gÇWÈ­k@g=I™X9 l@g `Lewl@göõ¶x|ml@g+PýÌªl@g…Š»‹èl@gü¢bn—ÿl@gßöõ]06m@gâJ}Æ�m@c                 óÎ   € ^ RI pVP                  V 4      pVP                  4        Uu. uF  p\        V4      NK  	  pp\	        V^ ,          4      ^8X  g   Q hV\
        R&   R# u upi )r   N:NNN)ÚurllibÚurlopenÚ	readlinesÚfloatÚroundÚ_zeta_zeros)Úurlrv   r9   r   ÚLs   &    r   Ú_load_zeta_zerosr~   »   sU   € ÛØ�‰�sÓ€AØŸ;™;œ=Ó)™=�aŒˆqŽ™=€AÐ)ä��1•‹;˜"ÔÐÐØ„K�ƒNùò 	*s   ¨A"c           	      ó�  € \        V4      pV^ 8  d!   V P                  V) 4      P                  4       # V^ 8X  d   \        R4      hV\	        \
        4      8”  d   VR8:  d   \        V4       V\	        \
        4      8”  d   \        R4      hV P                  RV P                  V P                  \
        V^,
          ,          4      4      # )r   zn must be nonzeroi † zn too large for zetazerosr   )r&   ÚzetazeroÚ	conjugateÚ
ValueErrorÚlenr{   r~   rk   ÚmpcrC   re   )r   r   r|   s   &&&r   Úoldzetazeror…   Ã   sœ   € äˆA‹€AØˆ1„uØ�|‰|˜Q˜BÓ×)Ñ)Ó+Ð+ØˆA„vÜÐ,Ó-Ð-ØŒ3Œ{ÓÔ  V¤Ü˜ÔØŒ3Œ{ÓÔÜ!Ð"=Ó>Ð>Ø�7‰7�3˜Ÿ™ S§[¡[´+¸aÀ½cÕ2BÓCÓDÐDr   c           	      óÊ  € V^ 8X  d   V P                   # \        V4      R8”  de   V P                  V4      pRV P                  V P                  V4      4      ,          p\        V4      \        V4      V P                  ,          8  d   V# \        V4      R8  d=   V ;P
                  \        V P                  \        V4      ^4      ) 4      ,          un        V P                  ;rEV P                  V4      p^p\        V4      \        V4      V P                  ,          8”  dF   WV,          V,          pWEWpP                  V^,           4      ,          ,          ,          pV^,          pKo  V# )r   éè  r   g{®Gáz„?)r1   rh   ÚliÚsqrtÚepsr#   r&   r6   Úoner   Ú	_zeta_int)r   r   r   r:   r(   r7   ÚuÚks   &&      r   Úriemannrr�   Ð   sý   € àˆA„vØ�x‰xˆä
ˆ1ƒv�„}Ø�F‰F�1‹IˆØ�—‘�s—x‘x “{Ó#Õ#ˆÜˆq‹6”C˜“F˜3Ÿ7™7•NÔ"ØˆHÜ
ˆ1ƒv�„}à�Š”C˜Ÿ™¤ Q£¨Ó*Ð*Ó+Õ+�à�G‰G€O€AØ�‰ˆq‹	€AØ	€AÜ
ˆa‹&”3�q“6˜#Ÿ'™'•>Ô
!Ø�E�A�IˆØ	�!—m‘m A a¥CÓ(Õ(Õ)Õ)ˆØ	ˆQ�ŠØ€Hr   c                 ó^   € \        V4      pV^8  d   ^ # \        V P                  V4      4      # rJ   )r&   rƒ   Úlist_primes)r   r   s   &&r   Úprimepir’   ç   s)   € äˆA‹€AØˆ1„uÙÜˆs�‰˜qÓ!Ó"Ð"r   c                 ó”  € \        V4      pV^8  d   V P                  P                  # VR8  d+   V P                  P                  V P	                  V4      4      # V P                  V4      pV P                  VRR7      V P                  VRR7      ,          ^,          V P                  RR7      ,          pV P                  V P                  P                  V4      V,
          P                  RR7      pV P                  V P                  P                  V4      V,           P                  RR7      pV P                  P                  WE.4      # )r,   ia
  r�   )Úroundingr9   )r&   Ú_ivr1   Úmpfr’   rˆ   r‰   r   r   Úfloorr   Úceilr:   )r   r   ÚmidÚerrr   r:   s   &&    r   Úprimepi2r›   ï   sñ   € äˆA‹€AØˆ1„uØ�w‰w�|‰|ÐØˆ4„xØ�w‰w�{‰{˜3Ÿ;™; q›>Ó*Ð*Ø
�&‰&�‹)€Cà
�(‰(�1˜cˆ(Ó
" 3§6¡6¨!°S 6Ó#9Õ
9¸!Õ
;¸C¿F¹FÈC¸FÓ<PÕ
P€CØ�	‰	�3—7‘7—;‘;˜sÓ# CÕ'×*Ñ*°Sˆ	Ó9€AØ�‰�#—'‘'—+‘+˜cÓ" 3Õ&×)Ñ)°CˆÓ8€AØ�7‰7�;‰;˜�uÓÐr   c                 ó¼  a aa€ S P                  S4      '       d   S# S P                  S4      ^ 8:  d   \        R4      hS^8X  d   S P                  # SR8X  d'   S P	                  S P
                  S P                  4      # S P                  S4      pVS P                  8”  d
   RS,          # S P                  \        V4      ,           oV VV3R lpS P                  V4      # )r   z.prime zeta function defined only for re(s) > 0r   c               3   ó  <"  € SP                   p ^ p V^,          pSP                  V4      pV'       g   K%  SSn         VSP                  SP                  VS,          4      4      ,          V,          pV'       g   R# V Sn         Vx € Kx  5i)r   N)r#   Úmoebiusr   rm   )r)   rŽ   r�   r7   r   r(   Úwps       €€€r   rQ   Úprimezeta.<locals>.terms  sr   øé € Ø—8‘8ˆDð ˆAØØ�Q•�Ø—K‘K “N�ßÙØ�”Ø�c—f‘f˜SŸX™X a¨¥c›]Ó+Õ+¨AÕ-�ßÙà�”Ø”ùs   ƒ1B
µAB
Á;B
)
ÚisnanÚrer‚   r%   r„   r0   r   r#   r&   rn   )r   r(   ÚrrQ   rŸ   s   ff  @r   Ú	primezetar¤   ý   s«   ú€ à
‡y�y�‡|‚|ØˆØ
‡v�vˆaƒy�A„~ÜÐIÓJÐJØˆA„vØ�w‰wˆØˆC„xØ�w‰w�s—x‘x §¡Ó(Ð(Ø�‰ˆq‹	€AØˆ3�8‰8„|Ø�A�vˆà�X‰Xœ˜A›Õˆ÷	ð" ×Ñ˜eÓ$Ð$r   c                 ó
  a aa€ \        S4      oS^ 8  d   \        R4      hS^ 8X  g   S^8X  d   S^8”  d   S P                  S4      # SR8X  d7   S P                  ^^S,
          4      ^,
          S P                  S4      ,          # S^8:  dr   S^ 8X  d
   S^ ,          # S^8X  d
   SR,
          # S^8X  d&   ^S,          S^,
          ,          ^,           ^,          # S^8X  d   SSSR,
          ,          R,           ,          # S P	                  S4      '       d
   SS,          # S P                  S4      '       d   S# \        S4      ^8”  d(   V VV3R lpS P                  V4      SS,          ,          # V VV3R lpS P                  V4      # )r   z-Bernoulli polynomials only defined for n >= 0r   g      ø?c               3   ó(  <"  € SP                   p V x € SP                   S,          p^pVS8:  da   V S^,           V,
          ,          V,          V,          p V^8”  d   V^,          '       g   V SP                  V4      ,          x € V^,          pKg  R# 5i©r   N)r‹   Ú	bernoulli)r7   r£   rŽ   r   r   rN   s      €€€r   rQ   Úbernpoly.<locals>.terms6  sw   øé € Ø—‘ˆAØŠGØ—‘˜•	ˆAØˆAØ�q”&Ø�q˜•s˜1•u•I˜a•K •M�Ø˜Aœ ! a§%¤%Ø˜CŸM™M¨!Ó,Õ,Ò,Ø�Q•’ñ	 ùs   ƒA'BÁ+'Bc               3   ó2  <"  € SP                  S4      x € SP                  p ^pVS8:  dj   V S^,           V,
          ,          V,          S,          p SV,
          pV^8”  d   V^,          '       g   V SP                  V4      ,          x € V^,          pKp  R# 5ir§   )r¨   r‹   )r7   rŽ   Úmr   r   rN   s      €€€r   rQ   r©   B  s}   øé € Ø—-‘- Ó"Ò"Ø—‘ˆAØˆAØ�q”&Ø�q˜•s˜1•u•I˜a•K !•O�Ø�a•C�Ø˜Aœ ! a§%¤%Ø˜CŸM™M¨!Ó,Õ,Ò,Ø�Q•’ñ ùs   ƒA,BÁ0'B)r&   r‚   r¨   Úldexpr4   r¡   rh   rn   )r   r   rN   rQ   s   fff r   Úbernpolyr­   !  s@  ú€ ô 	ˆA‹€AØˆ1„uÜÐHÓIÐIØˆA„v�!�q”&˜Q œUØ�}‰}˜QÓÐØˆC„xØ—	‘	˜!˜A˜a�CÓ  Õ" C§M¡M°!Ó$4Õ4Ð4ØˆA„vØ�Œ6˜!˜q�&�=Ø�Œ6˜!˜c�'�>Ø�Œ6˜1˜Q�3  !¥�9 Q�;¨�/Ð)Ø�Œ6˜!˜Q  #¥�Y s�]Õ+Ð+Ø
‡y�y�‡|‚|Ø�A�vˆØ
‡y�y�‡|‚|ØˆÜ
ˆ1ƒv�„z÷		ð ×!Ñ! %Ó(¨1¨a­4Õ/Ð/÷		ð ×!Ñ! %Ó(Ð(r   c                 óÊ  a aa€ \        S4      oS^ 8  d   \        R4      hS^8:  d8   S^ 8X  d
   S^ ,          # S^8X  d
   SR,
          # S^8X  d   SS^,
          ,          # S P                  S4      '       d
   SS,          # S P                  S4      '       d   S# S^,           pS^ 8X  dL   RS P	                  ^V4      ^,
          ,          S P                  V4      ,          V,          S^ ,          ,          # S^8X  dL   ^S P	                  ^V4      ^,
          ,          S P                  V4      ,          V,          S^ ,          ,          # SR8X  dz   S^,          '       d   S P                  # S^d8  g6   SS P                  RS,          4      ,          S P                  R,          8  d#   S P	                  S P                  S4      S) 4      # V VV3R lpS P                  V4      V,          # )r   z)Euler polynomials only defined for n >= 0r   g.eÏT>úÝ?r-   c               3   ó|  <"  € SP                   p ^ pSP                  ^S^,           4      p SV,
          ^,           pV^8”  d   V^,          '       g)   ^V,
          SP                  V4      ,          V ,          x € V^,          pVS8”  d   R# V S,          SV,
          ^,           ,          V,          p VR,          pK‘  5i)r   r   N)r‹   r¬   r¨   )r7   rŽ   Úwr   r   r   rN   s       €€€r   rQ   Úeulerpoly.<locals>.termsg  s™   øé € Ø�G‰GˆØˆØ�I‰I�a˜˜!�ÓˆØØ�!•�A•ˆAØ˜”E˜a !ŸeœeØ˜•s˜CŸM™M¨!Ó,Õ,¨QÕ.Ò.Ø��FˆAØ�1ŒuÙØ�!•�Q�q•S˜•U•˜A•ˆAØ��HŠAùs   ƒAB<ÁA*B<éþÿÿÿ)r&   r‚   r4   r¡   r¬   r¨   r1   r   r#   Ú	_eulernumrn   )r   r   rN   r«   rQ   s   fff  r   Ú	eulerpolyr´   N  s|  ú€ äˆA‹€AØˆ1„uÜÐDÓEÐEØˆA„vØ�Œ6˜!˜q�&�=Ø�Œ6˜!˜c�'�>Ø�Œ6˜!˜Q˜q�S�'�>Ø
‡y�y�‡|‚|Ø�!�tˆØ
‡y�y�‡|‚|ØˆØ	ˆ!�€AØˆA„vØ�3—9‘9˜Q˜q“> !Õ#Õ$ S§]¡]°1Ó%5Õ5°aÕ7¸!¸Q½$Õ>Ð>ØˆA„vØ�#—)‘)˜A˜a“. Õ"Õ# C§M¡M°!Ó$4Õ4°QÕ6¸¸A½Õ=Ð=ØˆC„xØˆq�5Œ5Ø—8‘8ˆOàˆsŒ7�a˜Ÿ™ 
¨1¥Ó-Õ-°·±¸µÔ=Ø—9‘9˜SŸ]™]¨1Ó-°¨rÓ2Ð2÷ð ×Ñ˜eÓ$ qÕ(Ð(r   Fc                 ó$  € \        V4      pV'       d   \        V P                  V4      4      # V^d8  d!   V P                  V P                  V4      4      # V^,          '       d   V P                  # V P	                  V P                  VR4      V4      # )éd   r   )r&   r³   r–   r1   r¬   r´   )r   r   Úexacts   &&&r   Úeulernumr¸   v  so   € äˆA‹€AßÜ�3—=‘= Ó#Ó$Ð$Øˆ3„wØ�w‰w�s—}‘} QÓ'Ó(Ð(Øˆ1‡u„uØ�x‰xˆØ�9‰9�S—]‘] 1 SÓ)¨1Ó-Ð-r   c                 ó¸   € V P                   5pV P                  p^pTp WeV,          ,          pWG,          p\        V4      V8  d    V# Wb,          pV^,          pK>  r   )rŠ   r1   rh   )r   r(   rN   ÚtolÚlrŽ   ÚzkÚterms   &&&     r   Úpolylog_seriesr¾   ‚  s\   € Ø�7‰7ˆ(€CØ�‰€AØ	€AØ	
€BØ
Ø�q•D�yˆØ	�	ˆÜˆt‹9�sŒ?Øð €Hð 	�ˆØ	ˆQ�Šr   c                 óV  € V^ 8  d
   V^ ,          # RV P                   ,          pW1,          ) V P                  V4      ,          V P                  WP                  V4      V,          4      ,          pV P	                  V4      '       d   V^ 8  d   V P                  V4      pV P                  V4      ^ 8  g-   V P                  V4      ^ 8X  d`   V P                  V4      ^8¼  dJ   WCV P                  V4      V^,
          ,          ,          V P                  V^,
          4      ,          ,          pV# )r   y               @)r   Úfacr­   r   rj   r   r2   )r   r   rN   Útwopijr   s   &&&  r   Úpolylog_continuationrÂ   �  sÓ   € Øˆ1„uØ��sˆ
Ø�#—&‘&�[€FØ	�ˆ
�3—7‘7˜1“:Õ §¡¨Q·±°q³	¸&Õ0@Ó AÕA€AØ
×Ñ˜×Ò  A¤Ø�G‰G�A‹JˆØ
‡w�wˆqƒz�A„~˜#Ÿ'™' !›*¨œ/¨c¯g©g°a«j¸A¬oØ	�C—F‘F˜1“I  !¥Õ$Õ$ S§W¡W¨Q¨q­S£\Õ1Õ1ˆØ€Hr   c                 ó¬  € V P                   5pV^8”  Ed(   V P                  pV P                  V4      pV P                  p^ p W,
          ^8w  dV   V P	                  W,
          4      V,          V P                  V4      ,          pV'       d   \        V4      V8  d   MWH,          pWe,          pV^,          pKv  W@P                  V4      V^,
          ,          V P                  V^,
          4      ,          V P                  V^,
          4      V P                  V P                  V4      ) 4      ,
          ,          ,          pMèV^8  dÜ   V P                  V) 4      V P                  V4      ) V^,
          ,          ,          pV P                  V4      pV P                  p	^ p
 V P                  W¡,
          ^,           4      pV'       dL   W¹,          V P                  V
4      W¡,
          ^,           ,          ,          p\        V4      V8  d   M!WH,          pW•,          p	V
^,          p
K…  \        hV P                  V4      '       d   V^ 8  d   V P                  V4      pV# r   )rŠ   r1   r   r‹   rm   rÀ   rh   Úharmonicr¨   r‚   rj   r   )r   r   rN   rº   r»   ÚlogzÚlogmzr«   r½   ÚlogkzrŽ   r:   s   &&&         r   Úpolylog_unitcirclerÈ   ›  s¶  € Ø�7‰7ˆ(€CØˆ1…uØ�H‰HˆØ�v‰v�a‹yˆØ—‘ˆØˆØØ•˜ŒzØ—x‘x ¥“} uÕ,¨s¯w©w°q«zÕ9�ßœC ›I¨œOØØ•	�Ø�MˆEØ��FŠAØ	�V‰V�A‹Y˜˜1�Õ˜cŸg™g a¨¥c›lÕ*¨C¯L©L¸¸1½Ó,=¸c¿f¹fÀcÇfÁfÈQÃiÀZÓ>PÕ,PÕQÕQ‰Ø	
ˆQŒØ�G‰G�Q�B‹K˜#Ÿ&™& ›)˜ q¨¥sÕ+Õ+ˆØ�v‰v�a‹yˆØ—‘ˆØˆØØ—‘˜a�c !�eÓ$ˆAßØ•w §¡¨£
¨A­C°­EÕ 2Õ3�Ü�t“9˜s”?ØØ•	�Ø�MˆEØ��FŠAäÐØ
×Ñ˜×Ò  A¤Ø�G‰G�A‹JˆØ€Hr   c                 ó"  € V P                   pV P                  V4      p\        V4      ^8  gÎ   V P                  p^V,
          pV P                  V) 4      ^V P                  ,          V,          ,          pV P                  V4      WS,          V P                  VRV,           4      ,          WS) ,          V P                  VRV,
          4      ,          ,           ,          ^V P                  ,          V,          ,          # ^p^ p V P                  W,
          4      V,          p	\        V	4      V P                  8  d   M#W9,          pV^,          pWt,          pWx,          pK]  V P                  ^V,
          4      V) V^,
          ,          ,          V,           # )é   r   )r1   r   rh   r   r   Úgammarm   rŠ   )
r   r(   rN   r   r�   r   Úyr7   rŽ   r½   s
   &&&       r   Úpolylog_generalrÍ   ¿  s*  € Ø�‰€AØ�‰ˆq‹	€AÜˆq‹6�AŒ:Ø�E‰EˆØˆa�CˆØ�F‰F�A�2‹J˜˜#Ÿ&™&� �
Õ#ˆØ�y‰y˜‹|˜Q�T #§(¡(¨1¨S°­UÓ"3Õ3°a¸µe¸C¿H¹HÀQÀsÈ1ÅuÓ<MÕ6MÕMÕNÐPQÐRU×RXÑRXÕPXÐ[\Í}Õ\Ð\Ø	€AØ	€AØ
Ø�x‰x˜�‹}˜qÕ ˆÜˆt‹9�s—w‘wÔØØ	�	ˆØ	ˆQ�ˆØ	�ˆØ	�ŠØ�9‰9�Q�q•S‹>˜A˜2  1¥�+Õ%¨Õ)Ð)r   c           	      óZ  € V P                  V4      pV P                  V4      pV^8X  d   V P                  V4      # VR8X  d   V P                  V4      ) # V^ 8X  d   V^V,
          ,          # V^8X  d   V P                  ^V,
          4      ) # VR8X  d   V^V,
          ^,          ,          # \	        V4      R8:  g(   V P                  V4      '       g   \	        V4      R8  d   \        WV4      # \	        V4      R8¼  dj   V P                  V4      '       dS   RV^,           ,          \        W^V,          4      ,          \        V \        V P                  V4      4      V4      ,           # V P                  V4      '       d&   \        V \        V P                  V4      4      V4      # \        WV4      # )r   ç      è?gÍÌÌÌÌÌì?gffffffö?éÿÿÿÿ)r   rm   Úaltzetar   rh   r'   r¾   rÂ   r&   r¢   rÈ   rÍ   )r   r(   rN   s   &&&r   ÚpolylogrÒ   Ó  sK  € à�‰�A‹€AØ�‰�A‹€AØˆA„vØ�x‰x˜‹{ÐØˆB„wØ—‘˜A“ˆÐØˆA„vØ�!�A•#�wˆØˆA„vØ—‘�q˜•s“ˆ|ÐØˆB„wØ�!�A•#˜•�zÐÜ
ˆ1ƒv�„~˜cŸi™i¨Ÿlšl¬s°1«v¸¬|Ü˜c aÓ(Ð(Ü
ˆ1ƒv�„}˜Ÿ™ 1ŸšØ�a˜•c�{œ>¨#°!°Aµ#Ó6Õ6Ô9MÈcÔSVÐWZ×W]ÑW]Ð^_ÓW`ÓSaÐcdÓ9eÕeÐeØ
‡y�y�‡|‚|Ü! #¤s¨3¯6©6°!«9£~°qÓ9Ð9Ü˜3 1Ó%Ð%r   c                 óÞ  € V P                  V4      '       d(   V^ 8  d!   \        V4      ^,          ^8X  d
   V^ ,          # V'       d   V P                  V4      pMV P                  V4      pV P	                  V4      '       d8   V P	                  V4      '       d!   V P                  V P                  W4      4      # ^V,          pRV P                  W4      V P                  W4      ,
          ,          # )r   r.   )r'   r&   Úexpjpirl   rj   ÚimrÒ   ©r   r(   rN   r   r   r:   s   &&&&  r   Úclsinr×   é  s­   € à
‡y�y�‡|‚|˜˜Aœ¤# a£&¨1¥*°¤/Ø��sˆ
ß	Ø�J‰J�q‹M‰à�H‰H�Q‹KˆØ
×Ñ˜×Ò × 1Ñ 1°!× 4Ò 4Ø�v‰v�c—k‘k !Ó&Ó'Ð'Ø	ˆ!�€AØ�C—K‘K Ó$ s§{¡{°1Ó'7Õ7Õ8Ð8r   c                 óÞ  € V P                  V4      '       d(   V^ 8  d!   \        V4      ^,          ^ 8X  d
   V^ ,          # V'       d   V P                  V4      pMV P                  V4      pV P	                  V4      '       d8   V P	                  V4      '       d!   V P                  V P                  W4      4      # ^V,          pRV P                  W4      V P                  W4      ,           ,          # )r   r   )r'   r&   rÔ   rl   rj   r¢   rÒ   rÖ   s   &&&&  r   ÚclcosrÙ   ö  s­   € à
‡y�y�‡|‚|˜˜Aœ¤# a£&¨1¥*°¤/Ø��sˆ
ß	Ø�J‰J�q‹M‰à�H‰H�Q‹KˆØ
×Ñ˜×Ò × 1Ñ 1°!× 4Ò 4Ø�v‰v�c—k‘k !Ó&Ó'Ð'Ø	ˆ!�€AØ�—‘˜AÓ  3§;¡;¨qÓ#3Õ3Õ4Ð4r   c                 ón   €  V P                   ! V3/ VB #   \         d    T P                  T4      u # i ; ir>   )Ú_altzetark   Ú_altzeta_generic)r   r(   rp   s   &&,r   rÑ   rÑ     s;   € ð'Ø�|Š|˜AÑ( Ñ(Ð(øÜô 'Ø×#Ñ# AÓ&Ò&ð'ús   ‚ •4³4c                 ó¤   € V^8X  d   V P                   ^ V,          ,           # V P                  ^^V,
          4      ) V P                  V4      ,          # r   )Úln2Úpowm1rm   )r   r(   s   &&r   rÜ   rÜ   
  s@   € àˆA„vØ�w‰w˜˜1��}ÐØ�I‰I�a˜˜1�ÓÐ §¡¨£Õ+Ð+r   Nc                 óL  € \        V4      pV^8X  d%   V'       g   V'       g    V P                  ! V3/ VB # V P                  V4      pV P                  pVP                  R4      pVP                  R4      pV'       g7   V'       g/   V P                  R4      V P                  V4      ^ ,          ,
          # V^8X  d�   VR8w  d–   \        V P                  V4      4      p	\        V P                  V4      4      p
\        V	4      RV,          8”  d   ^
V
,          V8  d   V^8:  g   VR8X  d/     V'       d   \        R4       V P                  ! W3/ VB Wpn        # V^8X  d   V P                  # \        V4      pW°P                  8X  dI   V P                  V4      V P                  8X  d    V^ 8X  d   V P                  # V P                   # V^ ,          # V P#                  V4      '       d
   ^V,          # V P                  V4      ^V P                  ,          8”  d4   V^8X  d-   V'       g%   V P                  V P%                  ^V) 4      ,           # V P&                  ! WV3/ VB 5#   \         d     EL3i ; i  \         d    T'       d   \        R4        Mi ; i Ypn        ELD  Ypn        i ; i)	r   ÚmethodÚverboser   zeuler-maclaurinrG   zriemann-siegelz4zeta: Attempting to use the Riemann-Siegel algorithmz0zeta: Could not use the Riemann-Siegel algorithm)r&   Ú_zetark   r   r#   rg   r–   Ú_convert_paramrh   r2   r   ÚprintÚrs_zetar%   r¢   r‹   r1   r¡   ÚpowerÚ_hurwitz)r   r(   r   r8   rá   rp   r9   r#   râ   rÕ   r¢   Úabsss   &&&&&,      r   rm   rm     s  € äˆJ‹€AØˆA„v—qŸFð	Ø—9’9˜QÑ) &Ñ)Ð)ð 	�‰�A‹€AØ�8‰8€DØ�Z‰Z˜Ó!€FØ�j‰j˜Ó#€GßŸ
Ø�w‰w�s‹|˜c×0Ñ0°Ó3°AÕ6Õ6Ð6ØˆA„v�&Ð-Ô-Ü�—‘˜“‹_ˆÜ�—‘˜“‹_ˆô ˆr‹7�S˜•XÔ " R¥%¨$¤,°:À´?ØÐ&Ô&ð
 ðßÜÐTÔUØŸ;š; qÑ?¸Ñ?ð  •ØˆA„vØ�w‰wˆÜˆq‹6€DØ�w‰w„Ø�6‰6�!‹9˜Ÿ™ÔØ�AŒvØ—w‘w�Ø—8‘8ˆOØ��sˆ
Ø	�‰�4�ŠØ��sˆ
Ø
‡v�vˆaƒy�1�S—X‘X•:Ô ! q¤&·Ø�w‰w˜Ÿ™ 1 q bÓ)Õ)Ð)Ø�LŠL˜˜qÑ+ FÑ+Ð+Ð+øô] #ô 	Úð	ûô6 +ô ßÜÐPÔQÙðúð à–ø˜4•úsA   ¤I Ä+I. Ä3I. ÉI+É*I+É.JÊJÊJ ÊJÊJ ÊJ#c                 ó~  € V P                   pVP                  R 4      p ^
pV ;P                   V,          un         V P                  V4      w  r(V P                  V4      ^ 8  d(   V'       d   \	        R4        \        WW#V4      WPn         # V'       d   \	        R4        WW,           V n         \        WW#V^
,           V4      w  ršV P                  V	4      V P                  Wš,           4      ,
          pV'       d&   \	        RV	4       \	        RV
4       \	        RVR4       W·8  d   Wš,           WPn         # \        ^V,          \        V^,           ^dV,          4      4      pWtP                  R^dV,          4      8”  g   KÝ  V P                  R	4      h  \         d     Mi ; iT'       g   EK  \	        R4       EL*  YPn         i ; i)
râ   z#zeta: Attempting reflection formulazzeta: Reflection formula failedz)zeta: Using the Euler-Maclaurin algorithmzTerm 1:zTerm 2:zCancellation:ÚbitsÚmaxpreczzeta: too much cancellation)r#   rg   rä   r¢   rå   Ú_hurwitz_reflectionrk   Ú_hurwitz_emr   ÚmaxÚminÚNoConvergence)r   r(   r   r9   rp   r#   râ   Ú	extraprecÚatypeÚT1ÚT2Úcancellations   &&&&,       r   rè   rè   F  sl  € à�8‰8€DØ�j‰j˜Ó#€GðØˆ	Ø�Š�IÕ�à×%Ñ% aÓ(‰ˆØ�6‰6�!‹9�qŒ=ßÜÐ;Ô<ðÜ*¨3°1¸Ó?ð, �÷# ÜÐ=Ô>ØØÕ'ˆCŒHÜ  ¨¨t°B­w¸Ó@‰FˆBØŸ7™7 2›;¨¯©°µ«Õ7ˆLßÜ�i Ô$Ü�i Ô$Ü�o |°VÔ<ØÔ'Ø•wð �ô	    )¥¬S°ÀÕ1AÀ3ÀtÅ8Ó-LÓM�	ØŸz™z¨)°S¸µXÓ>Ö>Ø×+Ñ+Ð,IÓJÐJøô' 'ô ÙðúçŠwÜÐ7Ö8øð$ �úsP   ŸAF4 Á+F4 Á7F Â
F4 ÂBF4 Ä0AF4 Å:F4 ÆFÆF4 ÆFÆ
F4 Æ'F4 Æ4F<c                 óÈ  a aaa€ V^ 8w  d   \         hS P                  V4      pV) pS P                  V4      '       d:   \        V4      pV^ 8:  d(   S P	                  ^V,
          V4      V^,
          ,          # VR8X  g   VR8X  g   \         h^V,
          o^ p^ p	VoS P                  S4      ^8”  d%   S^,          oVSV,          ,          pV	^,          p	K:  S P                  S4      ^ 8:  d%   VSV,          ,          pS^,          oV	^,          p	K:   VP
                  w  p
oY©S,          ,          p
^T
u;8:  d	   S8:  g   Q h Q hS P                  TT TT3R l\        ^S^,           4       4       4      pT^S P                  S4      ,          ^S P                  ,          S,          S,          ,          ,          pY‹,          pT#    T\        T4      8X  g   Q h\        T4      p
^o LÄ; i)r   ÚQÚZc              3   ó°   <"  € T FK  pSP                  S^,          ^V,          S,          ,
          4      SP                  SVS34      ,          x € KM  	  R# 5i)r,   N)Úcospirè   )Ú.0rŽ   r:   r   Úqr7   s   & €€€€r   Ú	<genexpr>Ú&_hurwitz_reflection.<locals>.<genexpr>Ž  sH   øé € ð ÙˆAð —‘˜1˜Q�3˜q �s 1�u�9Ó% c§l¡l°1°a¸°UÓ&;×;Ò;Ûùs   ƒAA)
rk   r¢   Úisnpintr&   r­   Ú_mpq_ÚfsumÚrangerË   r   )r   r(   r   r9   ró   ÚresÚnegsr   r   ÚshiftÚprD   r:   rý   r7   s   f&&&&       @@@r   rí   rí   k  s›  û€ àˆA„vÜ!Ð!Ø
�&‰&�‹)€CØˆ2€Dà
‡{�{�1‡~‚~Ü�‹HˆØ�Œ6Ø—<‘<  !¥ QÓ'¨1¨Q­3Õ/Ð/Ø�SŒL˜E SœLÜ!Ð!Ø	ˆ!�€Aà	€AØ€EØ	€AØ
�&‰&�‹)�aŒ-Ø	ˆQ�ˆØ	ˆQ��W�ˆØ��
ŠØ
�&‰&�‹)�qŒ.Ø	ˆQ��W�ˆØ	ˆQ�ˆØ��
ŠðØ�w‰w‰ˆˆ1ð
 ˆq�…L€AØ�Ž;�QŒ;Ð‰;Ðˆ;Ø�‰÷ Ü�q˜˜1�”óó 	€Aàˆˆ3�9‰9�Q‹<�˜˜3Ÿ6™6� !� a�Õ	'Õ'€AØ…F€AØ€HøðØ”C˜“FŒ{Ðˆ{Ü�‹FˆØŠús   ÄF> Æ>!G!c           
      ó¨  € V P                  V4      pV) p^ pV^,          pTp	^ p
V P                  V4      '       d   \        V P                  V4      4      pV^,
          p V P	                  WV,           W‡,
          ^,
          V.4      ^ ,          ^ ,          pW¬,          p
W‚,           pV P                  V4      pWã,          pV.p^V,          p^V,          pWÑ) ,          pV'       d5   V P                  V^,           W¾,          4      W³^,           ,          ,          pM^W½V,          ,          ,          pVRV,          V,          ,          p^.pTp^p\        ^V	^,           4       EF•  p^V,          pV^8X  d   ^.pMV^,
          V^,
          .pV FÝ  p\        VV^,           4      pVV8:  d    VP                  VR,          V,          4       ^ .V^,           ,          p\        V4       F$  p^V,
          V,
          VV,          ,          VV&   K&  	  \        ^V^,           4       F;  pVV;;,          VV^,
          ,
          VV^,
          ,          ,          ,          uu&   K=  	  TpVV,          pKß  	  V P                  VV4      V,          V P                  V4      ,          V) ,          pVV,          pV P                  V4      V8  d   V
RV,          V,          3u # VV^,           V^,           ,          ,          pEK˜  	  V'       d    \        RWxRV P                  X4      RV4       Yˆ^,          r‡V P                  V4      ^ 8  g   EKä  W™^,          ,          p	EKö  )r   r   z
Sum range:zterm magnitudeÚ	tolerancerÐ   )r   r'   r&   r   Ú_zetasumr   Úgammaincr  rð   Úappendr   Úfdotr¨   r   rå   r¢   ) r   r(   r   r9   r#   râ   rº   ÚM1ÚM2ÚNÚlsumÚs1r»   ÚM2aÚlogM2aÚlogM2adÚlogsÚlogrÚrM2aÚM2asÚtailsumÚUr£   Úfactr   Új2Úupdsr«   ÚDÚUnÚir7   s    &&&&&&                          r   rî   rî   ”  s´  € à�‰�A‹€AØˆ%€Cà	
€BØ	��€BØ
€AØ€Dà
‡y�y�‡|‚|Ü�—‘˜“
‹OˆØ	
ˆ1�€BØ
à�L‰L˜˜q�D "¥%¨¥'¨A¨3Ó/°Õ2°1Õ5ˆð
 	�	ˆØ�dˆØ—‘˜“ˆØ•)ˆØˆyˆØ��xˆØ��uˆØ�R�yˆßØ—l‘l 1 Q¥3¨­	Ó2°R¸A½#µYÕ>‰Gà˜" R�i�Õ(ˆGØ�3˜•= 4Õ'Õ'ˆØˆCˆØˆØˆÜ�q˜!˜A�#—ˆAà�1•ˆBØ�AŒvØ�s‘à˜1�˜b �d�|�Û�Ü˜˜!˜A�#“J�Ø˜”6Ø—K‘K  R¥¨4¥Ô0Ø�S˜!˜A�#•Y�Ü ž�A¨Q¨q­S°­U°A°aµD­L B q£E™Ü  ! A¥#ž�A¨¨1¯°!°Q°qµSµ'¸1¸Q¸q½S½6Õ1AÕ(A­™Ø�Ø�T•	’ñ ð —‘˜˜DÓ! AÕ%¨¯©°bÓ(9Õ9¸D¸5ÕAˆAØ�q�LˆGØ�w‰w�q‹z˜CÔØ˜b 1�W wÕ.Ð.Ò.Ø�R˜•T˜B˜q�D•MÕ!‹Dñ) ÷* Ü�, Ð(8¸#¿'¹'À!»*ÀkÐSVÔWØ˜•TˆBØ�6‰6�!‹9�q�=Ø�A•�I‹Ar   c                óÒ  a aa
a€ \        S P                  V4      4      RS P                  ,          8  d    S P                  VSW4V4      # S P                  VRR7      oV^ .8g  p\        V4      ^8H  pV'       gŠ   V'       g0   S P                  VV3R l\        V^,           4       4       4      .. 3# V'       dK   V^ ,          o
S P                  VV V
V3R l\        V^,           4       4       4      pRS
,          V,          .. 3# \        V4      p	V'       g   \        V	^,           4      pV U
u. uF  p
S P                  NK  	  pp
V'       d   V U
u. uF  p
S P                  NK  	  pp
M. p\        V^,           4       EFK  pSV,           pVS,          pV'       d)   S P                  S P                  Wï,          ,          4      pV'       dÉ   S P                  V4      ) pV'       dN   VV	,          pV^ ;;,          VV,          ,          uu&   V'       d   V^ ;;,          XV,          ,          uu&   K³  Kµ  S P                  pV FL  o
VS
;;,          VV,          ,          uu&   V'       d   VS
;;,          XV,          ,          uu&   VV,          pKN  	  EK  V^ ;;,          V,          uu&   V'       g   EK6  V^ ;;,          X,          uu&   EKN  	  W¼3#   \         d     EL‘i ; iu up
i u up
i )a‹  
Returns [xd0,xd1,...,xdr], [yd0,yd1,...ydr] where

xdk = D^k     ( 1/a^s     +  1/(a+1)^s      +  ...  +  1/(a+n)^s     )
ydk = D^k conj( 1/a^(1-s) +  1/(a+1)^(1-s)  +  ...  +  1/(a+n)^(1-s) )

D^k = kth derivative with respect to s, k ranges over the given list of
derivatives (which should consist of either a single element
or a range 0,1,...r). If reflect=False, the ydks are not computed.
r   T©r·   c              3   óD   <"  € T F  pSV,           S,          x € K  	  R # 5ir>   rK   )rü   rŽ   r   r  s   & €€r   rþ   Ú_zetasum.<locals>.<genexpr>í  s   øé € Ð>±+¨Q˜a �c DŸ[š[³+ùs   ƒ c              3   óŒ   <"  € T F9  pSP                  SV,           4      S,          SV,           S,          ,          x € K;  	  R # 5ir>   )r   )rü   rŽ   r   r   r9   r  s   & €€€€r   rþ   r%  ð  s1   øé € ÐK¹{¸!˜Ÿ™  !¥› a�¨1¨Q­3°­+×5Ò5»{ùs   ƒAArÐ   )rh   r¢   r#   Ú_zetasum_fastrk   Úfnegrƒ   r  r   rï   r  r1   Úconjr‹   r   )r   r(   r   r   ÚderivativesÚreflectÚhave_derivativesÚhave_one_derivativer   Úmaxdr9   ÚxsÚysrŽ   r°   ÚxtermÚytermÚlogwr7   r  s   f&f&&&    `        @r   r
  r
  Õ  s?  û€ ô ˆ3�6‰6�!‹9ƒ~˜˜cŸh™h�Ô&ð	Ø×$Ñ$ Q¨¨1¸7ÓCÐCð �8‰8�A˜Tˆ8Ó"€DØ" q cÑ)ÐÜ˜kÓ*¨aÑ/ÐßßØ—H‘HÕ>´&¸¸1½´+Ó>Ó>Ð?ÀÐCÐCßØ˜A•ˆAØ—‘×K¼vÀaÈÅc¼{ÓKÓKˆAØ˜!•G˜a•K�= "Ð$Ð$Üˆ{Ó€DßÜ˜D �F“mˆÙ'Ó	(™K�qˆ#�(Œ(™K€BÐ	(ßÙ +Ó,¡˜1ˆc�hŒh¡ˆÐ,ˆàˆÜ�A�a•C�[ˆØ��EˆØ�T•	ˆßØ—H‘H˜SŸW™W¨­	Õ2Ó3ˆEßØ—F‘F˜1“I�:ˆDß"Ø˜t•|�Ø�1—˜ �Õ%“ßØ�q—E˜U T�\Õ)•Eñ ð —G‘G�Û$�AØ�q—E˜U Q�YÕ&“EßØ˜1Ÿ ¨¥Õ*›Ø˜•I’Aô	 %ð ˆq�E�U�N‹EßŠwØ�1—˜•–ñ- ð. ˆ6€MøôW #ô 	Úð	üò 
)ùâ,s   ¶K Ä+KÅK$ËKËKc           	      ó   € V P                  V4      p\        V4      p\        V4      pV^8”  d   \        R4      hV P                  p V ;P                  ^
,          un        V^8X  dx   RpV FT  pV'       g   K  V^8w  g   K  RpV P
                  5p	V ;P                  ^V^,           ,          ,          un        W,          pKV  	  V'       d   V P                  5W`n        # V P                  p
\        ^V^,           4       F¬  pW+V,          ,          '       g   K  V^8X  d_   W¢W´,          ,          V P                  WV3^4      V P                  WV34      V P                  V4      ,          ,
          ,          ,          p
K  W¢W´,          ,          V P                  WV34      ,          ,          p
K®  	  W¤V,          ,          p
W`n        V
5#   Y`n        i ; i)r,   zarbitrary order derivativesTF)r   rƒ   r&   rk   r#   rŠ   r%   r1   r  rm   r6   )r   r(   Úchir8   rý   r9   r#   Ú	have_poler   rt   rN   r  s   &&&&        r   Ú	dirichletr7    so  € à�‰�A‹€AÜˆC‹€AÜˆJ‹€AØˆ1„uÜ!Ð"?Ó@Ð@Ø�8‰8€DðØ�Š�B��Ø�Œ6ØˆIÛ�ß‘1˜˜ažØ %�IØŸ™˜�AØ—H’H  1 Q¥3¥Õ'•HØ•F’Añ ÷ ØŸ™�xð �ð �H‰HˆÜ�q˜˜1�–ˆAØ�Q•3�xŽxØ˜”6Ø˜Q�S� S§X¡X¨a°A°¸Ó%:ØŸ™  q EÓ*¨3¯7©7°1«:Õ5õ&6õ 7õ 7’Að ˜Q�S� C§H¡H¨Q°1°Ó$6Õ6Õ6’Añ ð 	
��T�	ˆàŒØˆ2€Iøð �ús,   Á,G Á8G ÂA	G ÃG Ã4G ÄB%G ÇGc                 óè  a aaa€ S P                   pVV VV3R  lpS P                  ;rgS P                  p^ p	W„8”  dG   Wg,          pV	^,          p	S P                  S P	                  V	4      4      oV! V	4      p\        V4      pKL  ^ p
VP                  R4      '       dÖ   S P                  S4      pRS P                  R,          ,          \        ^V4      ,          SVR,
          ,          ,          S P                  S^S P                  ,          ,          4      ,          S P                  RSS^,          ,          4      ,          \        S P                  S^,          4      4      ,          p
\        V
4      p
V5W©3# )c                 ót   <€ SP                  R S,          SS^,          ,          RR7      SS) ,          ,          # )r   T)Úregularized)r  )r   r   r   Úgammr(   s   &€€€€r   r?   Ú&secondzeta_main_term.<locals>.<lambda>7  s+   ø€ �#—,‘,˜s 1�u a¨¨a­¥i¸T�,ÓBÀ4È1È"Å:ÖMr   Úerrorr   rÐ   r/   )rŠ   r1   r%   rÕ   Úzetazero_memoizedrh   rg   r¢   r   rï   r6   r  rË   )r   r(   r   rp   rº   r   Útotsumr½   Úmgr   rš   Úsgr;  s   fff,        @r   Úsecondzeta_main_termrB  5  s  û€ Ø
�'‰'€CßM€AØ—H‘HÐ€FØ	�‰€BØ	€AØ
Œ(Ø�ˆØ	ˆQ�ˆØ�v‰v�c×+Ñ+¨AÓ.Ó/ˆÙ�‹tˆÜ�‹YŠØ
€CØ‡z�z�'×ÒØ�V‰V�A‹YˆØ�#—&‘&˜2•,Õœs 1 R›yÕ(¨¨R°­V­Õ4°S·W±W¸TÀ1ÀSÇVÁVÅ8½_Ó5MÕMØ�\‰\˜$  $¨¥'¥	Ó*õ+Ü+.¨s¯y©y¸¸1½«~Ó+>õ?ˆä�#‹hˆØˆ7�Cˆ?Ðr   c                 ó4  a aa€ S P                   pVV V3R  lpS P                  ;rgS P                  p^p	W„8”  g   V	^	8  d<   Wg,          pV	^,          p	V! V	4      pV^ 8X  d   S P                  pK;  \        V4      pKH  VP	                  R4      '       d   Tp
V5X
V	3# )c                 óÐ  <€ SP                  R ^S,
          ,          RSP                  V 4      ^,          ,          SR,          ,          4      R SP                  V 4      ,          S^,
          ,          ,          SP                  V 4      ,          SP                  V 4      ,          ^SP	                  R S,          4      ,          SP                  SP
                  4      ,          ,          # )r   r-   rÐ   )r  r6   Úmangoldtr‰   rË   r   ©r   r   r   r(   s   &€€€r   r?   Ú'secondzeta_prime_term.<locals>.<lambda>K  s›   ø€ �#—,‘,˜s A a¥C�y¨¨c¯g©g°a«j¸!­mÕ);¸aÀ"½gÕ)EÓFØ
ˆc�g‰g�a‹j�.˜A˜a�CÕ	 õ"Ø"%§,¡,¨q£/õ2Ø25·(±(¸1³+õ>à	
ˆ3�9‰9�S˜•UÓÕ	˜CŸH™H S§V¡VÓ,Õ	,ö.r   r=  )rŠ   r1   r%   rh   rg   )r   r(   r   rp   rº   r   r?  r½   r@  r   rš   s   fff,       r   Úsecondzeta_prime_termrH  I  sŽ   ú€ Ø
�'‰'€Cö	.€Að —H‘HÐ€FØ	�‰€BØ	€AØ
Œ(�a˜!”eØ�ˆØ	ˆQ�ˆÙ�‹tˆØ�1Œ9Ø—‘ŠBä�T“ŠBØ‡z�z�'×ÒØˆØˆ7�C˜ˆ?Ðr   c                 ó6  a aa€ S P                  S4      '       di   S P                  S4      ^ 8:  dS   \        \        S P                  S4      4      4      pV^,          '       g!   S P	                  R4      V) ^,          ,          # S P
                  pVV V3R lpS P                  pV! ^ 4      pS P                  p^ p	W„8”  d'   Wg,          pV	^,          p	V! V	4      p\        V4      pK,  SRS,          ,          V,          S P                  RS,          4      ,          p
V
# )r   z-0.25c                 óz   <€ R S,          V ,          V RS,          ,           SP                  V 4      ,          ,          # )r-   r   )rÀ   rF  s   &€€€r   r?   Ú%secondzeta_exp_term.<locals>.<lambda>c  s'   ø€ �4˜•6˜A•+  # a¥%¥¨¯©°«Õ3Ö4r   r   )
r'   r¢   r&   rz   r–   rŠ   r1   r%   rh   rË   )r   r(   r   r«   rº   r   r?  r½   r@  r   r   s   fff        r   Úsecondzeta_exp_termrL  ]  sÖ   ú€ Ø
‡y�y�‡|‚|˜Ÿ™˜q›	 QœÜ”�c—f‘f˜Q“iÓ Ó!ˆØ�1�uŒuØ—7‘7˜7Ó# q b¨!¥eÕ,Ð,Ø
�'‰'€CÞ4€AØ�X‰X€FÙˆQ‹4€DØ	�‰€BØ	€AØ
Œ(Ø�ˆØ	ˆQ�ˆÙ�‹tˆÜ�‹YŠØ	ˆC��E�
�6Õ˜#Ÿ)™) C¨¥EÓ*Õ*€AØ€Hr   c           	      óÎ  a aa€ SR S^,
          ,          ,          ^S P                  S P                  4      ,          S P                  R S,          4      ,          ,          pS P                  V4      pS ;P                  V,          un        SR S^,
          ,          ,          ^S P                  S P                  4      ,          S P                  R S,          4      ,          ,          pS P
                  pVV V3R lpS P                  pS P                  p	^p
V! V
4      p\        V4      pWÆ8”  dH   WÉ8:  dB   W‹,          pV
^,          p
V! V
4      pW‹,          pV
^,          p
V! V
4      pTp	\        V4      pKM  W‹,          pRS^,
          R,          ,          S P                  S P                  ^S P                  ^,          ,          S,          4      ,           S^,
          R,          ,          ,           pWMV,           ,          p^ pVP                  R4      '       d¶   WÆ8”  d   WÉ8:  g�   WÆ8:  dA   S P                  ^
4      \        S P                  \        WF,          4      ^
4      4      ,          pWÉ8”  dA   S P                  ^
4      \        S P                  \        WI,          4      ^
4      4      ,          p\        VS P
                  R,          4      pS ;P                  V,          un        V5V3# )r   c                 ó   <€ SP                  V R 4      ^SP                  S4      ,          V ,          ,          SP                  RV ,          4      ,          SV ,           ^,
          SP                  V 4      ,          ,          # )rÏ   r   )r­   r‰   rË   rÀ   rF  s   &€€€r   r?   Ú*secondzeta_singular_term.<locals>.<lambda>v  sX   ø€ �#—,‘,˜q Ó&¨¨#¯(©(°1«+­¸Õ'9Õ9Ø
‡y�y��Q•ÓõØ˜A�#˜a�% §¡¨£Õ+ö-r   r=  ç      ð?r²   rÐ   )r‰   r   rË   r   r#   rŠ   r1   r%   rh   r"   r6   rg   r–   r&   rï   )r   r(   r   rp   Úfactorrò   rº   r   r?  Úmg1r   r½   Úmg2ÚpoleÚstrš   s   fff,            r   Úsecondzeta_singular_termrV  p  s  ú€ Ø��a˜•c•�^˜Q˜sŸx™x¨¯©Ó/Õ/°·	±	¸#¸a½%Ó0@Õ@ÕA€FØ—‘˜“€IØ‡H‚H�	Õ…HØ��a˜•c•�^˜Q˜sŸx™x¨¯©Ó/Õ/°·	±	¸#¸a½%Ó0@Õ@ÕA€FØ
�'‰'€Cö	-€Aà�X‰X€FØ
�'‰'€CØ	€AÙˆQ‹4€DÜ
ˆd‹)€CØ
Œ)˜œ
Ø�ˆØ	ˆQ�ˆÙ�‹tˆØ�ˆØ	ˆA�ˆÙ�‹tˆØˆÜ�$‹iŠØ
…N€FØˆq��s�b�k�>˜3Ÿ9™9 S§W¡W¨R°·±¸µ	­\¸!­^Ó%<Õ<¸qÀ½sÀb½kÕIÕI€DØ	�f•Õ	€BØ
€CØ‡z�z�'×ÒØ” ¤ØŒzØ—g‘g˜b“k¤3 s§w¡w¬s°6µ:«¸rÓ'BÓ#CÕC�ØŒyØ—g‘g˜b“k¤3 s§w¡w¬s°6µ:«¸rÓ'BÓ#CÕC�Ü�#�s—w‘w˜r•zÓ"ˆØ‡H‚H�	Õ…HØˆ3�ˆ8€Or   c           	     óx  € V P                  V4      pV P                  V4      pV P                  pV P                  V4      '       dÏ   V P                  V4      ^8:  d¹   \	        V^,
          4      VR,          8  d   V P
                  # \        \        V P                  V4      4      4      pV^,          '       d   V P
                  # RV) ^,          ,          V P                  ^V P                  V) RR7      ,
          ^V) ^,           ,          4      ,          # V P                  p \        WV4      p\        V P                  V4      ^ 4      pV ;P                  V^,           ,          un
        \        WVRRR7      w  ršp\        WVRRR7      w  rÍp\!        WVRR7      w  pp\        WV4      pW­,           V,           pWœ,
          V,           V,
          pVP#                  R4      '       dK   \%        RV	4       \%        R	VR
4       \%        RV4       \%        R	VR4       \%        RV4       \%        RV4       W`n
        VP#                  R4      '       dc   \        V P                  \	        V4      4      ^ 4      p\        V^V,          ,          V P                  R,          ^V,          ,          4      pV5V3# V5#   Y`n
        i ; i)a—  
Evaluates the secondary zeta function `Z(s)`, defined for
`\mathrm{Re}(s)>1` by

.. math ::

    Z(s) = \sum_{n=1}^{\infty} \frac{1}{\tau_n^s}

where `\frac12+i\tau_n` runs through the zeros of `\zeta(s)` with
imaginary part positive.

`Z(s)` extends to a meromorphic function on `\mathbb{C}`  with a
double pole at `s=1` and  simple poles at the points `-2n` for
`n=0`,  1, 2, ...

**Examples**

    >>> from mpmath import *
    >>> mp.pretty = True; mp.dps = 15
    >>> secondzeta(2)
    0.023104993115419
    >>> xi = lambda s: 0.5*s*(s-1)*pi**(-0.5*s)*gamma(0.5*s)*zeta(s)
    >>> Xi = lambda t: xi(0.5+t*j)
    >>> chop(-0.5*diff(Xi,0,n=2)/Xi(0))
    0.023104993115419

We may ask for an approximate error value::

    >>> secondzeta(0.5+100j, error=True)
    ((-0.216272011276718 - 0.844952708937228j), 2.22044604925031e-16)

The function has poles at the negative odd integers,
and dyadic rational values at the negative even integers::

    >>> mp.dps = 30
    >>> secondzeta(-8)
    -0.67236328125
    >>> secondzeta(-7)
    +inf

**Implementation notes**

The function is computed as sum of four terms `Z(s)=A(s)-P(s)+E(s)-S(s)`
respectively main, prime, exponential and singular terms.
The main term `A(s)` is computed from the zeros of zeta.
The prime term depends on the von Mangoldt function.
The singular term is responsible for the poles of the function.

The four terms depends on a small parameter `a`. We may change the
value of `a`. Theoretically this has no effect on the sum of the four
terms, but in practice may be important.

A smaller value of the parameter `a` makes `A(s)` depend on
a smaller number of zeros of zeta, but `P(s)`  uses more values of
von Mangoldt function.

We may also add a verbose option to obtain data about the
values of the four terms.

    >>> mp.dps = 10
    >>> secondzeta(0.5 + 40j, error=True, verbose=True)
    main term = (-30190318549.138656312556 - 13964804384.624622876523j)
        computed using 19 zeros of zeta
    prime term = (132717176.89212754625045 + 188980555.17563978290601j)
        computed using 9 values of the von Mangoldt function
    exponential term = (542447428666.07179812536 + 362434922978.80192435203j)
    singular term = (512124392939.98154322355 + 348281138038.65531023921j)
    ((0.059471043 + 0.3463514534j), 1.455191523e-11)

    >>> secondzeta(0.5 + 40j, a=0.04, error=True, verbose=True)
    main term = (-151962888.19606243907725 - 217930683.90210294051982j)
        computed using 9 zeros of zeta
    prime term = (2476659342.3038722372461 + 28711581821.921627163136j)
        computed using 37 values of the von Mangoldt function
    exponential term = (178506047114.7838188264 + 819674143244.45677330576j)
    singular term = (175877424884.22441310708 + 790744630738.28669174871j)
    ((0.059471043 + 0.3463514534j), 1.455191523e-11)

Notice the great cancellation between the four terms. Changing `a`, the
four terms are very different numbers but the cancellation gives
the good value of Z(s).

**References**

A. Voros, Zeta functions for the Riemann zeros, Ann. Institute Fourier,
53, (2003) 665--699.

A. Voros, Zeta functions over Zeros of Zeta Functions, Lecture Notes
of the Unione Matematica Italiana, Springer, 2009.
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Gives the Lerch transcendent, defined for `|z| < 1` and
`\Re{a} > 0` by

.. math ::

    \Phi(z,s,a) = \sum_{k=0}^{\infty} \frac{z^k}{(a+k)^s}

and generally by the recurrence `\Phi(z,s,a) = z \Phi(z,s,a+1) + a^{-s}`
along with the integral representation valid for `\Re{a} > 0`

.. math ::

    \Phi(z,s,a) = \frac{1}{2 a^s} +
            \int_0^{\infty} \frac{z^t}{(a+t)^s} dt -
            2 \int_0^{\infty} \frac{\sin(t \log z - s
                \operatorname{arctan}(t/a)}{(a^2 + t^2)^{s/2}
                (e^{2 \pi t}-1)} dt.

The Lerch transcendent generalizes the Hurwitz zeta function :func:`zeta`
(`z = 1`) and the polylogarithm :func:`polylog` (`a = 1`).

**Examples**

Several evaluations in terms of simpler functions::

    >>> from mpmath import *
    >>> mp.dps = 25; mp.pretty = True
    >>> lerchphi(-1,2,0.5); 4*catalan
    3.663862376708876060218414
    3.663862376708876060218414
    >>> diff(lerchphi, (-1,-2,1), (0,1,0)); 7*zeta(3)/(4*pi**2)
    0.2131391994087528954617607
    0.2131391994087528954617607
    >>> lerchphi(-4,1,1); log(5)/4
    0.4023594781085250936501898
    0.4023594781085250936501898
    >>> lerchphi(-3+2j,1,0.5); 2*atanh(sqrt(-3+2j))/sqrt(-3+2j)
    (1.142423447120257137774002 + 0.2118232380980201350495795j)
    (1.142423447120257137774002 + 0.2118232380980201350495795j)

Evaluation works for complex arguments and `|z| \ge 1`::

    >>> lerchphi(1+2j, 3-j, 4+2j)
    (0.002025009957009908600539469 + 0.003327897536813558807438089j)
    >>> lerchphi(-2,2,-2.5)
    -12.28676272353094275265944
    >>> lerchphi(10,10,10)
    (-4.462130727102185701817349e-11 - 1.575172198981096218823481e-12j)
    >>> lerchphi(10,10,-10.5)
    (112658784011940.5605789002 - 498113185.5756221777743631j)

Some degenerate cases::

    >>> lerchphi(0,1,2)
    0.5
    >>> lerchphi(0,1,-2)
    -0.5

Reduction to simpler functions::

    >>> lerchphi(1, 4.25+1j, 1)
    (1.044674457556746668033975 - 0.04674508654012658932271226j)
    >>> zeta(4.25+1j)
    (1.044674457556746668033975 - 0.04674508654012658932271226j)
    >>> lerchphi(1 - 0.5**10, 4.25+1j, 1)
    (1.044629338021507546737197 - 0.04667768813963388181708101j)
    >>> lerchphi(3, 4, 1)
    (1.249503297023366545192592 - 0.2314252413375664776474462j)
    >>> polylog(4, 3) / 3
    (1.249503297023366545192592 - 0.2314252413375664776474462j)
    >>> lerchphi(3, 4, 1 - 0.5**10)
    (1.253978063946663945672674 - 0.2316736622836535468765376j)

**References**

1. [DLMF]_ section 25.14

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          4      S^,          V ^,          ,           S,          SP                  SV ,          4      ,          ,          # rJ   )ÚsinÚatanÚexpm1)r7   r   r   rD   rt   r£   r(   s   &€€€€€€r   r?   Úlerchphi.<locals>.<lambda>}  sP   ø€ �#—'‘'˜!˜CŸH™H Q q¥S›M�/¨!¨A­#Õ-Ó.Ø
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