Properties

Degree 96
Conductor $ 29^{48} $
Sign $1$
Motivic weight 2
Primitive no
Self-dual yes
Analytic rank 0

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 16·2-s − 12·3-s + 121·4-s − 14·5-s + 192·6-s − 10·7-s − 564·8-s + 65·9-s + 224·10-s − 8·11-s − 1.45e3·12-s − 14·13-s + 160·14-s + 168·15-s + 1.74e3·16-s − 26·17-s − 1.04e3·18-s + 2·19-s − 1.69e3·20-s + 120·21-s + 128·22-s + 56·23-s + 6.76e3·24-s − 19·25-s + 224·26-s − 156·27-s − 1.21e3·28-s + ⋯
L(s)  = 1  − 8·2-s − 4·3-s + 30.2·4-s − 2.79·5-s + 32·6-s − 1.42·7-s − 70.5·8-s + 65/9·9-s + 22.3·10-s − 0.727·11-s − 121·12-s − 1.07·13-s + 80/7·14-s + 56/5·15-s + 109.·16-s − 1.52·17-s − 57.7·18-s + 2/19·19-s − 84.6·20-s + 40/7·21-s + 5.81·22-s + 2.43·23-s + 282·24-s − 0.759·25-s + 8.61·26-s − 5.77·27-s − 43.2·28-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut &\left(29^{48}\right)^{s/2} \, \Gamma_{\C}(s)^{48} \, L(s)\cr =\mathstrut & \,\Lambda(3-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut &\left(29^{48}\right)^{s/2} \, \Gamma_{\C}(s+1)^{48} \, L(s)\cr =\mathstrut & \,\Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(96\)
\( N \)  =  \(29^{48}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(2\)
character  :  induced by $\chi_{29} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(96,\ 29^{48} ,\ ( \ : [1]^{48} ),\ 1 )$
$L(\frac{3}{2})$  $\approx$  $1.18733e-6$
$L(\frac12)$  $\approx$  $1.18733e-6$
$L(2)$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \neq 29$, \(F_p\) is a polynomial of degree 96. If $p = 29$, then $F_p$ is a polynomial of degree at most 95.
$p$$F_p$
bad29 \( 1 + 170T + 1.64e4T^{2} + 1.15e6T^{3} + 6.75e7T^{4} + 3.39e9T^{5} + 1.49e11T^{6} + 5.78e12T^{7} + 1.96e14T^{8} + 5.75e15T^{9} + 1.34e17T^{10} + 1.80e18T^{11} - 3.61e19T^{12} - 4.07e21T^{13} - 2.04e23T^{14} - 7.82e24T^{15} - 2.51e26T^{16} - 6.88e27T^{17} - 1.54e29T^{18} - 2.38e30T^{19} + 5.70e30T^{20} + 2.37e33T^{21} + 1.23e35T^{22} + 4.66e36T^{23} + 1.46e38T^{24} + 3.92e39T^{25} + 8.75e40T^{26} + 1.41e42T^{27} + 2.85e42T^{28} - 1.00e45T^{29} - 5.47e46T^{30} - 2.04e48T^{31} - 6.29e49T^{32} - 1.64e51T^{33} - 3.61e52T^{34} - 6.06e53T^{35} - 4.52e54T^{36} + 1.90e56T^{37} + 1.18e58T^{38} + 4.28e59T^{39}+O(T^{40}) \)
good2 \( 1 + p^{4} T + 135 T^{2} + 197 p^{2} T^{3} + 111 p^{5} T^{4} + 1643 p^{3} T^{5} + 41739 T^{6} + 29633 p^{2} T^{7} + 78709 p^{2} T^{8} + 204397 p^{2} T^{9} + 1067567 p T^{10} + 175195 p^{5} T^{11} + 1815039 p^{3} T^{12} + 9114211 p^{2} T^{13} + 88397463 T^{14} + 13057715 p^{4} T^{15} + 488323459 T^{16} + 71231285 p^{4} T^{17} + 1327396905 p T^{18} + 1531728025 p^{2} T^{19} + 13919718399 T^{20} + 972461967 p^{5} T^{21} + 34412748821 p T^{22} + 37906073985 p^{2} T^{23} + 166932494989 p T^{24} + 183384212547 p^{2} T^{25} + 800528189889 p T^{26} + 865156445521 p^{2} T^{27} + 7411178034935 T^{28} + 246614249921 p^{6} T^{29} + 33566605723417 T^{30} + 17855922361445 p^{2} T^{31} + 75927672355019 p T^{32} + 80400348114177 p^{2} T^{33} + 676768432312661 T^{34} + 353728268225691 p^{2} T^{35} + 1472463936933815 p T^{36} + 764813411152363 p^{3} T^{37} + 3177093668793159 p^{2} T^{38} + 6593080706599709 p^{2} T^{39} + 27282480931142745 p T^{40} + 3510281784395543 p^{5} T^{41} + 229914341232963609 T^{42} + 29268993727661163 p^{4} T^{43} + 950885806896528813 T^{44} + 240893643149222809 p^{3} T^{45} + 974484947948979345 p^{2} T^{46} + 491095797463312935 p^{4} T^{47} + 15760107896406852297 T^{48} + 491095797463312935 p^{6} T^{49} + 974484947948979345 p^{6} T^{50} + 240893643149222809 p^{9} T^{51} + 950885806896528813 p^{8} T^{52} + 29268993727661163 p^{14} T^{53} + 229914341232963609 p^{12} T^{54} + 3510281784395543 p^{19} T^{55} + 27282480931142745 p^{17} T^{56} + 6593080706599709 p^{20} T^{57} + 3177093668793159 p^{22} T^{58} + 764813411152363 p^{25} T^{59} + 1472463936933815 p^{25} T^{60} + 353728268225691 p^{28} T^{61} + 676768432312661 p^{28} T^{62} + 80400348114177 p^{32} T^{63} + 75927672355019 p^{33} T^{64} + 17855922361445 p^{36} T^{65} + 33566605723417 p^{36} T^{66} + 246614249921 p^{44} T^{67} + 7411178034935 p^{40} T^{68} + 865156445521 p^{44} T^{69} + 800528189889 p^{45} T^{70} + 183384212547 p^{48} T^{71} + 166932494989 p^{49} T^{72} + 37906073985 p^{52} T^{73} + 34412748821 p^{53} T^{74} + 972461967 p^{59} T^{75} + 13919718399 p^{56} T^{76} + 1531728025 p^{60} T^{77} + 1327396905 p^{61} T^{78} + 71231285 p^{66} T^{79} + 488323459 p^{64} T^{80} + 13057715 p^{70} T^{81} + 88397463 p^{68} T^{82} + 9114211 p^{72} T^{83} + 1815039 p^{75} T^{84} + 175195 p^{79} T^{85} + 1067567 p^{77} T^{86} + 204397 p^{80} T^{87} + 78709 p^{82} T^{88} + 29633 p^{84} T^{89} + 41739 p^{84} T^{90} + 1643 p^{89} T^{91} + 111 p^{93} T^{92} + 197 p^{92} T^{93} + 135 p^{92} T^{94} + p^{98} T^{95} + p^{96} T^{96} \)
3 \( 1 + 4 p T + 79 T^{2} + 4 p^{4} T^{3} + 748 T^{4} - 494 T^{5} - 130 p^{4} T^{6} - 39118 T^{7} - 48541 T^{8} + 128030 T^{9} + 482833 T^{10} - 1998106 T^{11} - 16783124 T^{12} - 46528514 T^{13} + 1792499 p^{2} T^{14} + 536701510 T^{15} + 601721455 p T^{16} + 837911930 T^{17} - 9844192510 T^{18} - 22942460714 T^{19} + 27230985275 p T^{20} + 60653437636 p^{2} T^{21} + 1087074533884 T^{22} - 739642668232 p T^{23} - 15067957643603 T^{24} - 26011599498430 T^{25} + 53989809057728 T^{26} + 266577710253134 T^{27} + 2137734182375 p^{3} T^{28} - 3281217034587490 T^{29} - 3137678687282555 p T^{30} - 4065963991456922 T^{31} + 52011350594684165 T^{32} + 18587655913801294 p T^{33} - 312758690152830590 T^{34} - 1216210783148992826 T^{35} + 3463181333041964018 T^{36} + 8039407714287583988 p T^{37} + 48467602177176134969 T^{38} - 97610757960846102472 T^{39} - \)\(15\!\cdots\!00\)\( p T^{40} + \)\(11\!\cdots\!36\)\( T^{41} + \)\(72\!\cdots\!95\)\( T^{42} + \)\(59\!\cdots\!60\)\( p T^{43} - \)\(14\!\cdots\!58\)\( T^{44} - \)\(24\!\cdots\!82\)\( T^{45} - \)\(50\!\cdots\!52\)\( T^{46} + \)\(45\!\cdots\!90\)\( p^{2} T^{47} + \)\(51\!\cdots\!02\)\( T^{48} + \)\(45\!\cdots\!90\)\( p^{4} T^{49} - \)\(50\!\cdots\!52\)\( p^{4} T^{50} - \)\(24\!\cdots\!82\)\( p^{6} T^{51} - \)\(14\!\cdots\!58\)\( p^{8} T^{52} + \)\(59\!\cdots\!60\)\( p^{11} T^{53} + \)\(72\!\cdots\!95\)\( p^{12} T^{54} + \)\(11\!\cdots\!36\)\( p^{14} T^{55} - \)\(15\!\cdots\!00\)\( p^{17} T^{56} - 97610757960846102472 p^{18} T^{57} + 48467602177176134969 p^{20} T^{58} + 8039407714287583988 p^{23} T^{59} + 3463181333041964018 p^{24} T^{60} - 1216210783148992826 p^{26} T^{61} - 312758690152830590 p^{28} T^{62} + 18587655913801294 p^{31} T^{63} + 52011350594684165 p^{32} T^{64} - 4065963991456922 p^{34} T^{65} - 3137678687282555 p^{37} T^{66} - 3281217034587490 p^{38} T^{67} + 2137734182375 p^{43} T^{68} + 266577710253134 p^{42} T^{69} + 53989809057728 p^{44} T^{70} - 26011599498430 p^{46} T^{71} - 15067957643603 p^{48} T^{72} - 739642668232 p^{51} T^{73} + 1087074533884 p^{52} T^{74} + 60653437636 p^{56} T^{75} + 27230985275 p^{57} T^{76} - 22942460714 p^{58} T^{77} - 9844192510 p^{60} T^{78} + 837911930 p^{62} T^{79} + 601721455 p^{65} T^{80} + 536701510 p^{66} T^{81} + 1792499 p^{70} T^{82} - 46528514 p^{70} T^{83} - 16783124 p^{72} T^{84} - 1998106 p^{74} T^{85} + 482833 p^{76} T^{86} + 128030 p^{78} T^{87} - 48541 p^{80} T^{88} - 39118 p^{82} T^{89} - 130 p^{88} T^{90} - 494 p^{86} T^{91} + 748 p^{88} T^{92} + 4 p^{94} T^{93} + 79 p^{92} T^{94} + 4 p^{95} T^{95} + p^{96} T^{96} \)
5 \( 1 + 14 T + 43 p T^{2} + 1834 T^{3} + 3573 p T^{4} + 123158 T^{5} + 1009721 T^{6} + 6275206 T^{7} + 45581307 T^{8} + 254470454 T^{9} + 1643397299 T^{10} + 8282968722 T^{11} + 48998384761 T^{12} + 226606387504 T^{13} + 9995099042 p^{3} T^{14} + 5328052129288 T^{15} + 28456162978762 T^{16} + 119252280051548 T^{17} + 136912490888736 p T^{18} + 3196076151689568 T^{19} + 20754317597334414 T^{20} + 110475331409659104 T^{21} + 741356750745407536 T^{22} + 4062946872613183716 T^{23} + 25607985795088903054 T^{24} + \)\(13\!\cdots\!52\)\( T^{25} + \)\(77\!\cdots\!94\)\( T^{26} + \)\(37\!\cdots\!72\)\( T^{27} + \)\(19\!\cdots\!83\)\( T^{28} + \)\(36\!\cdots\!86\)\( p^{2} T^{29} + \)\(88\!\cdots\!69\)\( p T^{30} + \)\(20\!\cdots\!46\)\( T^{31} + \)\(97\!\cdots\!81\)\( T^{32} + \)\(47\!\cdots\!54\)\( T^{33} + \)\(24\!\cdots\!31\)\( T^{34} + \)\(13\!\cdots\!02\)\( T^{35} + \)\(76\!\cdots\!19\)\( T^{36} + \)\(44\!\cdots\!58\)\( T^{37} + \)\(24\!\cdots\!93\)\( T^{38} + \)\(13\!\cdots\!54\)\( T^{39} + \)\(72\!\cdots\!23\)\( T^{40} + \)\(37\!\cdots\!32\)\( T^{41} + \)\(72\!\cdots\!24\)\( p^{2} T^{42} + \)\(87\!\cdots\!44\)\( T^{43} + \)\(38\!\cdots\!64\)\( T^{44} + \)\(18\!\cdots\!56\)\( T^{45} + \)\(77\!\cdots\!16\)\( T^{46} + \)\(75\!\cdots\!08\)\( p T^{47} + \)\(17\!\cdots\!76\)\( T^{48} + \)\(75\!\cdots\!08\)\( p^{3} T^{49} + \)\(77\!\cdots\!16\)\( p^{4} T^{50} + \)\(18\!\cdots\!56\)\( p^{6} T^{51} + \)\(38\!\cdots\!64\)\( p^{8} T^{52} + \)\(87\!\cdots\!44\)\( p^{10} T^{53} + \)\(72\!\cdots\!24\)\( p^{14} T^{54} + \)\(37\!\cdots\!32\)\( p^{14} T^{55} + \)\(72\!\cdots\!23\)\( p^{16} T^{56} + \)\(13\!\cdots\!54\)\( p^{18} T^{57} + \)\(24\!\cdots\!93\)\( p^{20} T^{58} + \)\(44\!\cdots\!58\)\( p^{22} T^{59} + \)\(76\!\cdots\!19\)\( p^{24} T^{60} + \)\(13\!\cdots\!02\)\( p^{26} T^{61} + \)\(24\!\cdots\!31\)\( p^{28} T^{62} + \)\(47\!\cdots\!54\)\( p^{30} T^{63} + \)\(97\!\cdots\!81\)\( p^{32} T^{64} + \)\(20\!\cdots\!46\)\( p^{34} T^{65} + \)\(88\!\cdots\!69\)\( p^{37} T^{66} + \)\(36\!\cdots\!86\)\( p^{40} T^{67} + \)\(19\!\cdots\!83\)\( p^{40} T^{68} + \)\(37\!\cdots\!72\)\( p^{42} T^{69} + \)\(77\!\cdots\!94\)\( p^{44} T^{70} + \)\(13\!\cdots\!52\)\( p^{46} T^{71} + 25607985795088903054 p^{48} T^{72} + 4062946872613183716 p^{50} T^{73} + 741356750745407536 p^{52} T^{74} + 110475331409659104 p^{54} T^{75} + 20754317597334414 p^{56} T^{76} + 3196076151689568 p^{58} T^{77} + 136912490888736 p^{61} T^{78} + 119252280051548 p^{62} T^{79} + 28456162978762 p^{64} T^{80} + 5328052129288 p^{66} T^{81} + 9995099042 p^{71} T^{82} + 226606387504 p^{70} T^{83} + 48998384761 p^{72} T^{84} + 8282968722 p^{74} T^{85} + 1643397299 p^{76} T^{86} + 254470454 p^{78} T^{87} + 45581307 p^{80} T^{88} + 6275206 p^{82} T^{89} + 1009721 p^{84} T^{90} + 123158 p^{86} T^{91} + 3573 p^{89} T^{92} + 1834 p^{90} T^{93} + 43 p^{93} T^{94} + 14 p^{94} T^{95} + p^{96} T^{96} \)
7 \( 1 + 10 T - 57 p T^{2} - 4236 T^{3} + 88012 T^{4} + 961236 T^{5} - 289276 p^{2} T^{6} - 155492966 T^{7} + 265275399 p T^{8} + 20094259140 T^{9} - 208698108449 T^{10} - 2199633719140 T^{11} + 20758293283115 T^{12} + 210987582526270 T^{13} - 1865195037242882 T^{14} - 18111952270636036 T^{15} + 153584251691602171 T^{16} + 1410417772300341874 T^{17} - 11707484613660681198 T^{18} - \)\(10\!\cdots\!46\)\( T^{19} + \)\(11\!\cdots\!81\)\( p T^{20} + \)\(65\!\cdots\!40\)\( T^{21} - \)\(55\!\cdots\!76\)\( T^{22} - \)\(39\!\cdots\!98\)\( T^{23} + \)\(34\!\cdots\!90\)\( T^{24} + \)\(21\!\cdots\!42\)\( T^{25} - \)\(29\!\cdots\!80\)\( p T^{26} - \)\(10\!\cdots\!08\)\( T^{27} + \)\(11\!\cdots\!89\)\( T^{28} + \)\(70\!\cdots\!38\)\( p T^{29} - \)\(85\!\cdots\!42\)\( p T^{30} - \)\(18\!\cdots\!14\)\( T^{31} + \)\(29\!\cdots\!61\)\( T^{32} + \)\(53\!\cdots\!40\)\( T^{33} - \)\(13\!\cdots\!02\)\( T^{34} - \)\(38\!\cdots\!02\)\( T^{35} + \)\(56\!\cdots\!64\)\( T^{36} - \)\(85\!\cdots\!82\)\( T^{37} - \)\(21\!\cdots\!74\)\( T^{38} + \)\(81\!\cdots\!00\)\( T^{39} + \)\(67\!\cdots\!27\)\( T^{40} - \)\(70\!\cdots\!30\)\( p T^{41} - \)\(15\!\cdots\!86\)\( T^{42} + \)\(33\!\cdots\!62\)\( p T^{43} + \)\(44\!\cdots\!35\)\( T^{44} - \)\(81\!\cdots\!68\)\( T^{45} + \)\(21\!\cdots\!04\)\( T^{46} + \)\(14\!\cdots\!90\)\( T^{47} - \)\(14\!\cdots\!86\)\( T^{48} + \)\(14\!\cdots\!90\)\( p^{2} T^{49} + \)\(21\!\cdots\!04\)\( p^{4} T^{50} - \)\(81\!\cdots\!68\)\( p^{6} T^{51} + \)\(44\!\cdots\!35\)\( p^{8} T^{52} + \)\(33\!\cdots\!62\)\( p^{11} T^{53} - \)\(15\!\cdots\!86\)\( p^{12} T^{54} - \)\(70\!\cdots\!30\)\( p^{15} T^{55} + \)\(67\!\cdots\!27\)\( p^{16} T^{56} + \)\(81\!\cdots\!00\)\( p^{18} T^{57} - \)\(21\!\cdots\!74\)\( p^{20} T^{58} - \)\(85\!\cdots\!82\)\( p^{22} T^{59} + \)\(56\!\cdots\!64\)\( p^{24} T^{60} - \)\(38\!\cdots\!02\)\( p^{26} T^{61} - \)\(13\!\cdots\!02\)\( p^{28} T^{62} + \)\(53\!\cdots\!40\)\( p^{30} T^{63} + \)\(29\!\cdots\!61\)\( p^{32} T^{64} - \)\(18\!\cdots\!14\)\( p^{34} T^{65} - \)\(85\!\cdots\!42\)\( p^{37} T^{66} + \)\(70\!\cdots\!38\)\( p^{39} T^{67} + \)\(11\!\cdots\!89\)\( p^{40} T^{68} - \)\(10\!\cdots\!08\)\( p^{42} T^{69} - \)\(29\!\cdots\!80\)\( p^{45} T^{70} + \)\(21\!\cdots\!42\)\( p^{46} T^{71} + \)\(34\!\cdots\!90\)\( p^{48} T^{72} - \)\(39\!\cdots\!98\)\( p^{50} T^{73} - \)\(55\!\cdots\!76\)\( p^{52} T^{74} + \)\(65\!\cdots\!40\)\( p^{54} T^{75} + \)\(11\!\cdots\!81\)\( p^{57} T^{76} - \)\(10\!\cdots\!46\)\( p^{58} T^{77} - 11707484613660681198 p^{60} T^{78} + 1410417772300341874 p^{62} T^{79} + 153584251691602171 p^{64} T^{80} - 18111952270636036 p^{66} T^{81} - 1865195037242882 p^{68} T^{82} + 210987582526270 p^{70} T^{83} + 20758293283115 p^{72} T^{84} - 2199633719140 p^{74} T^{85} - 208698108449 p^{76} T^{86} + 20094259140 p^{78} T^{87} + 265275399 p^{81} T^{88} - 155492966 p^{82} T^{89} - 289276 p^{86} T^{90} + 961236 p^{86} T^{91} + 88012 p^{88} T^{92} - 4236 p^{90} T^{93} - 57 p^{93} T^{94} + 10 p^{94} T^{95} + p^{96} T^{96} \)
11 \( 1 + 8 T + 375 T^{2} - 3272 T^{3} + 2241 T^{4} - 2162458 T^{5} + 5347517 T^{6} - 157814138 T^{7} + 7201185249 T^{8} + 9877404696 T^{9} + 917857593905 T^{10} - 13867055165844 T^{11} - 55986550658351 T^{12} - 261730948803364 p T^{13} + 12885112707636822 T^{14} + 123497906965345760 T^{15} + 6429964586798756808 T^{16} + 18962958071140584144 T^{17} - 56515688687913417570 T^{18} - \)\(99\!\cdots\!24\)\( T^{19} - \)\(10\!\cdots\!34\)\( T^{20} - \)\(41\!\cdots\!72\)\( T^{21} + \)\(93\!\cdots\!42\)\( T^{22} + \)\(23\!\cdots\!84\)\( T^{23} + \)\(16\!\cdots\!72\)\( T^{24} + \)\(26\!\cdots\!04\)\( T^{25} - \)\(33\!\cdots\!30\)\( T^{26} - \)\(34\!\cdots\!04\)\( T^{27} - \)\(30\!\cdots\!73\)\( T^{28} + \)\(29\!\cdots\!32\)\( T^{29} + \)\(51\!\cdots\!07\)\( T^{30} + \)\(73\!\cdots\!92\)\( T^{31} + \)\(13\!\cdots\!03\)\( T^{32} - \)\(51\!\cdots\!58\)\( T^{33} - \)\(11\!\cdots\!53\)\( T^{34} - \)\(61\!\cdots\!30\)\( T^{35} + \)\(21\!\cdots\!43\)\( T^{36} + \)\(13\!\cdots\!88\)\( T^{37} + \)\(14\!\cdots\!69\)\( T^{38} + \)\(43\!\cdots\!00\)\( T^{39} - \)\(10\!\cdots\!25\)\( T^{40} - \)\(21\!\cdots\!84\)\( T^{41} - \)\(12\!\cdots\!40\)\( T^{42} + \)\(17\!\cdots\!20\)\( T^{43} + \)\(24\!\cdots\!56\)\( p T^{44} + \)\(22\!\cdots\!56\)\( T^{45} + \)\(13\!\cdots\!08\)\( T^{46} - \)\(24\!\cdots\!36\)\( p T^{47} - \)\(31\!\cdots\!20\)\( T^{48} - \)\(24\!\cdots\!36\)\( p^{3} T^{49} + \)\(13\!\cdots\!08\)\( p^{4} T^{50} + \)\(22\!\cdots\!56\)\( p^{6} T^{51} + \)\(24\!\cdots\!56\)\( p^{9} T^{52} + \)\(17\!\cdots\!20\)\( p^{10} T^{53} - \)\(12\!\cdots\!40\)\( p^{12} T^{54} - \)\(21\!\cdots\!84\)\( p^{14} T^{55} - \)\(10\!\cdots\!25\)\( p^{16} T^{56} + \)\(43\!\cdots\!00\)\( p^{18} T^{57} + \)\(14\!\cdots\!69\)\( p^{20} T^{58} + \)\(13\!\cdots\!88\)\( p^{22} T^{59} + \)\(21\!\cdots\!43\)\( p^{24} T^{60} - \)\(61\!\cdots\!30\)\( p^{26} T^{61} - \)\(11\!\cdots\!53\)\( p^{28} T^{62} - \)\(51\!\cdots\!58\)\( p^{30} T^{63} + \)\(13\!\cdots\!03\)\( p^{32} T^{64} + \)\(73\!\cdots\!92\)\( p^{34} T^{65} + \)\(51\!\cdots\!07\)\( p^{36} T^{66} + \)\(29\!\cdots\!32\)\( p^{38} T^{67} - \)\(30\!\cdots\!73\)\( p^{40} T^{68} - \)\(34\!\cdots\!04\)\( p^{42} T^{69} - \)\(33\!\cdots\!30\)\( p^{44} T^{70} + \)\(26\!\cdots\!04\)\( p^{46} T^{71} + \)\(16\!\cdots\!72\)\( p^{48} T^{72} + \)\(23\!\cdots\!84\)\( p^{50} T^{73} + \)\(93\!\cdots\!42\)\( p^{52} T^{74} - \)\(41\!\cdots\!72\)\( p^{54} T^{75} - \)\(10\!\cdots\!34\)\( p^{56} T^{76} - \)\(99\!\cdots\!24\)\( p^{58} T^{77} - 56515688687913417570 p^{60} T^{78} + 18962958071140584144 p^{62} T^{79} + 6429964586798756808 p^{64} T^{80} + 123497906965345760 p^{66} T^{81} + 12885112707636822 p^{68} T^{82} - 261730948803364 p^{71} T^{83} - 55986550658351 p^{72} T^{84} - 13867055165844 p^{74} T^{85} + 917857593905 p^{76} T^{86} + 9877404696 p^{78} T^{87} + 7201185249 p^{80} T^{88} - 157814138 p^{82} T^{89} + 5347517 p^{84} T^{90} - 2162458 p^{86} T^{91} + 2241 p^{88} T^{92} - 3272 p^{90} T^{93} + 375 p^{92} T^{94} + 8 p^{94} T^{95} + p^{96} T^{96} \)
13 \( 1 + 14 T + 87 p T^{2} + 7182 T^{3} + 513722 T^{4} - 21532 p T^{5} + 140514018 T^{6} - 1045192008 T^{7} + 28981359188 T^{8} - 395007332664 T^{9} + 4185090585476 T^{10} - 85414188632556 T^{11} + 195249953381758 T^{12} - 7527599672416894 T^{13} - 84934009480748863 T^{14} + 1866452920027596922 T^{15} - 35643886055783294108 T^{16} + \)\(87\!\cdots\!46\)\( T^{17} - \)\(86\!\cdots\!77\)\( T^{18} + \)\(18\!\cdots\!58\)\( T^{19} - \)\(13\!\cdots\!10\)\( T^{20} + \)\(23\!\cdots\!60\)\( T^{21} - \)\(10\!\cdots\!86\)\( T^{22} + \)\(65\!\cdots\!88\)\( T^{23} + \)\(13\!\cdots\!01\)\( T^{24} - \)\(38\!\cdots\!70\)\( T^{25} + \)\(63\!\cdots\!87\)\( T^{26} - \)\(94\!\cdots\!58\)\( T^{27} + \)\(10\!\cdots\!90\)\( T^{28} - \)\(11\!\cdots\!18\)\( T^{29} + \)\(63\!\cdots\!13\)\( T^{30} - \)\(61\!\cdots\!34\)\( T^{31} - \)\(73\!\cdots\!61\)\( T^{32} - \)\(10\!\cdots\!40\)\( T^{33} - \)\(64\!\cdots\!34\)\( T^{34} - \)\(18\!\cdots\!12\)\( T^{35} + \)\(69\!\cdots\!87\)\( T^{36} - \)\(92\!\cdots\!88\)\( T^{37} + \)\(27\!\cdots\!60\)\( T^{38} - \)\(21\!\cdots\!80\)\( T^{39} + \)\(52\!\cdots\!67\)\( T^{40} - \)\(27\!\cdots\!64\)\( T^{41} + \)\(35\!\cdots\!20\)\( p T^{42} + \)\(62\!\cdots\!52\)\( T^{43} - \)\(50\!\cdots\!65\)\( T^{44} + \)\(12\!\cdots\!10\)\( T^{45} - \)\(29\!\cdots\!93\)\( T^{46} + \)\(34\!\cdots\!42\)\( T^{47} - \)\(64\!\cdots\!72\)\( T^{48} + \)\(34\!\cdots\!42\)\( p^{2} T^{49} - \)\(29\!\cdots\!93\)\( p^{4} T^{50} + \)\(12\!\cdots\!10\)\( p^{6} T^{51} - \)\(50\!\cdots\!65\)\( p^{8} T^{52} + \)\(62\!\cdots\!52\)\( p^{10} T^{53} + \)\(35\!\cdots\!20\)\( p^{13} T^{54} - \)\(27\!\cdots\!64\)\( p^{14} T^{55} + \)\(52\!\cdots\!67\)\( p^{16} T^{56} - \)\(21\!\cdots\!80\)\( p^{18} T^{57} + \)\(27\!\cdots\!60\)\( p^{20} T^{58} - \)\(92\!\cdots\!88\)\( p^{22} T^{59} + \)\(69\!\cdots\!87\)\( p^{24} T^{60} - \)\(18\!\cdots\!12\)\( p^{26} T^{61} - \)\(64\!\cdots\!34\)\( p^{28} T^{62} - \)\(10\!\cdots\!40\)\( p^{30} T^{63} - \)\(73\!\cdots\!61\)\( p^{32} T^{64} - \)\(61\!\cdots\!34\)\( p^{34} T^{65} + \)\(63\!\cdots\!13\)\( p^{36} T^{66} - \)\(11\!\cdots\!18\)\( p^{38} T^{67} + \)\(10\!\cdots\!90\)\( p^{40} T^{68} - \)\(94\!\cdots\!58\)\( p^{42} T^{69} + \)\(63\!\cdots\!87\)\( p^{44} T^{70} - \)\(38\!\cdots\!70\)\( p^{46} T^{71} + \)\(13\!\cdots\!01\)\( p^{48} T^{72} + \)\(65\!\cdots\!88\)\( p^{50} T^{73} - \)\(10\!\cdots\!86\)\( p^{52} T^{74} + \)\(23\!\cdots\!60\)\( p^{54} T^{75} - \)\(13\!\cdots\!10\)\( p^{56} T^{76} + \)\(18\!\cdots\!58\)\( p^{58} T^{77} - \)\(86\!\cdots\!77\)\( p^{60} T^{78} + \)\(87\!\cdots\!46\)\( p^{62} T^{79} - 35643886055783294108 p^{64} T^{80} + 1866452920027596922 p^{66} T^{81} - 84934009480748863 p^{68} T^{82} - 7527599672416894 p^{70} T^{83} + 195249953381758 p^{72} T^{84} - 85414188632556 p^{74} T^{85} + 4185090585476 p^{76} T^{86} - 395007332664 p^{78} T^{87} + 28981359188 p^{80} T^{88} - 1045192008 p^{82} T^{89} + 140514018 p^{84} T^{90} - 21532 p^{87} T^{91} + 513722 p^{88} T^{92} + 7182 p^{90} T^{93} + 87 p^{93} T^{94} + 14 p^{94} T^{95} + p^{96} T^{96} \)
17 \( 1 + 26T + 338T^{2} + 1.08e4T^{3} + 3.76e5T^{4} + 8.81e6T^{5} + 1.60e8T^{6} + 4.07e9T^{7} + 9.66e10T^{8} + 1.90e12T^{9} + 3.92e13T^{10} + 8.31e14T^{11} + 1.72e16T^{12} + 3.35e17T^{13} + 6.64e18T^{14} + 1.28e20T^{15} + 2.45e21T^{16} + 4.71e22T^{17} + 8.69e23T^{18} + 1.59e25T^{19} + 2.91e26T^{20} + 5.31e27T^{21} + 9.25e28T^{22} + 1.62e30T^{23} + 2.80e31T^{24} + 4.71e32T^{25} + 7.86e33T^{26} + 1.27e35T^{27} + 2.02e36T^{28} + 3.08e37T^{29} + 4.61e38T^{30} + 6.24e39T^{31} + 7.61e40T^{32} + 8.44e41T^{33} + 4.06e42T^{34} - 1.22e44T^{35} - 5.33e45T^{36} - 1.43e47T^{37} - 3.42e48T^{38} - 7.45e49T^{39} - 1.52e51T^{40} - 3.04e52T^{41} - 5.78e53T^{42} - 1.08e55T^{43} - 2.00e56T^{44} - 3.60e57T^{45}+O(T^{46}) \)
19 \( 1 - 2T + 555T^{2} + 1.86e3T^{3} - 1.33e4T^{4} + 7.25e6T^{5} - 6.55e7T^{6} + 4.32e9T^{7} - 1.17e10T^{8} + 1.04e12T^{9} + 4.40e12T^{10} - 7.96e13T^{11} + 1.17e16T^{12} - 2.26e17T^{13} + 9.25e18T^{14} - 1.26e20T^{15} + 2.30e21T^{16} - 1.30e22T^{17} - 4.02e23T^{18} + 1.91e25T^{19} - 4.63e26T^{20} + 1.12e28T^{21} - 2.14e29T^{22} + 3.39e30T^{23} - 5.93e31T^{24} + 2.22e32T^{25} + 8.49e33T^{26} - 5.09e35T^{27} + 1.35e37T^{28} - 3.06e38T^{29} + 4.62e39T^{30} - 7.45e40T^{31} + 7.45e41T^{32} - 3.29e42T^{33} - 1.04e44T^{34} + 6.25e45T^{35} - 1.87e47T^{36} + 4.64e48T^{37} - 8.61e49T^{38} + 1.52e51T^{39} - 1.47e52T^{40} + 1.57e53T^{41} + 1.14e54T^{42} - 2.29e55T^{43} + 1.32e57T^{44}+O(T^{45}) \)
23 \( 1 - 56T + 177T^{2} + 3.90e4T^{3} - 3.54e5T^{4} - 6.63e6T^{5} + 1.98e7T^{6} - 2.13e10T^{7} + 6.02e11T^{8} + 1.71e13T^{9} - 5.38e14T^{10} - 5.08e15T^{11} + 9.76e16T^{12} - 3.87e17T^{13} + 1.82e20T^{14} + 7.36e20T^{15} - 1.93e23T^{16} + 7.16e22T^{17} + 7.83e25T^{18} + 5.70e24T^{19} + 6.23e27T^{20} - 2.20e29T^{21} - 3.05e31T^{22} + 2.95e32T^{23} + 1.90e34T^{24} - 1.92e35T^{25} - 2.59e36T^{26} + 2.16e37T^{27} - 3.99e39T^{28} + 5.77e40T^{29} + 3.24e42T^{30} - 5.33e43T^{31} - 6.81e44T^{32} + 1.32e46T^{33} - 4.70e47T^{34} + 9.04e48T^{35} + 4.68e50T^{36} - 1.06e52T^{37} - 1.55e53T^{38} + 4.53e54T^{39} - 2.49e55T^{40} + 2.18e56T^{41}+O(T^{42}) \)
31 \( 1 + 88T + 5.37e3T^{2} + 3.60e5T^{3} + 1.99e7T^{4} + 9.66e8T^{5} + 4.62e10T^{6} + 2.05e12T^{7} + 8.43e13T^{8} + 3.42e15T^{9} + 1.32e17T^{10} + 4.88e18T^{11} + 1.79e20T^{12} + 6.43e21T^{13} + 2.21e23T^{14} + 7.72e24T^{15} + 2.64e26T^{16} + 8.82e27T^{17} + 2.97e29T^{18} + 9.95e30T^{19} + 3.22e32T^{20} + 1.05e34T^{21} + 3.41e35T^{22} + 1.06e37T^{23} + 3.32e38T^{24} + 1.02e40T^{25} + 2.99e41T^{26} + 8.79e42T^{27} + 2.52e44T^{28} + 6.73e45T^{29} + 1.77e47T^{30} + 4.48e48T^{31} + 9.56e49T^{32} + 1.78e51T^{33} + 1.99e52T^{34} - 7.50e53T^{35} - 5.98e55T^{36} - 2.90e57T^{37} - 1.29e59T^{38}+O(T^{39}) \)
37 \( 1 + 56T + 4.66e3T^{2} + 1.32e5T^{3} + 5.13e6T^{4} - 2.61e8T^{5} - 1.47e10T^{6} - 1.46e12T^{7} - 3.43e13T^{8} - 1.59e15T^{9} + 5.51e16T^{10} + 2.78e18T^{11} + 3.04e20T^{12} + 7.14e21T^{13} + 3.53e23T^{14} - 7.23e24T^{15} - 3.54e26T^{16} - 4.59e28T^{17} - 1.08e30T^{18} - 5.40e31T^{19} + 7.56e32T^{20} + 4.11e34T^{21} + 5.74e36T^{22} + 1.45e38T^{23} + 6.90e39T^{24} - 5.06e40T^{25} - 4.13e42T^{26} - 5.96e44T^{27} - 1.65e46T^{28} - 7.27e47T^{29} + 1.75e48T^{30} + 4.03e50T^{31} + 5.46e52T^{32} + 1.71e54T^{33} + 6.80e55T^{34} + 1.86e56T^{35} - 3.50e58T^{36}+O(T^{37}) \)
41 \( 1 + 34T + 578T^{2} + 1.34e5T^{3} + 3.17e6T^{4} - 1.04e8T^{5} + 3.71e9T^{6} - 2.00e11T^{7} - 3.88e13T^{8} - 8.45e13T^{9} - 1.38e16T^{10} - 3.03e18T^{11} + 8.23e19T^{12} + 5.20e21T^{13} - 6.26e22T^{14} + 1.34e25T^{15} + 6.22e26T^{16} - 1.28e28T^{17} + 4.46e29T^{18} + 2.01e31T^{19} - 2.99e33T^{20} - 3.96e34T^{21} + 8.65e35T^{22} - 2.34e38T^{23} - 1.38e39T^{24} + 3.25e41T^{25} - 6.35e42T^{26} + 2.29e44T^{27} + 4.25e46T^{28} - 1.52e47T^{29} + 3.23e47T^{30} + 2.44e51T^{31} - 5.78e52T^{32} - 3.44e54T^{33} + 8.37e55T^{34} - 5.94e57T^{35}+O(T^{36}) \)
43 \( 1 - 176T + 1.10e4T^{2} - 1.60e5T^{3} - 1.60e7T^{4} + 1.57e9T^{5} - 1.15e11T^{6} + 4.18e12T^{7} + 1.48e14T^{8} - 2.14e16T^{9} + 1.03e18T^{10} - 3.45e19T^{11} + 5.35e18T^{12} + 1.35e23T^{13} - 9.48e24T^{14} + 2.65e26T^{15} + 7.85e26T^{16} - 6.45e29T^{17} + 5.68e31T^{18} - 2.43e33T^{19} + 1.75e34T^{20} + 3.64e36T^{21} - 2.78e38T^{22} + 1.36e40T^{23} - 2.98e41T^{24} - 1.41e43T^{25} + 1.50e45T^{26} - 6.94e46T^{27} + 1.93e48T^{28} + 2.18e49T^{29} - 6.32e51T^{30} + 3.57e53T^{31} - 1.04e55T^{32} + 2.08e55T^{33} + 2.12e58T^{34} - 1.56e60T^{35}+O(T^{36}) \)
47 \( 1 - 208T + 1.75e4T^{2} - 5.35e5T^{3} - 3.12e7T^{4} + 3.59e9T^{5} - 6.21e10T^{6} - 8.93e12T^{7} + 5.48e14T^{8} + 7.72e15T^{9} - 2.15e18T^{10} + 6.89e19T^{11} + 3.05e21T^{12} - 2.84e23T^{13} + 2.48e24T^{14} + 6.02e26T^{15} - 2.43e28T^{16} - 6.88e29T^{17} + 7.66e31T^{18} - 8.14e32T^{19} - 1.45e35T^{20} + 6.08e36T^{21} + 1.22e38T^{22} - 1.42e40T^{23} + 1.03e41T^{24} + 2.39e43T^{25} - 6.71e44T^{26} - 3.37e46T^{27} + 1.68e48T^{28} + 3.43e49T^{29} - 2.68e51T^{30} - 9.40e52T^{31} + 9.23e54T^{32} + 1.59e56T^{33} - 3.18e58T^{34}+O(T^{35}) \)
53 \( 1 + 14T - 1.77e4T^{2} + 3.03e5T^{3} + 1.58e8T^{4} - 6.32e9T^{5} - 8.00e11T^{6} + 4.96e13T^{7} + 2.32e15T^{8} - 2.07e17T^{9} - 4.20e18T^{10} + 4.92e20T^{11} + 1.15e22T^{12} - 7.67e23T^{13} - 4.90e25T^{14} + 1.22e27T^{15} + 6.29e28T^{16} + 4.33e30T^{17} + 2.95e32T^{18} - 6.60e34T^{19} - 6.48e35T^{20} + 3.06e38T^{21} - 2.20e39T^{22} - 7.83e41T^{23} + 5.27e42T^{24} + 1.59e45T^{25} + 1.86e46T^{26} - 3.45e48T^{27} - 8.29e49T^{28} + 2.77e51T^{29} + 2.94e53T^{30} + 2.44e55T^{31} - 1.74e57T^{32} - 1.20e59T^{33}+O(T^{34}) \)
59 \( 1 + 44T + 1.14e5T^{2} + 5.03e6T^{3} + 6.45e9T^{4} + 2.85e11T^{5} + 2.41e14T^{6} + 1.06e16T^{7} + 6.70e18T^{8} + 2.95e20T^{9} + 1.47e23T^{10} + 6.47e24T^{11} + 2.68e27T^{12} + 1.17e29T^{13} + 4.14e31T^{14} + 1.79e33T^{15} + 5.54e35T^{16} + 2.38e37T^{17} + 6.54e39T^{18} + 2.77e41T^{19} + 6.87e43T^{20} + 2.88e45T^{21} + 6.49e47T^{22} + 2.69e49T^{23} + 5.57e51T^{24} + 2.27e53T^{25} + 4.36e55T^{26} + 1.75e57T^{27} + 3.14e59T^{28} + 1.24e61T^{29} + 2.08e63T^{30} + 8.07e64T^{31} + 1.28e67T^{32}+O(T^{33}) \)
61 \( 1 + 30T + 7.93e3T^{2} + 1.89e6T^{3} + 1.04e8T^{4} + 1.58e10T^{5} + 1.94e12T^{6} + 1.46e14T^{7} + 1.59e16T^{8} + 1.46e18T^{9} + 1.22e20T^{10} + 1.09e22T^{11} + 8.86e23T^{12} + 7.29e25T^{13} + 5.77e27T^{14} + 4.41e29T^{15} + 3.40e31T^{16} + 2.50e33T^{17} + 1.83e35T^{18} + 1.32e37T^{19} + 9.25e38T^{20} + 6.50e40T^{21} + 4.45e42T^{22} + 2.99e44T^{23} + 2.01e46T^{24} + 1.32e48T^{25} + 8.62e49T^{26} + 5.57e51T^{27} + 3.52e53T^{28} + 2.21e55T^{29} + 1.37e57T^{30} + 8.41e58T^{31} + 5.11e60T^{32}+O(T^{33}) \)
67 \( 1 + 574T + 1.83e5T^{2} + 4.17e7T^{3} + 7.45e9T^{4} + 1.09e12T^{5} + 1.33e14T^{6} + 1.37e16T^{7} + 1.18e18T^{8} + 8.08e19T^{9} + 3.75e21T^{10} + 8.54e21T^{11} - 2.14e25T^{12} - 2.85e27T^{13} - 2.50e29T^{14} - 1.72e31T^{15} - 1.02e33T^{16} - 5.93e34T^{17} - 4.06e36T^{18} - 3.26e38T^{19} - 2.45e40T^{20} - 1.30e42T^{21} - 9.57e42T^{22} + 7.81e45T^{23} + 1.12e48T^{24} + 1.02e50T^{25} + 7.00e51T^{26} + 3.73e53T^{27} + 1.57e55T^{28} + 5.91e56T^{29} + 3.20e58T^{30} + 2.66e60T^{31}+O(T^{32}) \)
71 \( 1 - 224T + 3.68e4T^{2} - 3.98e6T^{3} + 3.24e8T^{4} - 2.21e10T^{5} + 1.69e12T^{6} - 1.95e14T^{7} + 2.40e16T^{8} - 2.44e18T^{9} + 1.93e20T^{10} - 1.28e22T^{11} + 8.70e23T^{12} - 7.97e25T^{13} + 8.72e27T^{14} - 8.50e29T^{15} + 6.90e31T^{16} - 4.74e33T^{17} + 3.16e35T^{18} - 2.50e37T^{19} + 2.34e39T^{20} - 2.17e41T^{21} + 1.78e43T^{22} - 1.28e45T^{23} + 8.68e46T^{24} - 6.35e48T^{25} + 5.26e50T^{26} - 4.55e52T^{27} + 3.73e54T^{28} - 2.77e56T^{29} + 1.94e58T^{30} - 1.37e60T^{31}+O(T^{32}) \)
73 \( 1 + 22T + 3.04e4T^{2} - 3.43e5T^{3} + 4.88e8T^{4} - 2.77e10T^{5} + 6.09e12T^{6} - 6.05e14T^{7} + 7.45e16T^{8} - 8.57e18T^{9} + 9.09e20T^{10} - 9.94e22T^{11} + 1.01e25T^{12} - 1.04e27T^{13} + 1.02e29T^{14} - 1.00e31T^{15} + 9.49e32T^{16} - 8.91e34T^{17} + 8.15e36T^{18} - 7.31e38T^{19} + 6.50e40T^{20} - 5.61e42T^{21} + 4.81e44T^{22} - 4.03e46T^{23} + 3.32e48T^{24} - 2.69e50T^{25} + 2.14e52T^{26} - 1.67e54T^{27} + 1.28e56T^{28} - 9.59e57T^{29} + 7.00e59T^{30}+O(T^{31}) \)
79 \( 1 - 564T + 1.61e5T^{2} - 3.02e7T^{3} + 4.06e9T^{4} - 4.05e11T^{5} + 3.00e13T^{6} - 1.70e15T^{7} + 1.00e17T^{8} - 1.08e19T^{9} + 1.35e21T^{10} - 1.14e23T^{11} + 4.19e24T^{12} + 2.65e26T^{13} - 4.07e28T^{14} + 3.89e29T^{15} + 3.70e32T^{16} - 4.31e34T^{17} + 1.69e36T^{18} + 9.97e37T^{19} - 1.40e40T^{20} - 7.76e40T^{21} + 1.44e44T^{22} - 1.43e46T^{23} + 4.54e47T^{24} + 3.17e49T^{25} - 2.80e51T^{26} - 2.21e53T^{27} + 5.38e55T^{28} - 4.20e57T^{29} + 1.06e59T^{30}+O(T^{31}) \)
83 \( 1 + 126T - 2.97e4T^{2} - 3.22e6T^{3} + 5.37e8T^{4} + 4.91e10T^{5} - 6.27e12T^{6} - 5.24e14T^{7} + 6.21e16T^{8} + 4.97e18T^{9} - 6.04e20T^{10} - 4.29e22T^{11} + 5.88e24T^{12} + 3.59e26T^{13} - 4.98e28T^{14} - 2.74e30T^{15} + 3.57e32T^{16} + 1.65e34T^{17} - 2.26e36T^{18} - 6.54e37T^{19} + 1.38e40T^{20} + 5.70e40T^{21} - 8.81e43T^{22} + 1.71e45T^{23} + 4.47e47T^{24} - 2.48e49T^{25} - 9.30e50T^{26} + 2.59e53T^{27} - 7.41e54T^{28} - 2.37e57T^{29} + 8.84e58T^{30}+O(T^{31}) \)
89 \( 1 + 160T + 1.74e4T^{2} + 5.00e6T^{3} + 6.45e8T^{4} + 6.27e10T^{5} + 1.09e13T^{6} + 1.16e15T^{7} + 9.98e16T^{8} + 1.36e19T^{9} + 1.24e21T^{10} + 9.96e22T^{11} + 1.23e25T^{12} + 1.04e27T^{13} + 9.22e28T^{14} + 1.11e31T^{15} + 9.94e32T^{16} + 1.03e35T^{17} + 1.15e37T^{18} + 1.04e39T^{19} + 1.12e41T^{20} + 1.10e43T^{21} + 9.77e44T^{22} + 1.02e47T^{23} + 9.28e48T^{24} + 8.28e50T^{25} + 8.53e52T^{26} + 7.58e54T^{27} + 6.97e56T^{28} + 6.90e58T^{29}+O(T^{30}) \)
97 \( 1 - 604T + 2.20e5T^{2} - 6.01e7T^{3} + 1.32e10T^{4} - 2.45e12T^{5} + 3.94e14T^{6} - 5.54e16T^{7} + 6.86e18T^{8} - 7.42e20T^{9} + 6.96e22T^{10} - 5.50e24T^{11} + 3.49e26T^{12} - 1.58e28T^{13} + 4.09e29T^{14} - 2.49e31T^{15} + 8.06e33T^{16} - 1.56e36T^{17} + 1.70e38T^{18} - 9.74e39T^{19} - 5.27e41T^{20} + 2.23e44T^{21} - 3.30e46T^{22} + 3.53e48T^{23} - 2.72e50T^{24} + 1.59e52T^{25} - 5.51e53T^{26} - 1.14e55T^{27} + 3.42e57T^{28} - 8.90e59T^{29}+O(T^{30}) \)
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\[\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{96} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}\]

Imaginary part of the first few zeros on the critical line

−3.43132571465032652387404096427, −3.28148896817635677017620667670, −3.21028728649660128859876654384, −3.15619892179706629784387352715, −3.11852373935073293238531571467, −3.05971723940551040384636244451, −2.98020498858095359387189277077, −2.76279595876678456465117455869, −2.50462259204940996241159694855, −2.45063910181647824651044181471, −2.41121060941770110206489168690, −2.39585652968535222842806348355, −2.26853040547057054155404056923, −2.24269937766881496962787346783, −2.17461356851085325180502795350, −2.17403251881833238533724983390, −2.09619830703676726421077332286, −2.09147568901355725236710768500, −2.00214759199398563929312893731, −1.54582187350455528357458628921, −1.25813071232847850378125019971, −1.17954302663865593114647519843, −0.983428657714887671064535433412, −0.59471600967433110349041754503, −0.05084524685827240654317412974, 0.05084524685827240654317412974, 0.59471600967433110349041754503, 0.983428657714887671064535433412, 1.17954302663865593114647519843, 1.25813071232847850378125019971, 1.54582187350455528357458628921, 2.00214759199398563929312893731, 2.09147568901355725236710768500, 2.09619830703676726421077332286, 2.17403251881833238533724983390, 2.17461356851085325180502795350, 2.24269937766881496962787346783, 2.26853040547057054155404056923, 2.39585652968535222842806348355, 2.41121060941770110206489168690, 2.45063910181647824651044181471, 2.50462259204940996241159694855, 2.76279595876678456465117455869, 2.98020498858095359387189277077, 3.05971723940551040384636244451, 3.11852373935073293238531571467, 3.15619892179706629784387352715, 3.21028728649660128859876654384, 3.28148896817635677017620667670, 3.43132571465032652387404096427

Graph of the $Z$-function along the critical line

Plot not available for L-functions of degree greater than 10.