Properties

Label 36-177e18-1.1-c7e18-0-0
Degree $36$
Conductor $2.907\times 10^{40}$
Sign $1$
Analytic cond. $2.33314\times 10^{31}$
Root an. cond. $7.43586$
Motivic weight $7$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 24·2-s + 486·3-s − 185·4-s + 678·5-s + 1.16e4·6-s + 3.08e3·7-s − 8.70e3·8-s + 1.24e5·9-s + 1.62e4·10-s + 1.50e4·11-s − 8.99e4·12-s + 1.36e4·13-s + 7.39e4·14-s + 3.29e5·15-s − 3.05e4·16-s + 7.19e4·17-s + 2.99e6·18-s + 5.62e4·19-s − 1.25e5·20-s + 1.49e6·21-s + 3.61e5·22-s + 1.50e5·23-s − 4.22e6·24-s − 2.73e5·25-s + 3.27e5·26-s + 2.24e7·27-s − 5.69e5·28-s + ⋯
L(s)  = 1  + 2.12·2-s + 10.3·3-s − 1.44·4-s + 2.42·5-s + 22.0·6-s + 3.39·7-s − 6.00·8-s + 57·9-s + 5.14·10-s + 3.41·11-s − 15.0·12-s + 1.72·13-s + 7.20·14-s + 25.2·15-s − 1.86·16-s + 3.55·17-s + 120.·18-s + 1.88·19-s − 3.50·20-s + 35.2·21-s + 7.24·22-s + 2.57·23-s − 62.4·24-s − 3.50·25-s + 3.65·26-s + 219.·27-s − 4.90·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{18} \cdot 59^{18}\right)^{s/2} \, \Gamma_{\C}(s)^{18} \, L(s)\cr=\mathstrut & \,\Lambda(8-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{18} \cdot 59^{18}\right)^{s/2} \, \Gamma_{\C}(s+7/2)^{18} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(36\)
Conductor: \(3^{18} \cdot 59^{18}\)
Sign: $1$
Analytic conductor: \(2.33314\times 10^{31}\)
Root analytic conductor: \(7.43586\)
Motivic weight: \(7\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{177} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((36,\ 3^{18} \cdot 59^{18} ,\ ( \ : [7/2]^{18} ),\ 1 )\)

Particular Values

\(L(4)\) \(\approx\) \(8.808578241\times10^{6}\)
\(L(\frac12)\) \(\approx\) \(8.808578241\times10^{6}\)
\(L(\frac{9}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( ( 1 - p^{3} T )^{18} \)
59 \( ( 1 - p^{3} T )^{18} \)
good2 \( 1 - 3 p^{3} T + 761 T^{2} - 14001 T^{3} + 9329 p^{5} T^{4} - 4919479 T^{5} + 43350183 p T^{6} - 163404067 p^{3} T^{7} + 1281438849 p^{4} T^{8} - 9043393781 p^{5} T^{9} + 65286908867 p^{6} T^{10} - 217387834921 p^{8} T^{11} + 1471049043209 p^{9} T^{12} - 37302104460067 p^{8} T^{13} + 239107441134799 p^{9} T^{14} - 722039279144243 p^{11} T^{15} + 4401008621380775 p^{12} T^{16} - 6347690873730075 p^{15} T^{17} + 36908168297120903 p^{16} T^{18} - 6347690873730075 p^{22} T^{19} + 4401008621380775 p^{26} T^{20} - 722039279144243 p^{32} T^{21} + 239107441134799 p^{37} T^{22} - 37302104460067 p^{43} T^{23} + 1471049043209 p^{51} T^{24} - 217387834921 p^{57} T^{25} + 65286908867 p^{62} T^{26} - 9043393781 p^{68} T^{27} + 1281438849 p^{74} T^{28} - 163404067 p^{80} T^{29} + 43350183 p^{85} T^{30} - 4919479 p^{91} T^{31} + 9329 p^{103} T^{32} - 14001 p^{105} T^{33} + 761 p^{112} T^{34} - 3 p^{122} T^{35} + p^{126} T^{36} \)
5 \( 1 - 678 T + 733131 T^{2} - 405457892 T^{3} + 269376931213 T^{4} - 128780251471662 T^{5} + 13427900661291114 p T^{6} - 28682555673001892798 T^{7} + \)\(25\!\cdots\!77\)\( p T^{8} - \)\(79\!\cdots\!24\)\( p^{4} T^{9} + \)\(15\!\cdots\!47\)\( p^{3} T^{10} - \)\(22\!\cdots\!38\)\( p^{5} T^{11} + \)\(82\!\cdots\!47\)\( p^{5} T^{12} - \)\(10\!\cdots\!08\)\( p^{7} T^{13} + \)\(36\!\cdots\!66\)\( p^{7} T^{14} - \)\(22\!\cdots\!52\)\( p^{8} T^{15} + \)\(13\!\cdots\!06\)\( p^{9} T^{16} - \)\(80\!\cdots\!72\)\( p^{10} T^{17} + \)\(46\!\cdots\!48\)\( p^{11} T^{18} - \)\(80\!\cdots\!72\)\( p^{17} T^{19} + \)\(13\!\cdots\!06\)\( p^{23} T^{20} - \)\(22\!\cdots\!52\)\( p^{29} T^{21} + \)\(36\!\cdots\!66\)\( p^{35} T^{22} - \)\(10\!\cdots\!08\)\( p^{42} T^{23} + \)\(82\!\cdots\!47\)\( p^{47} T^{24} - \)\(22\!\cdots\!38\)\( p^{54} T^{25} + \)\(15\!\cdots\!47\)\( p^{59} T^{26} - \)\(79\!\cdots\!24\)\( p^{67} T^{27} + \)\(25\!\cdots\!77\)\( p^{71} T^{28} - 28682555673001892798 p^{77} T^{29} + 13427900661291114 p^{85} T^{30} - 128780251471662 p^{91} T^{31} + 269376931213 p^{98} T^{32} - 405457892 p^{105} T^{33} + 733131 p^{112} T^{34} - 678 p^{119} T^{35} + p^{126} T^{36} \)
7 \( 1 - 3081 T + 1619922 p T^{2} - 3600688184 p T^{3} + 56592815003477 T^{4} - 100310833155680354 T^{5} + \)\(17\!\cdots\!70\)\( T^{6} - \)\(37\!\cdots\!81\)\( p T^{7} + \)\(54\!\cdots\!96\)\( p T^{8} - \)\(50\!\cdots\!93\)\( T^{9} + \)\(65\!\cdots\!12\)\( T^{10} - \)\(77\!\cdots\!48\)\( T^{11} + \)\(91\!\cdots\!33\)\( T^{12} - \)\(99\!\cdots\!20\)\( T^{13} + \)\(10\!\cdots\!32\)\( T^{14} - \)\(10\!\cdots\!67\)\( T^{15} + \)\(10\!\cdots\!85\)\( T^{16} - \)\(10\!\cdots\!26\)\( T^{17} + \)\(96\!\cdots\!40\)\( T^{18} - \)\(10\!\cdots\!26\)\( p^{7} T^{19} + \)\(10\!\cdots\!85\)\( p^{14} T^{20} - \)\(10\!\cdots\!67\)\( p^{21} T^{21} + \)\(10\!\cdots\!32\)\( p^{28} T^{22} - \)\(99\!\cdots\!20\)\( p^{35} T^{23} + \)\(91\!\cdots\!33\)\( p^{42} T^{24} - \)\(77\!\cdots\!48\)\( p^{49} T^{25} + \)\(65\!\cdots\!12\)\( p^{56} T^{26} - \)\(50\!\cdots\!93\)\( p^{63} T^{27} + \)\(54\!\cdots\!96\)\( p^{71} T^{28} - \)\(37\!\cdots\!81\)\( p^{78} T^{29} + \)\(17\!\cdots\!70\)\( p^{84} T^{30} - 100310833155680354 p^{91} T^{31} + 56592815003477 p^{98} T^{32} - 3600688184 p^{106} T^{33} + 1619922 p^{113} T^{34} - 3081 p^{119} T^{35} + p^{126} T^{36} \)
11 \( 1 - 1370 p T + 264789526 T^{2} - 2870131397488 T^{3} + 31585681011136094 T^{4} - \)\(27\!\cdots\!30\)\( T^{5} + \)\(23\!\cdots\!22\)\( T^{6} - \)\(18\!\cdots\!58\)\( T^{7} + \)\(13\!\cdots\!30\)\( T^{8} - \)\(87\!\cdots\!68\)\( T^{9} + \)\(51\!\cdots\!46\)\( p T^{10} - \)\(34\!\cdots\!54\)\( T^{11} + \)\(19\!\cdots\!79\)\( T^{12} - \)\(10\!\cdots\!64\)\( T^{13} + \)\(57\!\cdots\!36\)\( T^{14} - \)\(29\!\cdots\!40\)\( T^{15} + \)\(14\!\cdots\!80\)\( T^{16} - \)\(66\!\cdots\!04\)\( T^{17} + \)\(30\!\cdots\!44\)\( T^{18} - \)\(66\!\cdots\!04\)\( p^{7} T^{19} + \)\(14\!\cdots\!80\)\( p^{14} T^{20} - \)\(29\!\cdots\!40\)\( p^{21} T^{21} + \)\(57\!\cdots\!36\)\( p^{28} T^{22} - \)\(10\!\cdots\!64\)\( p^{35} T^{23} + \)\(19\!\cdots\!79\)\( p^{42} T^{24} - \)\(34\!\cdots\!54\)\( p^{49} T^{25} + \)\(51\!\cdots\!46\)\( p^{57} T^{26} - \)\(87\!\cdots\!68\)\( p^{63} T^{27} + \)\(13\!\cdots\!30\)\( p^{70} T^{28} - \)\(18\!\cdots\!58\)\( p^{77} T^{29} + \)\(23\!\cdots\!22\)\( p^{84} T^{30} - \)\(27\!\cdots\!30\)\( p^{91} T^{31} + 31585681011136094 p^{98} T^{32} - 2870131397488 p^{105} T^{33} + 264789526 p^{112} T^{34} - 1370 p^{120} T^{35} + p^{126} T^{36} \)
13 \( 1 - 13662 T + 52726790 p T^{2} - 8579765696922 T^{3} + 231182144205292896 T^{4} - \)\(26\!\cdots\!58\)\( T^{5} + \)\(51\!\cdots\!36\)\( T^{6} - \)\(55\!\cdots\!70\)\( T^{7} + \)\(84\!\cdots\!56\)\( T^{8} - \)\(86\!\cdots\!18\)\( T^{9} + \)\(10\!\cdots\!70\)\( T^{10} - \)\(10\!\cdots\!10\)\( T^{11} + \)\(11\!\cdots\!99\)\( T^{12} - \)\(10\!\cdots\!24\)\( T^{13} + \)\(10\!\cdots\!08\)\( T^{14} - \)\(69\!\cdots\!00\)\( p T^{15} + \)\(81\!\cdots\!88\)\( T^{16} - \)\(65\!\cdots\!44\)\( T^{17} + \)\(54\!\cdots\!68\)\( T^{18} - \)\(65\!\cdots\!44\)\( p^{7} T^{19} + \)\(81\!\cdots\!88\)\( p^{14} T^{20} - \)\(69\!\cdots\!00\)\( p^{22} T^{21} + \)\(10\!\cdots\!08\)\( p^{28} T^{22} - \)\(10\!\cdots\!24\)\( p^{35} T^{23} + \)\(11\!\cdots\!99\)\( p^{42} T^{24} - \)\(10\!\cdots\!10\)\( p^{49} T^{25} + \)\(10\!\cdots\!70\)\( p^{56} T^{26} - \)\(86\!\cdots\!18\)\( p^{63} T^{27} + \)\(84\!\cdots\!56\)\( p^{70} T^{28} - \)\(55\!\cdots\!70\)\( p^{77} T^{29} + \)\(51\!\cdots\!36\)\( p^{84} T^{30} - \)\(26\!\cdots\!58\)\( p^{91} T^{31} + 231182144205292896 p^{98} T^{32} - 8579765696922 p^{105} T^{33} + 52726790 p^{113} T^{34} - 13662 p^{119} T^{35} + p^{126} T^{36} \)
17 \( 1 - 71919 T + 326961090 p T^{2} - 280249055484188 T^{3} + 796294733696017499 p T^{4} - \)\(53\!\cdots\!38\)\( T^{5} + \)\(20\!\cdots\!50\)\( T^{6} - \)\(68\!\cdots\!35\)\( T^{7} + \)\(21\!\cdots\!92\)\( T^{8} - \)\(64\!\cdots\!27\)\( T^{9} + \)\(18\!\cdots\!48\)\( T^{10} - \)\(48\!\cdots\!52\)\( T^{11} + \)\(12\!\cdots\!79\)\( T^{12} - \)\(30\!\cdots\!76\)\( T^{13} + \)\(70\!\cdots\!24\)\( T^{14} - \)\(16\!\cdots\!71\)\( T^{15} + \)\(35\!\cdots\!49\)\( T^{16} - \)\(74\!\cdots\!62\)\( T^{17} + \)\(15\!\cdots\!52\)\( T^{18} - \)\(74\!\cdots\!62\)\( p^{7} T^{19} + \)\(35\!\cdots\!49\)\( p^{14} T^{20} - \)\(16\!\cdots\!71\)\( p^{21} T^{21} + \)\(70\!\cdots\!24\)\( p^{28} T^{22} - \)\(30\!\cdots\!76\)\( p^{35} T^{23} + \)\(12\!\cdots\!79\)\( p^{42} T^{24} - \)\(48\!\cdots\!52\)\( p^{49} T^{25} + \)\(18\!\cdots\!48\)\( p^{56} T^{26} - \)\(64\!\cdots\!27\)\( p^{63} T^{27} + \)\(21\!\cdots\!92\)\( p^{70} T^{28} - \)\(68\!\cdots\!35\)\( p^{77} T^{29} + \)\(20\!\cdots\!50\)\( p^{84} T^{30} - \)\(53\!\cdots\!38\)\( p^{91} T^{31} + 796294733696017499 p^{99} T^{32} - 280249055484188 p^{105} T^{33} + 326961090 p^{113} T^{34} - 71919 p^{119} T^{35} + p^{126} T^{36} \)
19 \( 1 - 56231 T + 6913011087 T^{2} - 255539629397459 T^{3} + 18608088375208192030 T^{4} - \)\(43\!\cdots\!03\)\( T^{5} + \)\(26\!\cdots\!05\)\( T^{6} - \)\(27\!\cdots\!91\)\( T^{7} + \)\(26\!\cdots\!46\)\( T^{8} + \)\(93\!\cdots\!99\)\( T^{9} + \)\(31\!\cdots\!83\)\( T^{10} - \)\(66\!\cdots\!19\)\( T^{11} + \)\(45\!\cdots\!24\)\( T^{12} - \)\(31\!\cdots\!83\)\( T^{13} + \)\(51\!\cdots\!95\)\( T^{14} - \)\(21\!\cdots\!55\)\( T^{15} + \)\(42\!\cdots\!03\)\( T^{16} + \)\(89\!\cdots\!18\)\( T^{17} + \)\(33\!\cdots\!12\)\( T^{18} + \)\(89\!\cdots\!18\)\( p^{7} T^{19} + \)\(42\!\cdots\!03\)\( p^{14} T^{20} - \)\(21\!\cdots\!55\)\( p^{21} T^{21} + \)\(51\!\cdots\!95\)\( p^{28} T^{22} - \)\(31\!\cdots\!83\)\( p^{35} T^{23} + \)\(45\!\cdots\!24\)\( p^{42} T^{24} - \)\(66\!\cdots\!19\)\( p^{49} T^{25} + \)\(31\!\cdots\!83\)\( p^{56} T^{26} + \)\(93\!\cdots\!99\)\( p^{63} T^{27} + \)\(26\!\cdots\!46\)\( p^{70} T^{28} - \)\(27\!\cdots\!91\)\( p^{77} T^{29} + \)\(26\!\cdots\!05\)\( p^{84} T^{30} - \)\(43\!\cdots\!03\)\( p^{91} T^{31} + 18608088375208192030 p^{98} T^{32} - 255539629397459 p^{105} T^{33} + 6913011087 p^{112} T^{34} - 56231 p^{119} T^{35} + p^{126} T^{36} \)
23 \( 1 - 6523 p T + 39534739883 T^{2} - 4517141236191609 T^{3} + \)\(70\!\cdots\!38\)\( T^{4} - \)\(65\!\cdots\!17\)\( T^{5} + \)\(76\!\cdots\!25\)\( T^{6} - \)\(59\!\cdots\!41\)\( T^{7} + \)\(57\!\cdots\!94\)\( T^{8} - \)\(37\!\cdots\!03\)\( T^{9} + \)\(31\!\cdots\!71\)\( T^{10} - \)\(17\!\cdots\!01\)\( T^{11} + \)\(12\!\cdots\!88\)\( T^{12} - \)\(56\!\cdots\!85\)\( T^{13} + \)\(40\!\cdots\!87\)\( T^{14} - \)\(13\!\cdots\!65\)\( T^{15} + \)\(10\!\cdots\!15\)\( T^{16} - \)\(26\!\cdots\!18\)\( T^{17} + \)\(30\!\cdots\!08\)\( T^{18} - \)\(26\!\cdots\!18\)\( p^{7} T^{19} + \)\(10\!\cdots\!15\)\( p^{14} T^{20} - \)\(13\!\cdots\!65\)\( p^{21} T^{21} + \)\(40\!\cdots\!87\)\( p^{28} T^{22} - \)\(56\!\cdots\!85\)\( p^{35} T^{23} + \)\(12\!\cdots\!88\)\( p^{42} T^{24} - \)\(17\!\cdots\!01\)\( p^{49} T^{25} + \)\(31\!\cdots\!71\)\( p^{56} T^{26} - \)\(37\!\cdots\!03\)\( p^{63} T^{27} + \)\(57\!\cdots\!94\)\( p^{70} T^{28} - \)\(59\!\cdots\!41\)\( p^{77} T^{29} + \)\(76\!\cdots\!25\)\( p^{84} T^{30} - \)\(65\!\cdots\!17\)\( p^{91} T^{31} + \)\(70\!\cdots\!38\)\( p^{98} T^{32} - 4517141236191609 p^{105} T^{33} + 39534739883 p^{112} T^{34} - 6523 p^{120} T^{35} + p^{126} T^{36} \)
29 \( 1 - 591285 T + 365010542031 T^{2} - 139853784708019775 T^{3} + \)\(51\!\cdots\!84\)\( T^{4} - \)\(14\!\cdots\!81\)\( T^{5} + \)\(41\!\cdots\!33\)\( T^{6} - \)\(97\!\cdots\!47\)\( T^{7} + \)\(22\!\cdots\!18\)\( T^{8} - \)\(44\!\cdots\!39\)\( T^{9} + \)\(85\!\cdots\!79\)\( T^{10} - \)\(14\!\cdots\!91\)\( T^{11} + \)\(25\!\cdots\!90\)\( T^{12} - \)\(38\!\cdots\!45\)\( T^{13} + \)\(59\!\cdots\!75\)\( T^{14} - \)\(29\!\cdots\!87\)\( p T^{15} + \)\(12\!\cdots\!19\)\( T^{16} - \)\(16\!\cdots\!98\)\( T^{17} + \)\(21\!\cdots\!00\)\( T^{18} - \)\(16\!\cdots\!98\)\( p^{7} T^{19} + \)\(12\!\cdots\!19\)\( p^{14} T^{20} - \)\(29\!\cdots\!87\)\( p^{22} T^{21} + \)\(59\!\cdots\!75\)\( p^{28} T^{22} - \)\(38\!\cdots\!45\)\( p^{35} T^{23} + \)\(25\!\cdots\!90\)\( p^{42} T^{24} - \)\(14\!\cdots\!91\)\( p^{49} T^{25} + \)\(85\!\cdots\!79\)\( p^{56} T^{26} - \)\(44\!\cdots\!39\)\( p^{63} T^{27} + \)\(22\!\cdots\!18\)\( p^{70} T^{28} - \)\(97\!\cdots\!47\)\( p^{77} T^{29} + \)\(41\!\cdots\!33\)\( p^{84} T^{30} - \)\(14\!\cdots\!81\)\( p^{91} T^{31} + \)\(51\!\cdots\!84\)\( p^{98} T^{32} - 139853784708019775 p^{105} T^{33} + 365010542031 p^{112} T^{34} - 591285 p^{119} T^{35} + p^{126} T^{36} \)
31 \( 1 - 426733 T + 356827683131 T^{2} - 128185087840611257 T^{3} + \)\(61\!\cdots\!68\)\( T^{4} - \)\(19\!\cdots\!65\)\( T^{5} + \)\(67\!\cdots\!89\)\( T^{6} - \)\(18\!\cdots\!29\)\( T^{7} + \)\(54\!\cdots\!78\)\( T^{8} - \)\(13\!\cdots\!15\)\( T^{9} + \)\(33\!\cdots\!99\)\( T^{10} - \)\(76\!\cdots\!73\)\( T^{11} + \)\(17\!\cdots\!22\)\( T^{12} - \)\(35\!\cdots\!25\)\( T^{13} + \)\(71\!\cdots\!43\)\( T^{14} - \)\(13\!\cdots\!73\)\( T^{15} + \)\(25\!\cdots\!59\)\( T^{16} - \)\(44\!\cdots\!26\)\( T^{17} + \)\(74\!\cdots\!92\)\( T^{18} - \)\(44\!\cdots\!26\)\( p^{7} T^{19} + \)\(25\!\cdots\!59\)\( p^{14} T^{20} - \)\(13\!\cdots\!73\)\( p^{21} T^{21} + \)\(71\!\cdots\!43\)\( p^{28} T^{22} - \)\(35\!\cdots\!25\)\( p^{35} T^{23} + \)\(17\!\cdots\!22\)\( p^{42} T^{24} - \)\(76\!\cdots\!73\)\( p^{49} T^{25} + \)\(33\!\cdots\!99\)\( p^{56} T^{26} - \)\(13\!\cdots\!15\)\( p^{63} T^{27} + \)\(54\!\cdots\!78\)\( p^{70} T^{28} - \)\(18\!\cdots\!29\)\( p^{77} T^{29} + \)\(67\!\cdots\!89\)\( p^{84} T^{30} - \)\(19\!\cdots\!65\)\( p^{91} T^{31} + \)\(61\!\cdots\!68\)\( p^{98} T^{32} - 128185087840611257 p^{105} T^{33} + 356827683131 p^{112} T^{34} - 426733 p^{119} T^{35} + p^{126} T^{36} \)
37 \( 1 - 7703 T + 1033524094348 T^{2} + 916981333928126 p T^{3} + \)\(53\!\cdots\!13\)\( T^{4} + \)\(34\!\cdots\!40\)\( T^{5} + \)\(18\!\cdots\!38\)\( T^{6} + \)\(16\!\cdots\!53\)\( T^{7} + \)\(47\!\cdots\!12\)\( T^{8} + \)\(50\!\cdots\!77\)\( T^{9} + \)\(98\!\cdots\!48\)\( T^{10} + \)\(11\!\cdots\!38\)\( T^{11} + \)\(16\!\cdots\!69\)\( T^{12} + \)\(20\!\cdots\!90\)\( T^{13} + \)\(23\!\cdots\!94\)\( T^{14} + \)\(28\!\cdots\!21\)\( T^{15} + \)\(28\!\cdots\!29\)\( T^{16} + \)\(32\!\cdots\!78\)\( T^{17} + \)\(28\!\cdots\!04\)\( T^{18} + \)\(32\!\cdots\!78\)\( p^{7} T^{19} + \)\(28\!\cdots\!29\)\( p^{14} T^{20} + \)\(28\!\cdots\!21\)\( p^{21} T^{21} + \)\(23\!\cdots\!94\)\( p^{28} T^{22} + \)\(20\!\cdots\!90\)\( p^{35} T^{23} + \)\(16\!\cdots\!69\)\( p^{42} T^{24} + \)\(11\!\cdots\!38\)\( p^{49} T^{25} + \)\(98\!\cdots\!48\)\( p^{56} T^{26} + \)\(50\!\cdots\!77\)\( p^{63} T^{27} + \)\(47\!\cdots\!12\)\( p^{70} T^{28} + \)\(16\!\cdots\!53\)\( p^{77} T^{29} + \)\(18\!\cdots\!38\)\( p^{84} T^{30} + \)\(34\!\cdots\!40\)\( p^{91} T^{31} + \)\(53\!\cdots\!13\)\( p^{98} T^{32} + 916981333928126 p^{106} T^{33} + 1033524094348 p^{112} T^{34} - 7703 p^{119} T^{35} + p^{126} T^{36} \)
41 \( 1 - 770959 T + 2443099898286 T^{2} - 1660386833592575028 T^{3} + \)\(29\!\cdots\!07\)\( T^{4} - \)\(17\!\cdots\!62\)\( T^{5} + \)\(22\!\cdots\!14\)\( T^{6} - \)\(30\!\cdots\!27\)\( p T^{7} + \)\(12\!\cdots\!48\)\( T^{8} - \)\(64\!\cdots\!07\)\( T^{9} + \)\(55\!\cdots\!52\)\( T^{10} - \)\(25\!\cdots\!76\)\( T^{11} + \)\(19\!\cdots\!95\)\( T^{12} - \)\(85\!\cdots\!00\)\( T^{13} + \)\(58\!\cdots\!52\)\( T^{14} - \)\(23\!\cdots\!87\)\( T^{15} + \)\(14\!\cdots\!61\)\( T^{16} - \)\(53\!\cdots\!98\)\( T^{17} + \)\(30\!\cdots\!16\)\( T^{18} - \)\(53\!\cdots\!98\)\( p^{7} T^{19} + \)\(14\!\cdots\!61\)\( p^{14} T^{20} - \)\(23\!\cdots\!87\)\( p^{21} T^{21} + \)\(58\!\cdots\!52\)\( p^{28} T^{22} - \)\(85\!\cdots\!00\)\( p^{35} T^{23} + \)\(19\!\cdots\!95\)\( p^{42} T^{24} - \)\(25\!\cdots\!76\)\( p^{49} T^{25} + \)\(55\!\cdots\!52\)\( p^{56} T^{26} - \)\(64\!\cdots\!07\)\( p^{63} T^{27} + \)\(12\!\cdots\!48\)\( p^{70} T^{28} - \)\(30\!\cdots\!27\)\( p^{78} T^{29} + \)\(22\!\cdots\!14\)\( p^{84} T^{30} - \)\(17\!\cdots\!62\)\( p^{91} T^{31} + \)\(29\!\cdots\!07\)\( p^{98} T^{32} - 1660386833592575028 p^{105} T^{33} + 2443099898286 p^{112} T^{34} - 770959 p^{119} T^{35} + p^{126} T^{36} \)
43 \( 1 - 793050 T + 3578356047914 T^{2} - 2430358434264038576 T^{3} + \)\(59\!\cdots\!94\)\( T^{4} - \)\(34\!\cdots\!90\)\( T^{5} + \)\(62\!\cdots\!78\)\( T^{6} - \)\(31\!\cdots\!62\)\( T^{7} + \)\(45\!\cdots\!94\)\( T^{8} - \)\(19\!\cdots\!16\)\( T^{9} + \)\(25\!\cdots\!70\)\( T^{10} - \)\(93\!\cdots\!30\)\( T^{11} + \)\(11\!\cdots\!31\)\( T^{12} - \)\(34\!\cdots\!12\)\( T^{13} + \)\(40\!\cdots\!48\)\( T^{14} - \)\(10\!\cdots\!12\)\( T^{15} + \)\(12\!\cdots\!56\)\( T^{16} - \)\(30\!\cdots\!36\)\( T^{17} + \)\(36\!\cdots\!32\)\( T^{18} - \)\(30\!\cdots\!36\)\( p^{7} T^{19} + \)\(12\!\cdots\!56\)\( p^{14} T^{20} - \)\(10\!\cdots\!12\)\( p^{21} T^{21} + \)\(40\!\cdots\!48\)\( p^{28} T^{22} - \)\(34\!\cdots\!12\)\( p^{35} T^{23} + \)\(11\!\cdots\!31\)\( p^{42} T^{24} - \)\(93\!\cdots\!30\)\( p^{49} T^{25} + \)\(25\!\cdots\!70\)\( p^{56} T^{26} - \)\(19\!\cdots\!16\)\( p^{63} T^{27} + \)\(45\!\cdots\!94\)\( p^{70} T^{28} - \)\(31\!\cdots\!62\)\( p^{77} T^{29} + \)\(62\!\cdots\!78\)\( p^{84} T^{30} - \)\(34\!\cdots\!90\)\( p^{91} T^{31} + \)\(59\!\cdots\!94\)\( p^{98} T^{32} - 2430358434264038576 p^{105} T^{33} + 3578356047914 p^{112} T^{34} - 793050 p^{119} T^{35} + p^{126} T^{36} \)
47 \( 1 - 1410373 T + 5763697399605 T^{2} - 7463106534160997625 T^{3} + \)\(16\!\cdots\!34\)\( T^{4} - \)\(19\!\cdots\!73\)\( T^{5} + \)\(32\!\cdots\!71\)\( T^{6} - \)\(35\!\cdots\!89\)\( T^{7} + \)\(47\!\cdots\!50\)\( T^{8} - \)\(47\!\cdots\!59\)\( T^{9} + \)\(54\!\cdots\!57\)\( T^{10} - \)\(49\!\cdots\!37\)\( T^{11} + \)\(50\!\cdots\!04\)\( T^{12} - \)\(42\!\cdots\!77\)\( T^{13} + \)\(38\!\cdots\!65\)\( T^{14} - \)\(30\!\cdots\!41\)\( T^{15} + \)\(25\!\cdots\!23\)\( T^{16} - \)\(18\!\cdots\!22\)\( T^{17} + \)\(13\!\cdots\!96\)\( T^{18} - \)\(18\!\cdots\!22\)\( p^{7} T^{19} + \)\(25\!\cdots\!23\)\( p^{14} T^{20} - \)\(30\!\cdots\!41\)\( p^{21} T^{21} + \)\(38\!\cdots\!65\)\( p^{28} T^{22} - \)\(42\!\cdots\!77\)\( p^{35} T^{23} + \)\(50\!\cdots\!04\)\( p^{42} T^{24} - \)\(49\!\cdots\!37\)\( p^{49} T^{25} + \)\(54\!\cdots\!57\)\( p^{56} T^{26} - \)\(47\!\cdots\!59\)\( p^{63} T^{27} + \)\(47\!\cdots\!50\)\( p^{70} T^{28} - \)\(35\!\cdots\!89\)\( p^{77} T^{29} + \)\(32\!\cdots\!71\)\( p^{84} T^{30} - \)\(19\!\cdots\!73\)\( p^{91} T^{31} + \)\(16\!\cdots\!34\)\( p^{98} T^{32} - 7463106534160997625 p^{105} T^{33} + 5763697399605 p^{112} T^{34} - 1410373 p^{119} T^{35} + p^{126} T^{36} \)
53 \( 1 - 1037934 T + 8591895245123 T^{2} - 10895948617615661820 T^{3} + \)\(39\!\cdots\!25\)\( T^{4} - \)\(53\!\cdots\!50\)\( T^{5} + \)\(12\!\cdots\!22\)\( T^{6} - \)\(17\!\cdots\!50\)\( T^{7} + \)\(32\!\cdots\!57\)\( T^{8} - \)\(41\!\cdots\!36\)\( T^{9} + \)\(67\!\cdots\!31\)\( T^{10} - \)\(78\!\cdots\!66\)\( T^{11} + \)\(11\!\cdots\!19\)\( T^{12} - \)\(12\!\cdots\!84\)\( T^{13} + \)\(16\!\cdots\!10\)\( T^{14} - \)\(17\!\cdots\!84\)\( T^{15} + \)\(22\!\cdots\!30\)\( T^{16} - \)\(22\!\cdots\!16\)\( T^{17} + \)\(27\!\cdots\!68\)\( T^{18} - \)\(22\!\cdots\!16\)\( p^{7} T^{19} + \)\(22\!\cdots\!30\)\( p^{14} T^{20} - \)\(17\!\cdots\!84\)\( p^{21} T^{21} + \)\(16\!\cdots\!10\)\( p^{28} T^{22} - \)\(12\!\cdots\!84\)\( p^{35} T^{23} + \)\(11\!\cdots\!19\)\( p^{42} T^{24} - \)\(78\!\cdots\!66\)\( p^{49} T^{25} + \)\(67\!\cdots\!31\)\( p^{56} T^{26} - \)\(41\!\cdots\!36\)\( p^{63} T^{27} + \)\(32\!\cdots\!57\)\( p^{70} T^{28} - \)\(17\!\cdots\!50\)\( p^{77} T^{29} + \)\(12\!\cdots\!22\)\( p^{84} T^{30} - \)\(53\!\cdots\!50\)\( p^{91} T^{31} + \)\(39\!\cdots\!25\)\( p^{98} T^{32} - 10895948617615661820 p^{105} T^{33} + 8591895245123 p^{112} T^{34} - 1037934 p^{119} T^{35} + p^{126} T^{36} \)
61 \( 1 + 1374623 T + 27159502236915 T^{2} + 30076954692047341253 T^{3} + \)\(36\!\cdots\!96\)\( T^{4} + \)\(32\!\cdots\!71\)\( T^{5} + \)\(31\!\cdots\!69\)\( T^{6} + \)\(22\!\cdots\!93\)\( T^{7} + \)\(21\!\cdots\!14\)\( T^{8} + \)\(12\!\cdots\!85\)\( T^{9} + \)\(11\!\cdots\!91\)\( T^{10} + \)\(52\!\cdots\!49\)\( T^{11} + \)\(54\!\cdots\!98\)\( T^{12} + \)\(18\!\cdots\!27\)\( T^{13} + \)\(21\!\cdots\!39\)\( T^{14} + \)\(58\!\cdots\!89\)\( T^{15} + \)\(80\!\cdots\!27\)\( T^{16} + \)\(17\!\cdots\!30\)\( T^{17} + \)\(26\!\cdots\!40\)\( T^{18} + \)\(17\!\cdots\!30\)\( p^{7} T^{19} + \)\(80\!\cdots\!27\)\( p^{14} T^{20} + \)\(58\!\cdots\!89\)\( p^{21} T^{21} + \)\(21\!\cdots\!39\)\( p^{28} T^{22} + \)\(18\!\cdots\!27\)\( p^{35} T^{23} + \)\(54\!\cdots\!98\)\( p^{42} T^{24} + \)\(52\!\cdots\!49\)\( p^{49} T^{25} + \)\(11\!\cdots\!91\)\( p^{56} T^{26} + \)\(12\!\cdots\!85\)\( p^{63} T^{27} + \)\(21\!\cdots\!14\)\( p^{70} T^{28} + \)\(22\!\cdots\!93\)\( p^{77} T^{29} + \)\(31\!\cdots\!69\)\( p^{84} T^{30} + \)\(32\!\cdots\!71\)\( p^{91} T^{31} + \)\(36\!\cdots\!96\)\( p^{98} T^{32} + 30076954692047341253 p^{105} T^{33} + 27159502236915 p^{112} T^{34} + 1374623 p^{119} T^{35} + p^{126} T^{36} \)
67 \( 1 + 2436904 T + 73607202603217 T^{2} + \)\(18\!\cdots\!44\)\( T^{3} + \)\(26\!\cdots\!43\)\( T^{4} + \)\(71\!\cdots\!56\)\( T^{5} + \)\(64\!\cdots\!62\)\( T^{6} + \)\(17\!\cdots\!04\)\( T^{7} + \)\(11\!\cdots\!55\)\( T^{8} + \)\(30\!\cdots\!28\)\( T^{9} + \)\(15\!\cdots\!77\)\( T^{10} + \)\(41\!\cdots\!72\)\( T^{11} + \)\(17\!\cdots\!19\)\( T^{12} + \)\(45\!\cdots\!12\)\( T^{13} + \)\(16\!\cdots\!30\)\( T^{14} + \)\(40\!\cdots\!92\)\( T^{15} + \)\(12\!\cdots\!90\)\( T^{16} + \)\(29\!\cdots\!84\)\( T^{17} + \)\(83\!\cdots\!16\)\( T^{18} + \)\(29\!\cdots\!84\)\( p^{7} T^{19} + \)\(12\!\cdots\!90\)\( p^{14} T^{20} + \)\(40\!\cdots\!92\)\( p^{21} T^{21} + \)\(16\!\cdots\!30\)\( p^{28} T^{22} + \)\(45\!\cdots\!12\)\( p^{35} T^{23} + \)\(17\!\cdots\!19\)\( p^{42} T^{24} + \)\(41\!\cdots\!72\)\( p^{49} T^{25} + \)\(15\!\cdots\!77\)\( p^{56} T^{26} + \)\(30\!\cdots\!28\)\( p^{63} T^{27} + \)\(11\!\cdots\!55\)\( p^{70} T^{28} + \)\(17\!\cdots\!04\)\( p^{77} T^{29} + \)\(64\!\cdots\!62\)\( p^{84} T^{30} + \)\(71\!\cdots\!56\)\( p^{91} T^{31} + \)\(26\!\cdots\!43\)\( p^{98} T^{32} + \)\(18\!\cdots\!44\)\( p^{105} T^{33} + 73607202603217 p^{112} T^{34} + 2436904 p^{119} T^{35} + p^{126} T^{36} \)
71 \( 1 - 14289172 T + 200864762815856 T^{2} - \)\(17\!\cdots\!42\)\( T^{3} + \)\(15\!\cdots\!82\)\( T^{4} - \)\(10\!\cdots\!48\)\( T^{5} + \)\(63\!\cdots\!52\)\( T^{6} - \)\(33\!\cdots\!00\)\( T^{7} + \)\(17\!\cdots\!78\)\( T^{8} - \)\(79\!\cdots\!26\)\( T^{9} + \)\(35\!\cdots\!24\)\( T^{10} - \)\(13\!\cdots\!52\)\( T^{11} + \)\(54\!\cdots\!31\)\( T^{12} - \)\(19\!\cdots\!24\)\( T^{13} + \)\(68\!\cdots\!08\)\( T^{14} - \)\(22\!\cdots\!36\)\( T^{15} + \)\(72\!\cdots\!52\)\( T^{16} - \)\(22\!\cdots\!80\)\( T^{17} + \)\(69\!\cdots\!52\)\( T^{18} - \)\(22\!\cdots\!80\)\( p^{7} T^{19} + \)\(72\!\cdots\!52\)\( p^{14} T^{20} - \)\(22\!\cdots\!36\)\( p^{21} T^{21} + \)\(68\!\cdots\!08\)\( p^{28} T^{22} - \)\(19\!\cdots\!24\)\( p^{35} T^{23} + \)\(54\!\cdots\!31\)\( p^{42} T^{24} - \)\(13\!\cdots\!52\)\( p^{49} T^{25} + \)\(35\!\cdots\!24\)\( p^{56} T^{26} - \)\(79\!\cdots\!26\)\( p^{63} T^{27} + \)\(17\!\cdots\!78\)\( p^{70} T^{28} - \)\(33\!\cdots\!00\)\( p^{77} T^{29} + \)\(63\!\cdots\!52\)\( p^{84} T^{30} - \)\(10\!\cdots\!48\)\( p^{91} T^{31} + \)\(15\!\cdots\!82\)\( p^{98} T^{32} - \)\(17\!\cdots\!42\)\( p^{105} T^{33} + 200864762815856 p^{112} T^{34} - 14289172 p^{119} T^{35} + p^{126} T^{36} \)
73 \( 1 - 5482515 T + 99963666816463 T^{2} - \)\(56\!\cdots\!01\)\( T^{3} + \)\(52\!\cdots\!42\)\( T^{4} - \)\(27\!\cdots\!19\)\( T^{5} + \)\(18\!\cdots\!05\)\( T^{6} - \)\(90\!\cdots\!37\)\( T^{7} + \)\(48\!\cdots\!38\)\( T^{8} - \)\(22\!\cdots\!73\)\( T^{9} + \)\(10\!\cdots\!39\)\( T^{10} - \)\(42\!\cdots\!33\)\( T^{11} + \)\(17\!\cdots\!72\)\( T^{12} - \)\(69\!\cdots\!83\)\( T^{13} + \)\(26\!\cdots\!91\)\( T^{14} - \)\(97\!\cdots\!01\)\( T^{15} + \)\(34\!\cdots\!55\)\( T^{16} - \)\(11\!\cdots\!26\)\( T^{17} + \)\(40\!\cdots\!40\)\( T^{18} - \)\(11\!\cdots\!26\)\( p^{7} T^{19} + \)\(34\!\cdots\!55\)\( p^{14} T^{20} - \)\(97\!\cdots\!01\)\( p^{21} T^{21} + \)\(26\!\cdots\!91\)\( p^{28} T^{22} - \)\(69\!\cdots\!83\)\( p^{35} T^{23} + \)\(17\!\cdots\!72\)\( p^{42} T^{24} - \)\(42\!\cdots\!33\)\( p^{49} T^{25} + \)\(10\!\cdots\!39\)\( p^{56} T^{26} - \)\(22\!\cdots\!73\)\( p^{63} T^{27} + \)\(48\!\cdots\!38\)\( p^{70} T^{28} - \)\(90\!\cdots\!37\)\( p^{77} T^{29} + \)\(18\!\cdots\!05\)\( p^{84} T^{30} - \)\(27\!\cdots\!19\)\( p^{91} T^{31} + \)\(52\!\cdots\!42\)\( p^{98} T^{32} - \)\(56\!\cdots\!01\)\( p^{105} T^{33} + 99963666816463 p^{112} T^{34} - 5482515 p^{119} T^{35} + p^{126} T^{36} \)
79 \( 1 - 19786414 T + 455950835401310 T^{2} - \)\(62\!\cdots\!80\)\( T^{3} + \)\(86\!\cdots\!10\)\( T^{4} - \)\(92\!\cdots\!22\)\( T^{5} + \)\(96\!\cdots\!38\)\( T^{6} - \)\(86\!\cdots\!34\)\( T^{7} + \)\(73\!\cdots\!98\)\( T^{8} - \)\(56\!\cdots\!56\)\( T^{9} + \)\(40\!\cdots\!50\)\( T^{10} - \)\(27\!\cdots\!22\)\( T^{11} + \)\(17\!\cdots\!43\)\( T^{12} - \)\(10\!\cdots\!28\)\( T^{13} + \)\(57\!\cdots\!12\)\( T^{14} - \)\(30\!\cdots\!00\)\( T^{15} + \)\(15\!\cdots\!84\)\( T^{16} - \)\(72\!\cdots\!52\)\( T^{17} + \)\(32\!\cdots\!28\)\( T^{18} - \)\(72\!\cdots\!52\)\( p^{7} T^{19} + \)\(15\!\cdots\!84\)\( p^{14} T^{20} - \)\(30\!\cdots\!00\)\( p^{21} T^{21} + \)\(57\!\cdots\!12\)\( p^{28} T^{22} - \)\(10\!\cdots\!28\)\( p^{35} T^{23} + \)\(17\!\cdots\!43\)\( p^{42} T^{24} - \)\(27\!\cdots\!22\)\( p^{49} T^{25} + \)\(40\!\cdots\!50\)\( p^{56} T^{26} - \)\(56\!\cdots\!56\)\( p^{63} T^{27} + \)\(73\!\cdots\!98\)\( p^{70} T^{28} - \)\(86\!\cdots\!34\)\( p^{77} T^{29} + \)\(96\!\cdots\!38\)\( p^{84} T^{30} - \)\(92\!\cdots\!22\)\( p^{91} T^{31} + \)\(86\!\cdots\!10\)\( p^{98} T^{32} - \)\(62\!\cdots\!80\)\( p^{105} T^{33} + 455950835401310 p^{112} T^{34} - 19786414 p^{119} T^{35} + p^{126} T^{36} \)
83 \( 1 - 30227337 T + 680293024016706 T^{2} - \)\(10\!\cdots\!90\)\( T^{3} + \)\(14\!\cdots\!95\)\( T^{4} - \)\(17\!\cdots\!36\)\( T^{5} + \)\(17\!\cdots\!14\)\( T^{6} - \)\(15\!\cdots\!11\)\( T^{7} + \)\(13\!\cdots\!68\)\( T^{8} - \)\(98\!\cdots\!13\)\( T^{9} + \)\(69\!\cdots\!76\)\( T^{10} - \)\(45\!\cdots\!94\)\( T^{11} + \)\(28\!\cdots\!75\)\( T^{12} - \)\(16\!\cdots\!62\)\( T^{13} + \)\(95\!\cdots\!52\)\( T^{14} - \)\(51\!\cdots\!39\)\( T^{15} + \)\(27\!\cdots\!49\)\( T^{16} - \)\(14\!\cdots\!50\)\( T^{17} + \)\(76\!\cdots\!44\)\( T^{18} - \)\(14\!\cdots\!50\)\( p^{7} T^{19} + \)\(27\!\cdots\!49\)\( p^{14} T^{20} - \)\(51\!\cdots\!39\)\( p^{21} T^{21} + \)\(95\!\cdots\!52\)\( p^{28} T^{22} - \)\(16\!\cdots\!62\)\( p^{35} T^{23} + \)\(28\!\cdots\!75\)\( p^{42} T^{24} - \)\(45\!\cdots\!94\)\( p^{49} T^{25} + \)\(69\!\cdots\!76\)\( p^{56} T^{26} - \)\(98\!\cdots\!13\)\( p^{63} T^{27} + \)\(13\!\cdots\!68\)\( p^{70} T^{28} - \)\(15\!\cdots\!11\)\( p^{77} T^{29} + \)\(17\!\cdots\!14\)\( p^{84} T^{30} - \)\(17\!\cdots\!36\)\( p^{91} T^{31} + \)\(14\!\cdots\!95\)\( p^{98} T^{32} - \)\(10\!\cdots\!90\)\( p^{105} T^{33} + 680293024016706 p^{112} T^{34} - 30227337 p^{119} T^{35} + p^{126} T^{36} \)
89 \( 1 - 31061677 T + 802858236783735 T^{2} - \)\(14\!\cdots\!31\)\( T^{3} + \)\(23\!\cdots\!58\)\( T^{4} - \)\(31\!\cdots\!01\)\( T^{5} + \)\(40\!\cdots\!81\)\( T^{6} - \)\(45\!\cdots\!15\)\( T^{7} + \)\(49\!\cdots\!22\)\( T^{8} - \)\(48\!\cdots\!99\)\( T^{9} + \)\(46\!\cdots\!35\)\( T^{10} - \)\(41\!\cdots\!07\)\( T^{11} + \)\(35\!\cdots\!28\)\( T^{12} - \)\(28\!\cdots\!33\)\( T^{13} + \)\(22\!\cdots\!07\)\( T^{14} - \)\(17\!\cdots\!91\)\( T^{15} + \)\(12\!\cdots\!07\)\( T^{16} - \)\(87\!\cdots\!70\)\( T^{17} + \)\(59\!\cdots\!12\)\( T^{18} - \)\(87\!\cdots\!70\)\( p^{7} T^{19} + \)\(12\!\cdots\!07\)\( p^{14} T^{20} - \)\(17\!\cdots\!91\)\( p^{21} T^{21} + \)\(22\!\cdots\!07\)\( p^{28} T^{22} - \)\(28\!\cdots\!33\)\( p^{35} T^{23} + \)\(35\!\cdots\!28\)\( p^{42} T^{24} - \)\(41\!\cdots\!07\)\( p^{49} T^{25} + \)\(46\!\cdots\!35\)\( p^{56} T^{26} - \)\(48\!\cdots\!99\)\( p^{63} T^{27} + \)\(49\!\cdots\!22\)\( p^{70} T^{28} - \)\(45\!\cdots\!15\)\( p^{77} T^{29} + \)\(40\!\cdots\!81\)\( p^{84} T^{30} - \)\(31\!\cdots\!01\)\( p^{91} T^{31} + \)\(23\!\cdots\!58\)\( p^{98} T^{32} - \)\(14\!\cdots\!31\)\( p^{105} T^{33} + 802858236783735 p^{112} T^{34} - 31061677 p^{119} T^{35} + p^{126} T^{36} \)
97 \( 1 - 12084118 T + 589406717180899 T^{2} - \)\(70\!\cdots\!34\)\( T^{3} + \)\(17\!\cdots\!67\)\( T^{4} - \)\(20\!\cdots\!08\)\( T^{5} + \)\(36\!\cdots\!74\)\( T^{6} - \)\(39\!\cdots\!28\)\( T^{7} + \)\(57\!\cdots\!23\)\( T^{8} - \)\(57\!\cdots\!10\)\( T^{9} + \)\(73\!\cdots\!79\)\( T^{10} - \)\(69\!\cdots\!70\)\( T^{11} + \)\(79\!\cdots\!55\)\( T^{12} - \)\(71\!\cdots\!96\)\( T^{13} + \)\(75\!\cdots\!02\)\( T^{14} - \)\(65\!\cdots\!68\)\( T^{15} + \)\(65\!\cdots\!18\)\( T^{16} - \)\(55\!\cdots\!68\)\( T^{17} + \)\(54\!\cdots\!52\)\( T^{18} - \)\(55\!\cdots\!68\)\( p^{7} T^{19} + \)\(65\!\cdots\!18\)\( p^{14} T^{20} - \)\(65\!\cdots\!68\)\( p^{21} T^{21} + \)\(75\!\cdots\!02\)\( p^{28} T^{22} - \)\(71\!\cdots\!96\)\( p^{35} T^{23} + \)\(79\!\cdots\!55\)\( p^{42} T^{24} - \)\(69\!\cdots\!70\)\( p^{49} T^{25} + \)\(73\!\cdots\!79\)\( p^{56} T^{26} - \)\(57\!\cdots\!10\)\( p^{63} T^{27} + \)\(57\!\cdots\!23\)\( p^{70} T^{28} - \)\(39\!\cdots\!28\)\( p^{77} T^{29} + \)\(36\!\cdots\!74\)\( p^{84} T^{30} - \)\(20\!\cdots\!08\)\( p^{91} T^{31} + \)\(17\!\cdots\!67\)\( p^{98} T^{32} - \)\(70\!\cdots\!34\)\( p^{105} T^{33} + 589406717180899 p^{112} T^{34} - 12084118 p^{119} T^{35} + p^{126} T^{36} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{36} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−2.05319025205459750964437317412, −2.04371325526745881959284738884, −2.03200818194783062775943970416, −1.93132062995932247802800077444, −1.90648746126179472177499629214, −1.84728409290162336583703382106, −1.72667109364519435154698547922, −1.64908185379076646575161132804, −1.51789753634137289784327516974, −1.51432176893645747567166793556, −1.42180224023623939772885336295, −1.27412100385318772932881929018, −1.09749367616250774429494856929, −1.08300448984409725667560059164, −1.06893488905343463139062254596, −1.02844007919417945031662637728, −0.990596191157991163419761036133, −0.851874011131207138412055655842, −0.807665615614656166019043310415, −0.66240305464672685674841881425, −0.64447823028612992828617982427, −0.59870669636246793189453459867, −0.53852939831388295648646332943, −0.38451077005128205142020335772, −0.32814966425222134135344191912, 0.32814966425222134135344191912, 0.38451077005128205142020335772, 0.53852939831388295648646332943, 0.59870669636246793189453459867, 0.64447823028612992828617982427, 0.66240305464672685674841881425, 0.807665615614656166019043310415, 0.851874011131207138412055655842, 0.990596191157991163419761036133, 1.02844007919417945031662637728, 1.06893488905343463139062254596, 1.08300448984409725667560059164, 1.09749367616250774429494856929, 1.27412100385318772932881929018, 1.42180224023623939772885336295, 1.51432176893645747567166793556, 1.51789753634137289784327516974, 1.64908185379076646575161132804, 1.72667109364519435154698547922, 1.84728409290162336583703382106, 1.90648746126179472177499629214, 1.93132062995932247802800077444, 2.03200818194783062775943970416, 2.04371325526745881959284738884, 2.05319025205459750964437317412

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

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