Dirichlet series
| L(s) = 1 | − 579·2-s + 1.18e5·3-s − 4.91e5·4-s − 1.64e6·5-s − 6.83e7·6-s − 2.42e7·7-s + 3.64e8·8-s + 7.36e9·9-s + 9.52e8·10-s − 1.07e9·11-s − 5.80e10·12-s − 2.78e9·13-s + 1.40e10·14-s − 1.94e11·15-s + 6.18e10·16-s − 6.54e10·17-s − 4.26e12·18-s − 1.45e11·19-s + 8.08e11·20-s − 2.86e12·21-s + 6.20e11·22-s − 1.04e12·23-s + 4.30e13·24-s − 5.61e12·25-s + 1.61e12·26-s + 3.21e14·27-s + 1.19e13·28-s + ⋯ |
| L(s) = 1 | − 1.59·2-s + 10.3·3-s − 3.74·4-s − 1.88·5-s − 16.6·6-s − 1.59·7-s + 7.68·8-s + 57·9-s + 3.01·10-s − 1.50·11-s − 38.9·12-s − 0.947·13-s + 2.54·14-s − 19.5·15-s + 3.59·16-s − 2.27·17-s − 91.1·18-s − 1.96·19-s + 7.05·20-s − 16.5·21-s + 2.41·22-s − 2.78·23-s + 79.8·24-s − 7.36·25-s + 1.51·26-s + 219.·27-s + 5.97·28-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{18} \cdot 29^{18}\right)^{s/2} \, \Gamma_{\C}(s)^{18} \, L(s)\cr=\mathstrut & \,\Lambda(18-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{18} \cdot 29^{18}\right)^{s/2} \, \Gamma_{\C}(s+17/2)^{18} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(36\) |
| Conductor: | \(3^{18} \cdot 29^{18}\) |
| Sign: | $1$ |
| Analytic conductor: | \(4.41516\times 10^{39}\) |
| Root analytic conductor: | \(12.6254\) |
| Motivic weight: | \(17\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(18\) |
| Selberg data: | \((36,\ 3^{18} \cdot 29^{18} ,\ ( \ : [17/2]^{18} ),\ 1 )\) |
Particular Values
| \(L(9)\) | \(=\) | \(0\) |
| \(L(\frac12)\) | \(=\) | \(0\) |
| \(L(\frac{19}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 3 | \( ( 1 - p^{8} T )^{18} \) |
| 29 | \( ( 1 - p^{8} T )^{18} \) | |
| good | 2 | \( 1 + 579 T + 206655 p^{2} T^{2} + 49789597 p^{3} T^{3} + 45468602455 p^{3} T^{4} + 4837951348589 p^{5} T^{5} + 109876443335427 p^{10} T^{6} + 21412991843075511 p^{11} T^{7} + 13557668925260225159 p^{11} T^{8} + \)\(12\!\cdots\!47\)\( p^{13} T^{9} + \)\(88\!\cdots\!83\)\( p^{16} T^{10} + \)\(95\!\cdots\!95\)\( p^{21} T^{11} + \)\(79\!\cdots\!37\)\( p^{27} T^{12} + \)\(65\!\cdots\!87\)\( p^{29} T^{13} + \)\(20\!\cdots\!87\)\( p^{33} T^{14} + \)\(40\!\cdots\!73\)\( p^{37} T^{15} + \)\(30\!\cdots\!25\)\( p^{43} T^{16} + \)\(11\!\cdots\!03\)\( p^{46} T^{17} + \)\(12\!\cdots\!71\)\( p^{48} T^{18} + \)\(11\!\cdots\!03\)\( p^{63} T^{19} + \)\(30\!\cdots\!25\)\( p^{77} T^{20} + \)\(40\!\cdots\!73\)\( p^{88} T^{21} + \)\(20\!\cdots\!87\)\( p^{101} T^{22} + \)\(65\!\cdots\!87\)\( p^{114} T^{23} + \)\(79\!\cdots\!37\)\( p^{129} T^{24} + \)\(95\!\cdots\!95\)\( p^{140} T^{25} + \)\(88\!\cdots\!83\)\( p^{152} T^{26} + \)\(12\!\cdots\!47\)\( p^{166} T^{27} + 13557668925260225159 p^{181} T^{28} + 21412991843075511 p^{198} T^{29} + 109876443335427 p^{214} T^{30} + 4837951348589 p^{226} T^{31} + 45468602455 p^{241} T^{32} + 49789597 p^{258} T^{33} + 206655 p^{274} T^{34} + 579 p^{289} T^{35} + p^{306} T^{36} \) |
| 5 | \( 1 + 1644576 T + 8321323748589 T^{2} + 11835599920068100414 T^{3} + \)\(66\!\cdots\!24\)\( p T^{4} + \)\(42\!\cdots\!54\)\( T^{5} + \)\(87\!\cdots\!99\)\( T^{6} + \)\(10\!\cdots\!36\)\( T^{7} + \)\(16\!\cdots\!41\)\( T^{8} + \)\(36\!\cdots\!04\)\( p T^{9} + \)\(41\!\cdots\!22\)\( p^{4} T^{10} + \)\(85\!\cdots\!96\)\( p^{5} T^{11} + \)\(86\!\cdots\!87\)\( p^{8} T^{12} + \)\(16\!\cdots\!96\)\( p^{9} T^{13} + \)\(75\!\cdots\!81\)\( p^{11} T^{14} + \)\(21\!\cdots\!38\)\( p^{16} T^{15} + \)\(11\!\cdots\!79\)\( p^{15} T^{16} + \)\(77\!\cdots\!38\)\( p^{18} T^{17} + \)\(14\!\cdots\!06\)\( p^{19} T^{18} + \)\(77\!\cdots\!38\)\( p^{35} T^{19} + \)\(11\!\cdots\!79\)\( p^{49} T^{20} + \)\(21\!\cdots\!38\)\( p^{67} T^{21} + \)\(75\!\cdots\!81\)\( p^{79} T^{22} + \)\(16\!\cdots\!96\)\( p^{94} T^{23} + \)\(86\!\cdots\!87\)\( p^{110} T^{24} + \)\(85\!\cdots\!96\)\( p^{124} T^{25} + \)\(41\!\cdots\!22\)\( p^{140} T^{26} + \)\(36\!\cdots\!04\)\( p^{154} T^{27} + \)\(16\!\cdots\!41\)\( p^{170} T^{28} + \)\(10\!\cdots\!36\)\( p^{187} T^{29} + \)\(87\!\cdots\!99\)\( p^{204} T^{30} + \)\(42\!\cdots\!54\)\( p^{221} T^{31} + \)\(66\!\cdots\!24\)\( p^{239} T^{32} + 11835599920068100414 p^{255} T^{33} + 8321323748589 p^{272} T^{34} + 1644576 p^{289} T^{35} + p^{306} T^{36} \) | |
| 7 | \( 1 + 3471264 p T + 2290279506227422 T^{2} + \)\(44\!\cdots\!60\)\( T^{3} + \)\(23\!\cdots\!49\)\( T^{4} + \)\(54\!\cdots\!72\)\( p T^{5} + \)\(30\!\cdots\!24\)\( p^{2} T^{6} + \)\(60\!\cdots\!96\)\( p^{3} T^{7} + \)\(27\!\cdots\!14\)\( p^{4} T^{8} + \)\(48\!\cdots\!92\)\( p^{5} T^{9} + \)\(19\!\cdots\!56\)\( p^{6} T^{10} + \)\(30\!\cdots\!68\)\( p^{7} T^{11} + \)\(16\!\cdots\!18\)\( p^{9} T^{12} + \)\(34\!\cdots\!40\)\( p^{11} T^{13} + \)\(17\!\cdots\!48\)\( p^{13} T^{14} + \)\(34\!\cdots\!60\)\( p^{15} T^{15} + \)\(16\!\cdots\!37\)\( p^{17} T^{16} + \)\(33\!\cdots\!76\)\( p^{19} T^{17} + \)\(16\!\cdots\!02\)\( p^{21} T^{18} + \)\(33\!\cdots\!76\)\( p^{36} T^{19} + \)\(16\!\cdots\!37\)\( p^{51} T^{20} + \)\(34\!\cdots\!60\)\( p^{66} T^{21} + \)\(17\!\cdots\!48\)\( p^{81} T^{22} + \)\(34\!\cdots\!40\)\( p^{96} T^{23} + \)\(16\!\cdots\!18\)\( p^{111} T^{24} + \)\(30\!\cdots\!68\)\( p^{126} T^{25} + \)\(19\!\cdots\!56\)\( p^{142} T^{26} + \)\(48\!\cdots\!92\)\( p^{158} T^{27} + \)\(27\!\cdots\!14\)\( p^{174} T^{28} + \)\(60\!\cdots\!96\)\( p^{190} T^{29} + \)\(30\!\cdots\!24\)\( p^{206} T^{30} + \)\(54\!\cdots\!72\)\( p^{222} T^{31} + \)\(23\!\cdots\!49\)\( p^{238} T^{32} + \)\(44\!\cdots\!60\)\( p^{255} T^{33} + 2290279506227422 p^{272} T^{34} + 3471264 p^{290} T^{35} + p^{306} T^{36} \) | |
| 11 | \( 1 + 97480568 p T + 4399621964508619093 T^{2} + \)\(40\!\cdots\!28\)\( T^{3} + \)\(94\!\cdots\!19\)\( T^{4} + \)\(75\!\cdots\!08\)\( T^{5} + \)\(12\!\cdots\!58\)\( T^{6} + \)\(87\!\cdots\!16\)\( T^{7} + \)\(11\!\cdots\!37\)\( p T^{8} + \)\(55\!\cdots\!04\)\( p^{2} T^{9} + \)\(62\!\cdots\!23\)\( p^{3} T^{10} + \)\(20\!\cdots\!24\)\( p^{4} T^{11} + \)\(23\!\cdots\!89\)\( p^{5} T^{12} - \)\(28\!\cdots\!84\)\( p^{6} T^{13} + \)\(39\!\cdots\!82\)\( p^{7} T^{14} - \)\(58\!\cdots\!56\)\( p^{8} T^{15} - \)\(13\!\cdots\!02\)\( p^{9} T^{16} - \)\(42\!\cdots\!80\)\( p^{10} T^{17} - \)\(11\!\cdots\!04\)\( p^{11} T^{18} - \)\(42\!\cdots\!80\)\( p^{27} T^{19} - \)\(13\!\cdots\!02\)\( p^{43} T^{20} - \)\(58\!\cdots\!56\)\( p^{59} T^{21} + \)\(39\!\cdots\!82\)\( p^{75} T^{22} - \)\(28\!\cdots\!84\)\( p^{91} T^{23} + \)\(23\!\cdots\!89\)\( p^{107} T^{24} + \)\(20\!\cdots\!24\)\( p^{123} T^{25} + \)\(62\!\cdots\!23\)\( p^{139} T^{26} + \)\(55\!\cdots\!04\)\( p^{155} T^{27} + \)\(11\!\cdots\!37\)\( p^{171} T^{28} + \)\(87\!\cdots\!16\)\( p^{187} T^{29} + \)\(12\!\cdots\!58\)\( p^{204} T^{30} + \)\(75\!\cdots\!08\)\( p^{221} T^{31} + \)\(94\!\cdots\!19\)\( p^{238} T^{32} + \)\(40\!\cdots\!28\)\( p^{255} T^{33} + 4399621964508619093 p^{272} T^{34} + 97480568 p^{290} T^{35} + p^{306} T^{36} \) | |
| 13 | \( 1 + 2786209294 T + 87496129289828525951 T^{2} + \)\(21\!\cdots\!94\)\( T^{3} + \)\(37\!\cdots\!87\)\( T^{4} + \)\(65\!\cdots\!44\)\( p T^{5} + \)\(64\!\cdots\!98\)\( p^{2} T^{6} + \)\(10\!\cdots\!08\)\( p^{3} T^{7} + \)\(81\!\cdots\!39\)\( p^{4} T^{8} + \)\(11\!\cdots\!54\)\( p^{5} T^{9} + \)\(82\!\cdots\!35\)\( p^{6} T^{10} + \)\(11\!\cdots\!42\)\( p^{7} T^{11} + \)\(70\!\cdots\!99\)\( p^{8} T^{12} + \)\(87\!\cdots\!80\)\( p^{9} T^{13} + \)\(50\!\cdots\!74\)\( p^{10} T^{14} + \)\(58\!\cdots\!48\)\( p^{11} T^{15} + \)\(31\!\cdots\!50\)\( p^{12} T^{16} + \)\(26\!\cdots\!88\)\( p^{14} T^{17} + \)\(17\!\cdots\!48\)\( p^{14} T^{18} + \)\(26\!\cdots\!88\)\( p^{31} T^{19} + \)\(31\!\cdots\!50\)\( p^{46} T^{20} + \)\(58\!\cdots\!48\)\( p^{62} T^{21} + \)\(50\!\cdots\!74\)\( p^{78} T^{22} + \)\(87\!\cdots\!80\)\( p^{94} T^{23} + \)\(70\!\cdots\!99\)\( p^{110} T^{24} + \)\(11\!\cdots\!42\)\( p^{126} T^{25} + \)\(82\!\cdots\!35\)\( p^{142} T^{26} + \)\(11\!\cdots\!54\)\( p^{158} T^{27} + \)\(81\!\cdots\!39\)\( p^{174} T^{28} + \)\(10\!\cdots\!08\)\( p^{190} T^{29} + \)\(64\!\cdots\!98\)\( p^{206} T^{30} + \)\(65\!\cdots\!44\)\( p^{222} T^{31} + \)\(37\!\cdots\!87\)\( p^{238} T^{32} + \)\(21\!\cdots\!94\)\( p^{255} T^{33} + 87496129289828525951 p^{272} T^{34} + 2786209294 p^{289} T^{35} + p^{306} T^{36} \) | |
| 17 | \( 1 + 65414592362 T + \)\(86\!\cdots\!64\)\( T^{2} + \)\(49\!\cdots\!28\)\( T^{3} + \)\(37\!\cdots\!95\)\( T^{4} + \)\(19\!\cdots\!28\)\( T^{5} + \)\(10\!\cdots\!74\)\( T^{6} + \)\(49\!\cdots\!06\)\( T^{7} + \)\(23\!\cdots\!34\)\( T^{8} + \)\(57\!\cdots\!66\)\( p T^{9} + \)\(40\!\cdots\!10\)\( T^{10} + \)\(15\!\cdots\!84\)\( T^{11} + \)\(56\!\cdots\!48\)\( T^{12} + \)\(19\!\cdots\!46\)\( T^{13} + \)\(66\!\cdots\!20\)\( T^{14} + \)\(21\!\cdots\!94\)\( T^{15} + \)\(67\!\cdots\!09\)\( T^{16} + \)\(20\!\cdots\!74\)\( T^{17} + \)\(59\!\cdots\!98\)\( T^{18} + \)\(20\!\cdots\!74\)\( p^{17} T^{19} + \)\(67\!\cdots\!09\)\( p^{34} T^{20} + \)\(21\!\cdots\!94\)\( p^{51} T^{21} + \)\(66\!\cdots\!20\)\( p^{68} T^{22} + \)\(19\!\cdots\!46\)\( p^{85} T^{23} + \)\(56\!\cdots\!48\)\( p^{102} T^{24} + \)\(15\!\cdots\!84\)\( p^{119} T^{25} + \)\(40\!\cdots\!10\)\( p^{136} T^{26} + \)\(57\!\cdots\!66\)\( p^{154} T^{27} + \)\(23\!\cdots\!34\)\( p^{170} T^{28} + \)\(49\!\cdots\!06\)\( p^{187} T^{29} + \)\(10\!\cdots\!74\)\( p^{204} T^{30} + \)\(19\!\cdots\!28\)\( p^{221} T^{31} + \)\(37\!\cdots\!95\)\( p^{238} T^{32} + \)\(49\!\cdots\!28\)\( p^{255} T^{33} + \)\(86\!\cdots\!64\)\( p^{272} T^{34} + 65414592362 p^{289} T^{35} + p^{306} T^{36} \) | |
| 19 | \( 1 + 145836589920 T + \)\(54\!\cdots\!17\)\( T^{2} + \)\(61\!\cdots\!38\)\( T^{3} + \)\(13\!\cdots\!08\)\( T^{4} + \)\(11\!\cdots\!54\)\( T^{5} + \)\(19\!\cdots\!39\)\( T^{6} + \)\(13\!\cdots\!04\)\( T^{7} + \)\(10\!\cdots\!31\)\( p T^{8} + \)\(10\!\cdots\!84\)\( T^{9} + \)\(15\!\cdots\!38\)\( T^{10} + \)\(57\!\cdots\!48\)\( T^{11} + \)\(90\!\cdots\!07\)\( T^{12} + \)\(14\!\cdots\!76\)\( T^{13} + \)\(43\!\cdots\!57\)\( T^{14} - \)\(59\!\cdots\!50\)\( T^{15} + \)\(18\!\cdots\!19\)\( T^{16} - \)\(91\!\cdots\!58\)\( T^{17} + \)\(84\!\cdots\!30\)\( T^{18} - \)\(91\!\cdots\!58\)\( p^{17} T^{19} + \)\(18\!\cdots\!19\)\( p^{34} T^{20} - \)\(59\!\cdots\!50\)\( p^{51} T^{21} + \)\(43\!\cdots\!57\)\( p^{68} T^{22} + \)\(14\!\cdots\!76\)\( p^{85} T^{23} + \)\(90\!\cdots\!07\)\( p^{102} T^{24} + \)\(57\!\cdots\!48\)\( p^{119} T^{25} + \)\(15\!\cdots\!38\)\( p^{136} T^{26} + \)\(10\!\cdots\!84\)\( p^{153} T^{27} + \)\(10\!\cdots\!31\)\( p^{171} T^{28} + \)\(13\!\cdots\!04\)\( p^{187} T^{29} + \)\(19\!\cdots\!39\)\( p^{204} T^{30} + \)\(11\!\cdots\!54\)\( p^{221} T^{31} + \)\(13\!\cdots\!08\)\( p^{238} T^{32} + \)\(61\!\cdots\!38\)\( p^{255} T^{33} + \)\(54\!\cdots\!17\)\( p^{272} T^{34} + 145836589920 p^{289} T^{35} + p^{306} T^{36} \) | |
| 23 | \( 1 + 1047541452126 T + \)\(13\!\cdots\!94\)\( T^{2} + \)\(10\!\cdots\!02\)\( T^{3} + \)\(87\!\cdots\!73\)\( T^{4} + \)\(54\!\cdots\!60\)\( T^{5} + \)\(33\!\cdots\!96\)\( T^{6} + \)\(17\!\cdots\!56\)\( T^{7} + \)\(89\!\cdots\!48\)\( T^{8} + \)\(39\!\cdots\!80\)\( T^{9} + \)\(17\!\cdots\!60\)\( T^{10} + \)\(69\!\cdots\!16\)\( T^{11} + \)\(28\!\cdots\!84\)\( T^{12} + \)\(10\!\cdots\!48\)\( T^{13} + \)\(41\!\cdots\!96\)\( T^{14} + \)\(15\!\cdots\!84\)\( T^{15} + \)\(59\!\cdots\!94\)\( T^{16} + \)\(21\!\cdots\!16\)\( T^{17} + \)\(85\!\cdots\!12\)\( T^{18} + \)\(21\!\cdots\!16\)\( p^{17} T^{19} + \)\(59\!\cdots\!94\)\( p^{34} T^{20} + \)\(15\!\cdots\!84\)\( p^{51} T^{21} + \)\(41\!\cdots\!96\)\( p^{68} T^{22} + \)\(10\!\cdots\!48\)\( p^{85} T^{23} + \)\(28\!\cdots\!84\)\( p^{102} T^{24} + \)\(69\!\cdots\!16\)\( p^{119} T^{25} + \)\(17\!\cdots\!60\)\( p^{136} T^{26} + \)\(39\!\cdots\!80\)\( p^{153} T^{27} + \)\(89\!\cdots\!48\)\( p^{170} T^{28} + \)\(17\!\cdots\!56\)\( p^{187} T^{29} + \)\(33\!\cdots\!96\)\( p^{204} T^{30} + \)\(54\!\cdots\!60\)\( p^{221} T^{31} + \)\(87\!\cdots\!73\)\( p^{238} T^{32} + \)\(10\!\cdots\!02\)\( p^{255} T^{33} + \)\(13\!\cdots\!94\)\( p^{272} T^{34} + 1047541452126 p^{289} T^{35} + p^{306} T^{36} \) | |
| 31 | \( 1 - 1953543639594 T + \)\(21\!\cdots\!98\)\( T^{2} - \)\(20\!\cdots\!38\)\( T^{3} + \)\(22\!\cdots\!21\)\( T^{4} - \)\(47\!\cdots\!12\)\( T^{5} + \)\(15\!\cdots\!04\)\( T^{6} + \)\(50\!\cdots\!84\)\( T^{7} + \)\(84\!\cdots\!80\)\( T^{8} + \)\(53\!\cdots\!00\)\( T^{9} + \)\(36\!\cdots\!40\)\( T^{10} + \)\(29\!\cdots\!00\)\( T^{11} + \)\(13\!\cdots\!00\)\( T^{12} + \)\(11\!\cdots\!80\)\( T^{13} + \)\(40\!\cdots\!76\)\( T^{14} + \)\(35\!\cdots\!20\)\( T^{15} + \)\(11\!\cdots\!26\)\( T^{16} + \)\(92\!\cdots\!56\)\( T^{17} + \)\(26\!\cdots\!16\)\( T^{18} + \)\(92\!\cdots\!56\)\( p^{17} T^{19} + \)\(11\!\cdots\!26\)\( p^{34} T^{20} + \)\(35\!\cdots\!20\)\( p^{51} T^{21} + \)\(40\!\cdots\!76\)\( p^{68} T^{22} + \)\(11\!\cdots\!80\)\( p^{85} T^{23} + \)\(13\!\cdots\!00\)\( p^{102} T^{24} + \)\(29\!\cdots\!00\)\( p^{119} T^{25} + \)\(36\!\cdots\!40\)\( p^{136} T^{26} + \)\(53\!\cdots\!00\)\( p^{153} T^{27} + \)\(84\!\cdots\!80\)\( p^{170} T^{28} + \)\(50\!\cdots\!84\)\( p^{187} T^{29} + \)\(15\!\cdots\!04\)\( p^{204} T^{30} - \)\(47\!\cdots\!12\)\( p^{221} T^{31} + \)\(22\!\cdots\!21\)\( p^{238} T^{32} - \)\(20\!\cdots\!38\)\( p^{255} T^{33} + \)\(21\!\cdots\!98\)\( p^{272} T^{34} - 1953543639594 p^{289} T^{35} + p^{306} T^{36} \) | |
| 37 | \( 1 + 49075890498274 T + \)\(41\!\cdots\!67\)\( T^{2} + \)\(17\!\cdots\!68\)\( T^{3} + \)\(85\!\cdots\!16\)\( T^{4} + \)\(29\!\cdots\!16\)\( T^{5} + \)\(11\!\cdots\!49\)\( T^{6} + \)\(34\!\cdots\!34\)\( T^{7} + \)\(11\!\cdots\!29\)\( T^{8} + \)\(30\!\cdots\!88\)\( T^{9} + \)\(87\!\cdots\!66\)\( T^{10} + \)\(22\!\cdots\!40\)\( T^{11} + \)\(59\!\cdots\!39\)\( T^{12} + \)\(14\!\cdots\!38\)\( T^{13} + \)\(34\!\cdots\!23\)\( T^{14} + \)\(79\!\cdots\!68\)\( T^{15} + \)\(18\!\cdots\!35\)\( T^{16} + \)\(39\!\cdots\!14\)\( T^{17} + \)\(87\!\cdots\!90\)\( T^{18} + \)\(39\!\cdots\!14\)\( p^{17} T^{19} + \)\(18\!\cdots\!35\)\( p^{34} T^{20} + \)\(79\!\cdots\!68\)\( p^{51} T^{21} + \)\(34\!\cdots\!23\)\( p^{68} T^{22} + \)\(14\!\cdots\!38\)\( p^{85} T^{23} + \)\(59\!\cdots\!39\)\( p^{102} T^{24} + \)\(22\!\cdots\!40\)\( p^{119} T^{25} + \)\(87\!\cdots\!66\)\( p^{136} T^{26} + \)\(30\!\cdots\!88\)\( p^{153} T^{27} + \)\(11\!\cdots\!29\)\( p^{170} T^{28} + \)\(34\!\cdots\!34\)\( p^{187} T^{29} + \)\(11\!\cdots\!49\)\( p^{204} T^{30} + \)\(29\!\cdots\!16\)\( p^{221} T^{31} + \)\(85\!\cdots\!16\)\( p^{238} T^{32} + \)\(17\!\cdots\!68\)\( p^{255} T^{33} + \)\(41\!\cdots\!67\)\( p^{272} T^{34} + 49075890498274 p^{289} T^{35} + p^{306} T^{36} \) | |
| 41 | \( 1 + 10682443476904 T + \)\(24\!\cdots\!83\)\( T^{2} + \)\(46\!\cdots\!76\)\( T^{3} + \)\(28\!\cdots\!12\)\( T^{4} + \)\(75\!\cdots\!64\)\( T^{5} + \)\(22\!\cdots\!37\)\( T^{6} + \)\(69\!\cdots\!40\)\( T^{7} + \)\(13\!\cdots\!93\)\( T^{8} + \)\(43\!\cdots\!08\)\( T^{9} + \)\(61\!\cdots\!54\)\( T^{10} + \)\(19\!\cdots\!44\)\( T^{11} + \)\(24\!\cdots\!35\)\( T^{12} + \)\(74\!\cdots\!60\)\( T^{13} + \)\(81\!\cdots\!15\)\( T^{14} + \)\(23\!\cdots\!32\)\( T^{15} + \)\(24\!\cdots\!39\)\( T^{16} + \)\(66\!\cdots\!72\)\( T^{17} + \)\(66\!\cdots\!82\)\( T^{18} + \)\(66\!\cdots\!72\)\( p^{17} T^{19} + \)\(24\!\cdots\!39\)\( p^{34} T^{20} + \)\(23\!\cdots\!32\)\( p^{51} T^{21} + \)\(81\!\cdots\!15\)\( p^{68} T^{22} + \)\(74\!\cdots\!60\)\( p^{85} T^{23} + \)\(24\!\cdots\!35\)\( p^{102} T^{24} + \)\(19\!\cdots\!44\)\( p^{119} T^{25} + \)\(61\!\cdots\!54\)\( p^{136} T^{26} + \)\(43\!\cdots\!08\)\( p^{153} T^{27} + \)\(13\!\cdots\!93\)\( p^{170} T^{28} + \)\(69\!\cdots\!40\)\( p^{187} T^{29} + \)\(22\!\cdots\!37\)\( p^{204} T^{30} + \)\(75\!\cdots\!64\)\( p^{221} T^{31} + \)\(28\!\cdots\!12\)\( p^{238} T^{32} + \)\(46\!\cdots\!76\)\( p^{255} T^{33} + \)\(24\!\cdots\!83\)\( p^{272} T^{34} + 10682443476904 p^{289} T^{35} + p^{306} T^{36} \) | |
| 43 | \( 1 + 62941906896246 T + \)\(69\!\cdots\!81\)\( T^{2} + \)\(49\!\cdots\!90\)\( T^{3} + \)\(24\!\cdots\!56\)\( T^{4} + \)\(18\!\cdots\!02\)\( T^{5} + \)\(55\!\cdots\!43\)\( T^{6} + \)\(43\!\cdots\!82\)\( T^{7} + \)\(95\!\cdots\!77\)\( T^{8} + \)\(74\!\cdots\!28\)\( T^{9} + \)\(12\!\cdots\!66\)\( T^{10} + \)\(98\!\cdots\!20\)\( T^{11} + \)\(14\!\cdots\!67\)\( T^{12} + \)\(10\!\cdots\!10\)\( T^{13} + \)\(12\!\cdots\!65\)\( T^{14} + \)\(87\!\cdots\!42\)\( T^{15} + \)\(97\!\cdots\!83\)\( T^{16} + \)\(62\!\cdots\!76\)\( T^{17} + \)\(14\!\cdots\!30\)\( p T^{18} + \)\(62\!\cdots\!76\)\( p^{17} T^{19} + \)\(97\!\cdots\!83\)\( p^{34} T^{20} + \)\(87\!\cdots\!42\)\( p^{51} T^{21} + \)\(12\!\cdots\!65\)\( p^{68} T^{22} + \)\(10\!\cdots\!10\)\( p^{85} T^{23} + \)\(14\!\cdots\!67\)\( p^{102} T^{24} + \)\(98\!\cdots\!20\)\( p^{119} T^{25} + \)\(12\!\cdots\!66\)\( p^{136} T^{26} + \)\(74\!\cdots\!28\)\( p^{153} T^{27} + \)\(95\!\cdots\!77\)\( p^{170} T^{28} + \)\(43\!\cdots\!82\)\( p^{187} T^{29} + \)\(55\!\cdots\!43\)\( p^{204} T^{30} + \)\(18\!\cdots\!02\)\( p^{221} T^{31} + \)\(24\!\cdots\!56\)\( p^{238} T^{32} + \)\(49\!\cdots\!90\)\( p^{255} T^{33} + \)\(69\!\cdots\!81\)\( p^{272} T^{34} + 62941906896246 p^{289} T^{35} + p^{306} T^{36} \) | |
| 47 | \( 1 + 211975746927086 T + \)\(19\!\cdots\!68\)\( T^{2} + \)\(31\!\cdots\!40\)\( T^{3} + \)\(16\!\cdots\!03\)\( T^{4} + \)\(18\!\cdots\!52\)\( T^{5} + \)\(82\!\cdots\!98\)\( T^{6} + \)\(46\!\cdots\!22\)\( T^{7} + \)\(27\!\cdots\!78\)\( T^{8} - \)\(28\!\cdots\!74\)\( T^{9} + \)\(77\!\cdots\!70\)\( T^{10} - \)\(54\!\cdots\!48\)\( T^{11} + \)\(25\!\cdots\!96\)\( T^{12} - \)\(18\!\cdots\!46\)\( T^{13} + \)\(95\!\cdots\!56\)\( T^{14} - \)\(39\!\cdots\!46\)\( T^{15} + \)\(32\!\cdots\!45\)\( T^{16} - \)\(77\!\cdots\!10\)\( T^{17} + \)\(92\!\cdots\!62\)\( T^{18} - \)\(77\!\cdots\!10\)\( p^{17} T^{19} + \)\(32\!\cdots\!45\)\( p^{34} T^{20} - \)\(39\!\cdots\!46\)\( p^{51} T^{21} + \)\(95\!\cdots\!56\)\( p^{68} T^{22} - \)\(18\!\cdots\!46\)\( p^{85} T^{23} + \)\(25\!\cdots\!96\)\( p^{102} T^{24} - \)\(54\!\cdots\!48\)\( p^{119} T^{25} + \)\(77\!\cdots\!70\)\( p^{136} T^{26} - \)\(28\!\cdots\!74\)\( p^{153} T^{27} + \)\(27\!\cdots\!78\)\( p^{170} T^{28} + \)\(46\!\cdots\!22\)\( p^{187} T^{29} + \)\(82\!\cdots\!98\)\( p^{204} T^{30} + \)\(18\!\cdots\!52\)\( p^{221} T^{31} + \)\(16\!\cdots\!03\)\( p^{238} T^{32} + \)\(31\!\cdots\!40\)\( p^{255} T^{33} + \)\(19\!\cdots\!68\)\( p^{272} T^{34} + 211975746927086 p^{289} T^{35} + p^{306} T^{36} \) | |
| 53 | \( 1 + 1497687112995810 T + \)\(32\!\cdots\!90\)\( T^{2} + \)\(35\!\cdots\!50\)\( T^{3} + \)\(45\!\cdots\!21\)\( T^{4} + \)\(39\!\cdots\!00\)\( T^{5} + \)\(38\!\cdots\!36\)\( T^{6} + \)\(28\!\cdots\!24\)\( T^{7} + \)\(23\!\cdots\!20\)\( T^{8} + \)\(28\!\cdots\!36\)\( p T^{9} + \)\(10\!\cdots\!68\)\( T^{10} + \)\(60\!\cdots\!16\)\( T^{11} + \)\(37\!\cdots\!92\)\( T^{12} + \)\(19\!\cdots\!60\)\( T^{13} + \)\(11\!\cdots\!08\)\( T^{14} + \)\(54\!\cdots\!96\)\( T^{15} + \)\(28\!\cdots\!66\)\( T^{16} + \)\(12\!\cdots\!80\)\( T^{17} + \)\(62\!\cdots\!16\)\( T^{18} + \)\(12\!\cdots\!80\)\( p^{17} T^{19} + \)\(28\!\cdots\!66\)\( p^{34} T^{20} + \)\(54\!\cdots\!96\)\( p^{51} T^{21} + \)\(11\!\cdots\!08\)\( p^{68} T^{22} + \)\(19\!\cdots\!60\)\( p^{85} T^{23} + \)\(37\!\cdots\!92\)\( p^{102} T^{24} + \)\(60\!\cdots\!16\)\( p^{119} T^{25} + \)\(10\!\cdots\!68\)\( p^{136} T^{26} + \)\(28\!\cdots\!36\)\( p^{154} T^{27} + \)\(23\!\cdots\!20\)\( p^{170} T^{28} + \)\(28\!\cdots\!24\)\( p^{187} T^{29} + \)\(38\!\cdots\!36\)\( p^{204} T^{30} + \)\(39\!\cdots\!00\)\( p^{221} T^{31} + \)\(45\!\cdots\!21\)\( p^{238} T^{32} + \)\(35\!\cdots\!50\)\( p^{255} T^{33} + \)\(32\!\cdots\!90\)\( p^{272} T^{34} + 1497687112995810 p^{289} T^{35} + p^{306} T^{36} \) | |
| 59 | \( 1 + 3786142915442892 T + \)\(21\!\cdots\!79\)\( T^{2} + \)\(63\!\cdots\!16\)\( T^{3} + \)\(21\!\cdots\!96\)\( T^{4} + \)\(49\!\cdots\!28\)\( T^{5} + \)\(12\!\cdots\!61\)\( T^{6} + \)\(25\!\cdots\!40\)\( T^{7} + \)\(51\!\cdots\!41\)\( T^{8} + \)\(25\!\cdots\!12\)\( p^{2} T^{9} + \)\(15\!\cdots\!26\)\( T^{10} + \)\(24\!\cdots\!44\)\( T^{11} + \)\(38\!\cdots\!75\)\( T^{12} + \)\(53\!\cdots\!56\)\( T^{13} + \)\(74\!\cdots\!59\)\( T^{14} + \)\(95\!\cdots\!36\)\( T^{15} + \)\(20\!\cdots\!73\)\( p T^{16} + \)\(14\!\cdots\!28\)\( T^{17} + \)\(16\!\cdots\!30\)\( T^{18} + \)\(14\!\cdots\!28\)\( p^{17} T^{19} + \)\(20\!\cdots\!73\)\( p^{35} T^{20} + \)\(95\!\cdots\!36\)\( p^{51} T^{21} + \)\(74\!\cdots\!59\)\( p^{68} T^{22} + \)\(53\!\cdots\!56\)\( p^{85} T^{23} + \)\(38\!\cdots\!75\)\( p^{102} T^{24} + \)\(24\!\cdots\!44\)\( p^{119} T^{25} + \)\(15\!\cdots\!26\)\( p^{136} T^{26} + \)\(25\!\cdots\!12\)\( p^{155} T^{27} + \)\(51\!\cdots\!41\)\( p^{170} T^{28} + \)\(25\!\cdots\!40\)\( p^{187} T^{29} + \)\(12\!\cdots\!61\)\( p^{204} T^{30} + \)\(49\!\cdots\!28\)\( p^{221} T^{31} + \)\(21\!\cdots\!96\)\( p^{238} T^{32} + \)\(63\!\cdots\!16\)\( p^{255} T^{33} + \)\(21\!\cdots\!79\)\( p^{272} T^{34} + 3786142915442892 p^{289} T^{35} + p^{306} T^{36} \) | |
| 61 | \( 1 + 4799845010026850 T + \)\(34\!\cdots\!22\)\( T^{2} + \)\(12\!\cdots\!70\)\( T^{3} + \)\(53\!\cdots\!57\)\( T^{4} + \)\(16\!\cdots\!68\)\( T^{5} + \)\(51\!\cdots\!80\)\( T^{6} + \)\(13\!\cdots\!24\)\( T^{7} + \)\(34\!\cdots\!52\)\( T^{8} + \)\(12\!\cdots\!96\)\( p T^{9} + \)\(17\!\cdots\!32\)\( T^{10} + \)\(35\!\cdots\!24\)\( T^{11} + \)\(72\!\cdots\!48\)\( T^{12} + \)\(13\!\cdots\!60\)\( T^{13} + \)\(24\!\cdots\!32\)\( T^{14} + \)\(40\!\cdots\!24\)\( T^{15} + \)\(68\!\cdots\!66\)\( T^{16} + \)\(10\!\cdots\!76\)\( T^{17} + \)\(16\!\cdots\!16\)\( T^{18} + \)\(10\!\cdots\!76\)\( p^{17} T^{19} + \)\(68\!\cdots\!66\)\( p^{34} T^{20} + \)\(40\!\cdots\!24\)\( p^{51} T^{21} + \)\(24\!\cdots\!32\)\( p^{68} T^{22} + \)\(13\!\cdots\!60\)\( p^{85} T^{23} + \)\(72\!\cdots\!48\)\( p^{102} T^{24} + \)\(35\!\cdots\!24\)\( p^{119} T^{25} + \)\(17\!\cdots\!32\)\( p^{136} T^{26} + \)\(12\!\cdots\!96\)\( p^{154} T^{27} + \)\(34\!\cdots\!52\)\( p^{170} T^{28} + \)\(13\!\cdots\!24\)\( p^{187} T^{29} + \)\(51\!\cdots\!80\)\( p^{204} T^{30} + \)\(16\!\cdots\!68\)\( p^{221} T^{31} + \)\(53\!\cdots\!57\)\( p^{238} T^{32} + \)\(12\!\cdots\!70\)\( p^{255} T^{33} + \)\(34\!\cdots\!22\)\( p^{272} T^{34} + 4799845010026850 p^{289} T^{35} + p^{306} T^{36} \) | |
| 67 | \( 1 + 6091191569462588 T + \)\(11\!\cdots\!09\)\( T^{2} + \)\(58\!\cdots\!52\)\( T^{3} + \)\(65\!\cdots\!15\)\( T^{4} + \)\(29\!\cdots\!24\)\( T^{5} + \)\(24\!\cdots\!58\)\( T^{6} + \)\(97\!\cdots\!20\)\( T^{7} + \)\(68\!\cdots\!51\)\( T^{8} + \)\(24\!\cdots\!60\)\( T^{9} + \)\(15\!\cdots\!61\)\( T^{10} + \)\(50\!\cdots\!04\)\( T^{11} + \)\(28\!\cdots\!19\)\( T^{12} + \)\(86\!\cdots\!28\)\( T^{13} + \)\(44\!\cdots\!30\)\( T^{14} + \)\(12\!\cdots\!08\)\( T^{15} + \)\(60\!\cdots\!74\)\( T^{16} + \)\(15\!\cdots\!28\)\( T^{17} + \)\(71\!\cdots\!04\)\( T^{18} + \)\(15\!\cdots\!28\)\( p^{17} T^{19} + \)\(60\!\cdots\!74\)\( p^{34} T^{20} + \)\(12\!\cdots\!08\)\( p^{51} T^{21} + \)\(44\!\cdots\!30\)\( p^{68} T^{22} + \)\(86\!\cdots\!28\)\( p^{85} T^{23} + \)\(28\!\cdots\!19\)\( p^{102} T^{24} + \)\(50\!\cdots\!04\)\( p^{119} T^{25} + \)\(15\!\cdots\!61\)\( p^{136} T^{26} + \)\(24\!\cdots\!60\)\( p^{153} T^{27} + \)\(68\!\cdots\!51\)\( p^{170} T^{28} + \)\(97\!\cdots\!20\)\( p^{187} T^{29} + \)\(24\!\cdots\!58\)\( p^{204} T^{30} + \)\(29\!\cdots\!24\)\( p^{221} T^{31} + \)\(65\!\cdots\!15\)\( p^{238} T^{32} + \)\(58\!\cdots\!52\)\( p^{255} T^{33} + \)\(11\!\cdots\!09\)\( p^{272} T^{34} + 6091191569462588 p^{289} T^{35} + p^{306} T^{36} \) | |
| 71 | \( 1 + 1323586587759862 T + \)\(24\!\cdots\!02\)\( T^{2} + \)\(43\!\cdots\!46\)\( T^{3} + \)\(31\!\cdots\!65\)\( T^{4} + \)\(63\!\cdots\!24\)\( T^{5} + \)\(26\!\cdots\!52\)\( T^{6} + \)\(57\!\cdots\!76\)\( T^{7} + \)\(17\!\cdots\!04\)\( T^{8} + \)\(37\!\cdots\!68\)\( T^{9} + \)\(93\!\cdots\!12\)\( T^{10} + \)\(19\!\cdots\!80\)\( T^{11} + \)\(42\!\cdots\!52\)\( T^{12} + \)\(85\!\cdots\!48\)\( T^{13} + \)\(16\!\cdots\!32\)\( T^{14} + \)\(32\!\cdots\!40\)\( T^{15} + \)\(57\!\cdots\!22\)\( T^{16} + \)\(10\!\cdots\!48\)\( T^{17} + \)\(17\!\cdots\!92\)\( T^{18} + \)\(10\!\cdots\!48\)\( p^{17} T^{19} + \)\(57\!\cdots\!22\)\( p^{34} T^{20} + \)\(32\!\cdots\!40\)\( p^{51} T^{21} + \)\(16\!\cdots\!32\)\( p^{68} T^{22} + \)\(85\!\cdots\!48\)\( p^{85} T^{23} + \)\(42\!\cdots\!52\)\( p^{102} T^{24} + \)\(19\!\cdots\!80\)\( p^{119} T^{25} + \)\(93\!\cdots\!12\)\( p^{136} T^{26} + \)\(37\!\cdots\!68\)\( p^{153} T^{27} + \)\(17\!\cdots\!04\)\( p^{170} T^{28} + \)\(57\!\cdots\!76\)\( p^{187} T^{29} + \)\(26\!\cdots\!52\)\( p^{204} T^{30} + \)\(63\!\cdots\!24\)\( p^{221} T^{31} + \)\(31\!\cdots\!65\)\( p^{238} T^{32} + \)\(43\!\cdots\!46\)\( p^{255} T^{33} + \)\(24\!\cdots\!02\)\( p^{272} T^{34} + 1323586587759862 p^{289} T^{35} + p^{306} T^{36} \) | |
| 73 | \( 1 + 10862028982418798 T + \)\(31\!\cdots\!98\)\( T^{2} + \)\(39\!\cdots\!34\)\( T^{3} + \)\(59\!\cdots\!53\)\( T^{4} + \)\(69\!\cdots\!16\)\( T^{5} + \)\(83\!\cdots\!40\)\( T^{6} + \)\(85\!\cdots\!68\)\( T^{7} + \)\(89\!\cdots\!64\)\( T^{8} + \)\(83\!\cdots\!20\)\( T^{9} + \)\(77\!\cdots\!32\)\( T^{10} + \)\(67\!\cdots\!92\)\( T^{11} + \)\(57\!\cdots\!88\)\( T^{12} + \)\(46\!\cdots\!28\)\( T^{13} + \)\(37\!\cdots\!64\)\( T^{14} + \)\(28\!\cdots\!52\)\( T^{15} + \)\(21\!\cdots\!18\)\( T^{16} + \)\(15\!\cdots\!76\)\( T^{17} + \)\(10\!\cdots\!52\)\( T^{18} + \)\(15\!\cdots\!76\)\( p^{17} T^{19} + \)\(21\!\cdots\!18\)\( p^{34} T^{20} + \)\(28\!\cdots\!52\)\( p^{51} T^{21} + \)\(37\!\cdots\!64\)\( p^{68} T^{22} + \)\(46\!\cdots\!28\)\( p^{85} T^{23} + \)\(57\!\cdots\!88\)\( p^{102} T^{24} + \)\(67\!\cdots\!92\)\( p^{119} T^{25} + \)\(77\!\cdots\!32\)\( p^{136} T^{26} + \)\(83\!\cdots\!20\)\( p^{153} T^{27} + \)\(89\!\cdots\!64\)\( p^{170} T^{28} + \)\(85\!\cdots\!68\)\( p^{187} T^{29} + \)\(83\!\cdots\!40\)\( p^{204} T^{30} + \)\(69\!\cdots\!16\)\( p^{221} T^{31} + \)\(59\!\cdots\!53\)\( p^{238} T^{32} + \)\(39\!\cdots\!34\)\( p^{255} T^{33} + \)\(31\!\cdots\!98\)\( p^{272} T^{34} + 10862028982418798 p^{289} T^{35} + p^{306} T^{36} \) | |
| 79 | \( 1 + 22340202584513118 T + \)\(27\!\cdots\!46\)\( p T^{2} + \)\(43\!\cdots\!82\)\( T^{3} + \)\(22\!\cdots\!21\)\( T^{4} + \)\(41\!\cdots\!68\)\( T^{5} + \)\(15\!\cdots\!04\)\( T^{6} + \)\(26\!\cdots\!60\)\( T^{7} + \)\(80\!\cdots\!84\)\( T^{8} + \)\(12\!\cdots\!96\)\( T^{9} + \)\(32\!\cdots\!72\)\( T^{10} + \)\(47\!\cdots\!12\)\( T^{11} + \)\(10\!\cdots\!96\)\( T^{12} + \)\(14\!\cdots\!44\)\( T^{13} + \)\(29\!\cdots\!00\)\( T^{14} + \)\(37\!\cdots\!52\)\( T^{15} + \)\(67\!\cdots\!34\)\( T^{16} + \)\(79\!\cdots\!56\)\( T^{17} + \)\(13\!\cdots\!48\)\( T^{18} + \)\(79\!\cdots\!56\)\( p^{17} T^{19} + \)\(67\!\cdots\!34\)\( p^{34} T^{20} + \)\(37\!\cdots\!52\)\( p^{51} T^{21} + \)\(29\!\cdots\!00\)\( p^{68} T^{22} + \)\(14\!\cdots\!44\)\( p^{85} T^{23} + \)\(10\!\cdots\!96\)\( p^{102} T^{24} + \)\(47\!\cdots\!12\)\( p^{119} T^{25} + \)\(32\!\cdots\!72\)\( p^{136} T^{26} + \)\(12\!\cdots\!96\)\( p^{153} T^{27} + \)\(80\!\cdots\!84\)\( p^{170} T^{28} + \)\(26\!\cdots\!60\)\( p^{187} T^{29} + \)\(15\!\cdots\!04\)\( p^{204} T^{30} + \)\(41\!\cdots\!68\)\( p^{221} T^{31} + \)\(22\!\cdots\!21\)\( p^{238} T^{32} + \)\(43\!\cdots\!82\)\( p^{255} T^{33} + \)\(27\!\cdots\!46\)\( p^{273} T^{34} + 22340202584513118 p^{289} T^{35} + p^{306} T^{36} \) | |
| 83 | \( 1 + 21208963855685652 T + \)\(49\!\cdots\!66\)\( T^{2} + \)\(99\!\cdots\!76\)\( T^{3} + \)\(12\!\cdots\!53\)\( T^{4} + \)\(22\!\cdots\!32\)\( T^{5} + \)\(19\!\cdots\!64\)\( T^{6} + \)\(34\!\cdots\!40\)\( T^{7} + \)\(22\!\cdots\!68\)\( T^{8} + \)\(37\!\cdots\!20\)\( T^{9} + \)\(20\!\cdots\!96\)\( T^{10} + \)\(31\!\cdots\!00\)\( T^{11} + \)\(14\!\cdots\!36\)\( T^{12} + \)\(22\!\cdots\!36\)\( T^{13} + \)\(92\!\cdots\!76\)\( T^{14} + \)\(12\!\cdots\!40\)\( T^{15} + \)\(48\!\cdots\!26\)\( T^{16} + \)\(63\!\cdots\!28\)\( T^{17} + \)\(21\!\cdots\!36\)\( T^{18} + \)\(63\!\cdots\!28\)\( p^{17} T^{19} + \)\(48\!\cdots\!26\)\( p^{34} T^{20} + \)\(12\!\cdots\!40\)\( p^{51} T^{21} + \)\(92\!\cdots\!76\)\( p^{68} T^{22} + \)\(22\!\cdots\!36\)\( p^{85} T^{23} + \)\(14\!\cdots\!36\)\( p^{102} T^{24} + \)\(31\!\cdots\!00\)\( p^{119} T^{25} + \)\(20\!\cdots\!96\)\( p^{136} T^{26} + \)\(37\!\cdots\!20\)\( p^{153} T^{27} + \)\(22\!\cdots\!68\)\( p^{170} T^{28} + \)\(34\!\cdots\!40\)\( p^{187} T^{29} + \)\(19\!\cdots\!64\)\( p^{204} T^{30} + \)\(22\!\cdots\!32\)\( p^{221} T^{31} + \)\(12\!\cdots\!53\)\( p^{238} T^{32} + \)\(99\!\cdots\!76\)\( p^{255} T^{33} + \)\(49\!\cdots\!66\)\( p^{272} T^{34} + 21208963855685652 p^{289} T^{35} + p^{306} T^{36} \) | |
| 89 | \( 1 + 44576805912191294 T + \)\(13\!\cdots\!55\)\( T^{2} + \)\(56\!\cdots\!26\)\( T^{3} + \)\(93\!\cdots\!91\)\( T^{4} + \)\(36\!\cdots\!08\)\( T^{5} + \)\(43\!\cdots\!10\)\( T^{6} + \)\(15\!\cdots\!12\)\( T^{7} + \)\(14\!\cdots\!83\)\( T^{8} + \)\(51\!\cdots\!90\)\( T^{9} + \)\(41\!\cdots\!47\)\( T^{10} + \)\(13\!\cdots\!14\)\( T^{11} + \)\(95\!\cdots\!63\)\( T^{12} + \)\(29\!\cdots\!72\)\( T^{13} + \)\(18\!\cdots\!06\)\( T^{14} + \)\(55\!\cdots\!60\)\( T^{15} + \)\(31\!\cdots\!42\)\( T^{16} + \)\(88\!\cdots\!88\)\( T^{17} + \)\(47\!\cdots\!24\)\( T^{18} + \)\(88\!\cdots\!88\)\( p^{17} T^{19} + \)\(31\!\cdots\!42\)\( p^{34} T^{20} + \)\(55\!\cdots\!60\)\( p^{51} T^{21} + \)\(18\!\cdots\!06\)\( p^{68} T^{22} + \)\(29\!\cdots\!72\)\( p^{85} T^{23} + \)\(95\!\cdots\!63\)\( p^{102} T^{24} + \)\(13\!\cdots\!14\)\( p^{119} T^{25} + \)\(41\!\cdots\!47\)\( p^{136} T^{26} + \)\(51\!\cdots\!90\)\( p^{153} T^{27} + \)\(14\!\cdots\!83\)\( p^{170} T^{28} + \)\(15\!\cdots\!12\)\( p^{187} T^{29} + \)\(43\!\cdots\!10\)\( p^{204} T^{30} + \)\(36\!\cdots\!08\)\( p^{221} T^{31} + \)\(93\!\cdots\!91\)\( p^{238} T^{32} + \)\(56\!\cdots\!26\)\( p^{255} T^{33} + \)\(13\!\cdots\!55\)\( p^{272} T^{34} + 44576805912191294 p^{289} T^{35} + p^{306} T^{36} \) | |
| 97 | \( 1 - 80308547600659938 T + \)\(51\!\cdots\!54\)\( T^{2} - \)\(47\!\cdots\!78\)\( p T^{3} + \)\(13\!\cdots\!89\)\( T^{4} - \)\(13\!\cdots\!80\)\( T^{5} + \)\(24\!\cdots\!20\)\( T^{6} - \)\(24\!\cdots\!60\)\( T^{7} + \)\(34\!\cdots\!04\)\( T^{8} - \)\(34\!\cdots\!72\)\( T^{9} + \)\(40\!\cdots\!20\)\( T^{10} - \)\(38\!\cdots\!36\)\( T^{11} + \)\(39\!\cdots\!84\)\( T^{12} - \)\(36\!\cdots\!36\)\( T^{13} + \)\(33\!\cdots\!72\)\( T^{14} - \)\(28\!\cdots\!92\)\( T^{15} + \)\(24\!\cdots\!30\)\( T^{16} - \)\(19\!\cdots\!64\)\( T^{17} + \)\(15\!\cdots\!04\)\( T^{18} - \)\(19\!\cdots\!64\)\( p^{17} T^{19} + \)\(24\!\cdots\!30\)\( p^{34} T^{20} - \)\(28\!\cdots\!92\)\( p^{51} T^{21} + \)\(33\!\cdots\!72\)\( p^{68} T^{22} - \)\(36\!\cdots\!36\)\( p^{85} T^{23} + \)\(39\!\cdots\!84\)\( p^{102} T^{24} - \)\(38\!\cdots\!36\)\( p^{119} T^{25} + \)\(40\!\cdots\!20\)\( p^{136} T^{26} - \)\(34\!\cdots\!72\)\( p^{153} T^{27} + \)\(34\!\cdots\!04\)\( p^{170} T^{28} - \)\(24\!\cdots\!60\)\( p^{187} T^{29} + \)\(24\!\cdots\!20\)\( p^{204} T^{30} - \)\(13\!\cdots\!80\)\( p^{221} T^{31} + \)\(13\!\cdots\!89\)\( p^{238} T^{32} - \)\(47\!\cdots\!78\)\( p^{256} T^{33} + \)\(51\!\cdots\!54\)\( p^{272} T^{34} - 80308547600659938 p^{289} T^{35} + p^{306} T^{36} \) | |
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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
−2.34831308593674286644821946298, −2.27995413827332877951942485349, −2.18622124195217698648059467404, −2.09869108574561695964030822619, −2.07873124939382475519872691171, −2.07596400825383666134291484456, −2.05352880621118782382886899139, −2.01372985428575323929211856625, −1.80954596087920748882996353987, −1.75290547555435142989734879869, −1.68263403477821406452057364691, −1.60969510395211168289340926735, −1.50356136429955050294364882393, −1.41888859424017000601981984361, −1.34074117341611794092733879367, −1.26075151087329240871316331795, −1.24708409609609986697147419246, −1.24550723511135040002350564231, −1.17535219878217845328844405878, −1.13549369955634105264092113439, −1.09775595706186920670845389930, −1.00978741840839201820209775530, −0.991738784619389251234571229479, −0.863461773280104095696238908371, −0.77599789347901819212838930197, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0.77599789347901819212838930197, 0.863461773280104095696238908371, 0.991738784619389251234571229479, 1.00978741840839201820209775530, 1.09775595706186920670845389930, 1.13549369955634105264092113439, 1.17535219878217845328844405878, 1.24550723511135040002350564231, 1.24708409609609986697147419246, 1.26075151087329240871316331795, 1.34074117341611794092733879367, 1.41888859424017000601981984361, 1.50356136429955050294364882393, 1.60969510395211168289340926735, 1.68263403477821406452057364691, 1.75290547555435142989734879869, 1.80954596087920748882996353987, 2.01372985428575323929211856625, 2.05352880621118782382886899139, 2.07596400825383666134291484456, 2.07873124939382475519872691171, 2.09869108574561695964030822619, 2.18622124195217698648059467404, 2.27995413827332877951942485349, 2.34831308593674286644821946298
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.