Dirichlet series
| L(s) = 1 | − 1.02e3·2-s − 1.04e7·4-s − 3.22e7·5-s + 1.89e8·7-s + 1.15e10·8-s + 3.29e10·10-s − 1.46e11·11-s + 1.77e11·13-s − 1.93e11·14-s + 4.86e13·16-s − 9.30e12·17-s − 4.88e12·19-s + 3.38e14·20-s + 1.49e14·22-s − 3.56e14·23-s − 3.55e15·25-s − 1.81e14·26-s − 1.98e15·28-s + 2.77e15·29-s − 3.06e15·31-s − 6.10e16·32-s + 9.52e15·34-s − 6.11e15·35-s + 1.50e16·37-s + 4.99e15·38-s − 3.73e17·40-s − 4.98e15·41-s + ⋯ |
| L(s) = 1 | − 0.706·2-s − 5.00·4-s − 1.47·5-s + 0.253·7-s + 3.81·8-s + 1.04·10-s − 1.69·11-s + 0.357·13-s − 0.179·14-s + 11.0·16-s − 1.11·17-s − 0.182·19-s + 7.38·20-s + 1.19·22-s − 1.79·23-s − 7.46·25-s − 0.252·26-s − 1.26·28-s + 1.22·29-s − 0.671·31-s − 9.58·32-s + 0.791·34-s − 0.374·35-s + 0.514·37-s + 0.129·38-s − 5.63·40-s − 0.0580·41-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{80}\right)^{s/2} \, \Gamma_{\C}(s)^{20} \, L(s)\cr=\mathstrut & \,\Lambda(22-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{80}\right)^{s/2} \, \Gamma_{\C}(s+21/2)^{20} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(40\) |
| Conductor: | \(3^{80}\) |
| Sign: | $1$ |
| Analytic conductor: | \(1.24920\times 10^{47}\) |
| Root analytic conductor: | \(15.0458\) |
| Motivic weight: | \(21\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(20\) |
| Selberg data: | \((40,\ 3^{80} ,\ ( \ : [21/2]^{20} ),\ 1 )\) |
Particular Values
| \(L(11)\) | \(=\) | \(0\) |
| \(L(\frac12)\) | \(=\) | \(0\) |
| \(L(\frac{23}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 3 | \( 1 \) |
| good | 2 | \( 1 + 1023 T + 11263 p^{10} T^{2} + 1367828529 p^{3} T^{3} + 8959905979343 p^{3} T^{4} + 2056513344235929 p^{5} T^{5} + 1256910978568996677 p^{8} T^{6} + \)\(28\!\cdots\!13\)\( p^{10} T^{7} + \)\(57\!\cdots\!15\)\( p^{11} T^{8} + \)\(12\!\cdots\!81\)\( p^{13} T^{9} + \)\(13\!\cdots\!23\)\( p^{18} T^{10} + \)\(80\!\cdots\!81\)\( p^{22} T^{11} + \)\(30\!\cdots\!25\)\( p^{25} T^{12} + \)\(90\!\cdots\!49\)\( p^{30} T^{13} + \)\(19\!\cdots\!29\)\( p^{37} T^{14} + \)\(23\!\cdots\!89\)\( p^{40} T^{15} + \)\(18\!\cdots\!41\)\( p^{45} T^{16} + \)\(13\!\cdots\!05\)\( p^{52} T^{17} + \)\(26\!\cdots\!25\)\( p^{59} T^{18} + \)\(18\!\cdots\!95\)\( p^{66} T^{19} + \)\(55\!\cdots\!71\)\( p^{69} T^{20} + \)\(18\!\cdots\!95\)\( p^{87} T^{21} + \)\(26\!\cdots\!25\)\( p^{101} T^{22} + \)\(13\!\cdots\!05\)\( p^{115} T^{23} + \)\(18\!\cdots\!41\)\( p^{129} T^{24} + \)\(23\!\cdots\!89\)\( p^{145} T^{25} + \)\(19\!\cdots\!29\)\( p^{163} T^{26} + \)\(90\!\cdots\!49\)\( p^{177} T^{27} + \)\(30\!\cdots\!25\)\( p^{193} T^{28} + \)\(80\!\cdots\!81\)\( p^{211} T^{29} + \)\(13\!\cdots\!23\)\( p^{228} T^{30} + \)\(12\!\cdots\!81\)\( p^{244} T^{31} + \)\(57\!\cdots\!15\)\( p^{263} T^{32} + \)\(28\!\cdots\!13\)\( p^{283} T^{33} + 1256910978568996677 p^{302} T^{34} + 2056513344235929 p^{320} T^{35} + 8959905979343 p^{339} T^{36} + 1367828529 p^{360} T^{37} + 11263 p^{388} T^{38} + 1023 p^{399} T^{39} + p^{420} T^{40} \) |
| 5 | \( 1 + 32234853 T + 919598705491178 p T^{2} + \)\(13\!\cdots\!77\)\( T^{3} + \)\(10\!\cdots\!09\)\( T^{4} + \)\(54\!\cdots\!24\)\( p T^{5} + \)\(66\!\cdots\!26\)\( p^{2} T^{6} + \)\(30\!\cdots\!42\)\( p^{3} T^{7} + \)\(31\!\cdots\!72\)\( p^{4} T^{8} + \)\(26\!\cdots\!02\)\( p^{6} T^{9} + \)\(47\!\cdots\!46\)\( p^{8} T^{10} + \)\(37\!\cdots\!44\)\( p^{10} T^{11} + \)\(62\!\cdots\!31\)\( p^{12} T^{12} + \)\(90\!\cdots\!51\)\( p^{15} T^{13} + \)\(71\!\cdots\!82\)\( p^{16} T^{14} + \)\(48\!\cdots\!43\)\( p^{18} T^{15} + \)\(14\!\cdots\!23\)\( p^{21} T^{16} + \)\(46\!\cdots\!76\)\( p^{22} T^{17} + \)\(64\!\cdots\!88\)\( p^{24} T^{18} + \)\(39\!\cdots\!36\)\( p^{26} T^{19} + \)\(51\!\cdots\!48\)\( p^{28} T^{20} + \)\(39\!\cdots\!36\)\( p^{47} T^{21} + \)\(64\!\cdots\!88\)\( p^{66} T^{22} + \)\(46\!\cdots\!76\)\( p^{85} T^{23} + \)\(14\!\cdots\!23\)\( p^{105} T^{24} + \)\(48\!\cdots\!43\)\( p^{123} T^{25} + \)\(71\!\cdots\!82\)\( p^{142} T^{26} + \)\(90\!\cdots\!51\)\( p^{162} T^{27} + \)\(62\!\cdots\!31\)\( p^{180} T^{28} + \)\(37\!\cdots\!44\)\( p^{199} T^{29} + \)\(47\!\cdots\!46\)\( p^{218} T^{30} + \)\(26\!\cdots\!02\)\( p^{237} T^{31} + \)\(31\!\cdots\!72\)\( p^{256} T^{32} + \)\(30\!\cdots\!42\)\( p^{276} T^{33} + \)\(66\!\cdots\!26\)\( p^{296} T^{34} + \)\(54\!\cdots\!24\)\( p^{316} T^{35} + \)\(10\!\cdots\!09\)\( p^{336} T^{36} + \)\(13\!\cdots\!77\)\( p^{357} T^{37} + 919598705491178 p^{379} T^{38} + 32234853 p^{399} T^{39} + p^{420} T^{40} \) | |
| 7 | \( 1 - 27089137 p T + 4906569197481033720 T^{2} - \)\(10\!\cdots\!27\)\( p T^{3} + \)\(23\!\cdots\!29\)\( p^{2} T^{4} - \)\(37\!\cdots\!36\)\( p^{3} T^{5} + \)\(74\!\cdots\!94\)\( p^{4} T^{6} - \)\(14\!\cdots\!06\)\( p^{6} T^{7} + \)\(24\!\cdots\!12\)\( p^{7} T^{8} - \)\(30\!\cdots\!22\)\( p^{8} T^{9} + \)\(31\!\cdots\!82\)\( p^{8} T^{10} - \)\(37\!\cdots\!28\)\( p^{9} T^{11} + \)\(46\!\cdots\!39\)\( p^{10} T^{12} - \)\(53\!\cdots\!95\)\( p^{11} T^{13} + \)\(59\!\cdots\!04\)\( p^{12} T^{14} - \)\(59\!\cdots\!37\)\( p^{13} T^{15} + \)\(68\!\cdots\!23\)\( p^{14} T^{16} - \)\(49\!\cdots\!20\)\( p^{15} T^{17} + \)\(74\!\cdots\!88\)\( p^{16} T^{18} - \)\(37\!\cdots\!00\)\( p^{17} T^{19} + \)\(82\!\cdots\!72\)\( p^{18} T^{20} - \)\(37\!\cdots\!00\)\( p^{38} T^{21} + \)\(74\!\cdots\!88\)\( p^{58} T^{22} - \)\(49\!\cdots\!20\)\( p^{78} T^{23} + \)\(68\!\cdots\!23\)\( p^{98} T^{24} - \)\(59\!\cdots\!37\)\( p^{118} T^{25} + \)\(59\!\cdots\!04\)\( p^{138} T^{26} - \)\(53\!\cdots\!95\)\( p^{158} T^{27} + \)\(46\!\cdots\!39\)\( p^{178} T^{28} - \)\(37\!\cdots\!28\)\( p^{198} T^{29} + \)\(31\!\cdots\!82\)\( p^{218} T^{30} - \)\(30\!\cdots\!22\)\( p^{239} T^{31} + \)\(24\!\cdots\!12\)\( p^{259} T^{32} - \)\(14\!\cdots\!06\)\( p^{279} T^{33} + \)\(74\!\cdots\!94\)\( p^{298} T^{34} - \)\(37\!\cdots\!36\)\( p^{318} T^{35} + \)\(23\!\cdots\!29\)\( p^{338} T^{36} - \)\(10\!\cdots\!27\)\( p^{358} T^{37} + 4906569197481033720 p^{378} T^{38} - 27089137 p^{400} T^{39} + p^{420} T^{40} \) | |
| 11 | \( 1 + 146068576386 T + \)\(80\!\cdots\!80\)\( T^{2} + \)\(10\!\cdots\!48\)\( T^{3} + \)\(33\!\cdots\!85\)\( T^{4} + \)\(35\!\cdots\!12\)\( p T^{5} + \)\(90\!\cdots\!32\)\( T^{6} + \)\(98\!\cdots\!94\)\( T^{7} + \)\(18\!\cdots\!35\)\( T^{8} + \)\(18\!\cdots\!12\)\( T^{9} + \)\(30\!\cdots\!60\)\( T^{10} + \)\(26\!\cdots\!48\)\( p T^{11} + \)\(34\!\cdots\!82\)\( p^{2} T^{12} + \)\(27\!\cdots\!00\)\( p^{3} T^{13} + \)\(33\!\cdots\!28\)\( p^{4} T^{14} + \)\(22\!\cdots\!92\)\( p^{6} T^{15} + \)\(27\!\cdots\!33\)\( p^{6} T^{16} + \)\(19\!\cdots\!02\)\( p^{7} T^{17} + \)\(20\!\cdots\!40\)\( p^{8} T^{18} + \)\(13\!\cdots\!04\)\( p^{9} T^{19} + \)\(11\!\cdots\!09\)\( p^{11} T^{20} + \)\(13\!\cdots\!04\)\( p^{30} T^{21} + \)\(20\!\cdots\!40\)\( p^{50} T^{22} + \)\(19\!\cdots\!02\)\( p^{70} T^{23} + \)\(27\!\cdots\!33\)\( p^{90} T^{24} + \)\(22\!\cdots\!92\)\( p^{111} T^{25} + \)\(33\!\cdots\!28\)\( p^{130} T^{26} + \)\(27\!\cdots\!00\)\( p^{150} T^{27} + \)\(34\!\cdots\!82\)\( p^{170} T^{28} + \)\(26\!\cdots\!48\)\( p^{190} T^{29} + \)\(30\!\cdots\!60\)\( p^{210} T^{30} + \)\(18\!\cdots\!12\)\( p^{231} T^{31} + \)\(18\!\cdots\!35\)\( p^{252} T^{32} + \)\(98\!\cdots\!94\)\( p^{273} T^{33} + \)\(90\!\cdots\!32\)\( p^{294} T^{34} + \)\(35\!\cdots\!12\)\( p^{316} T^{35} + \)\(33\!\cdots\!85\)\( p^{336} T^{36} + \)\(10\!\cdots\!48\)\( p^{357} T^{37} + \)\(80\!\cdots\!80\)\( p^{378} T^{38} + 146068576386 p^{399} T^{39} + p^{420} T^{40} \) | |
| 13 | \( 1 - 13658921329 p T + \)\(23\!\cdots\!28\)\( T^{2} - \)\(61\!\cdots\!05\)\( T^{3} + \)\(29\!\cdots\!29\)\( T^{4} - \)\(92\!\cdots\!64\)\( T^{5} + \)\(25\!\cdots\!82\)\( T^{6} - \)\(88\!\cdots\!38\)\( T^{7} + \)\(16\!\cdots\!52\)\( T^{8} - \)\(47\!\cdots\!42\)\( p T^{9} + \)\(55\!\cdots\!54\)\( p^{2} T^{10} - \)\(12\!\cdots\!40\)\( p^{4} T^{11} + \)\(15\!\cdots\!35\)\( p^{4} T^{12} - \)\(42\!\cdots\!15\)\( p^{5} T^{13} + \)\(35\!\cdots\!20\)\( p^{6} T^{14} - \)\(96\!\cdots\!03\)\( p^{7} T^{15} + \)\(71\!\cdots\!75\)\( p^{8} T^{16} - \)\(18\!\cdots\!32\)\( p^{9} T^{17} + \)\(12\!\cdots\!56\)\( p^{10} T^{18} - \)\(31\!\cdots\!40\)\( p^{11} T^{19} + \)\(19\!\cdots\!16\)\( p^{12} T^{20} - \)\(31\!\cdots\!40\)\( p^{32} T^{21} + \)\(12\!\cdots\!56\)\( p^{52} T^{22} - \)\(18\!\cdots\!32\)\( p^{72} T^{23} + \)\(71\!\cdots\!75\)\( p^{92} T^{24} - \)\(96\!\cdots\!03\)\( p^{112} T^{25} + \)\(35\!\cdots\!20\)\( p^{132} T^{26} - \)\(42\!\cdots\!15\)\( p^{152} T^{27} + \)\(15\!\cdots\!35\)\( p^{172} T^{28} - \)\(12\!\cdots\!40\)\( p^{193} T^{29} + \)\(55\!\cdots\!54\)\( p^{212} T^{30} - \)\(47\!\cdots\!42\)\( p^{232} T^{31} + \)\(16\!\cdots\!52\)\( p^{252} T^{32} - \)\(88\!\cdots\!38\)\( p^{273} T^{33} + \)\(25\!\cdots\!82\)\( p^{294} T^{34} - \)\(92\!\cdots\!64\)\( p^{315} T^{35} + \)\(29\!\cdots\!29\)\( p^{336} T^{36} - \)\(61\!\cdots\!05\)\( p^{357} T^{37} + \)\(23\!\cdots\!28\)\( p^{378} T^{38} - 13658921329 p^{400} T^{39} + p^{420} T^{40} \) | |
| 17 | \( 1 + 9307801874799 T + \)\(75\!\cdots\!63\)\( T^{2} + \)\(34\!\cdots\!72\)\( p T^{3} + \)\(26\!\cdots\!37\)\( T^{4} + \)\(99\!\cdots\!81\)\( p T^{5} + \)\(21\!\cdots\!32\)\( p^{2} T^{6} + \)\(62\!\cdots\!71\)\( p^{3} T^{7} + \)\(12\!\cdots\!97\)\( p^{4} T^{8} + \)\(26\!\cdots\!42\)\( p^{5} T^{9} + \)\(53\!\cdots\!23\)\( p^{6} T^{10} + \)\(85\!\cdots\!97\)\( p^{7} T^{11} + \)\(20\!\cdots\!88\)\( p^{8} T^{12} + \)\(21\!\cdots\!31\)\( p^{9} T^{13} + \)\(67\!\cdots\!89\)\( p^{10} T^{14} + \)\(48\!\cdots\!86\)\( p^{11} T^{15} + \)\(20\!\cdots\!74\)\( p^{12} T^{16} + \)\(11\!\cdots\!78\)\( p^{13} T^{17} + \)\(56\!\cdots\!21\)\( p^{14} T^{18} + \)\(28\!\cdots\!15\)\( p^{15} T^{19} + \)\(14\!\cdots\!46\)\( p^{16} T^{20} + \)\(28\!\cdots\!15\)\( p^{36} T^{21} + \)\(56\!\cdots\!21\)\( p^{56} T^{22} + \)\(11\!\cdots\!78\)\( p^{76} T^{23} + \)\(20\!\cdots\!74\)\( p^{96} T^{24} + \)\(48\!\cdots\!86\)\( p^{116} T^{25} + \)\(67\!\cdots\!89\)\( p^{136} T^{26} + \)\(21\!\cdots\!31\)\( p^{156} T^{27} + \)\(20\!\cdots\!88\)\( p^{176} T^{28} + \)\(85\!\cdots\!97\)\( p^{196} T^{29} + \)\(53\!\cdots\!23\)\( p^{216} T^{30} + \)\(26\!\cdots\!42\)\( p^{236} T^{31} + \)\(12\!\cdots\!97\)\( p^{256} T^{32} + \)\(62\!\cdots\!71\)\( p^{276} T^{33} + \)\(21\!\cdots\!32\)\( p^{296} T^{34} + \)\(99\!\cdots\!81\)\( p^{316} T^{35} + \)\(26\!\cdots\!37\)\( p^{336} T^{36} + \)\(34\!\cdots\!72\)\( p^{358} T^{37} + \)\(75\!\cdots\!63\)\( p^{378} T^{38} + 9307801874799 p^{399} T^{39} + p^{420} T^{40} \) | |
| 19 | \( 1 + 4884366861977 T + \)\(70\!\cdots\!37\)\( T^{2} + \)\(14\!\cdots\!62\)\( p T^{3} + \)\(70\!\cdots\!79\)\( p^{2} T^{4} + \)\(10\!\cdots\!99\)\( p^{3} T^{5} + \)\(47\!\cdots\!48\)\( p^{4} T^{6} + \)\(40\!\cdots\!83\)\( p^{5} T^{7} + \)\(23\!\cdots\!55\)\( p^{6} T^{8} + \)\(58\!\cdots\!52\)\( p^{7} T^{9} + \)\(97\!\cdots\!09\)\( p^{8} T^{10} - \)\(36\!\cdots\!25\)\( p^{9} T^{11} + \)\(33\!\cdots\!76\)\( p^{10} T^{12} - \)\(32\!\cdots\!29\)\( p^{11} T^{13} + \)\(51\!\cdots\!77\)\( p^{13} T^{14} - \)\(14\!\cdots\!24\)\( p^{13} T^{15} + \)\(25\!\cdots\!80\)\( p^{14} T^{16} - \)\(46\!\cdots\!80\)\( p^{15} T^{17} + \)\(58\!\cdots\!47\)\( p^{16} T^{18} - \)\(11\!\cdots\!99\)\( p^{17} T^{19} + \)\(12\!\cdots\!50\)\( p^{18} T^{20} - \)\(11\!\cdots\!99\)\( p^{38} T^{21} + \)\(58\!\cdots\!47\)\( p^{58} T^{22} - \)\(46\!\cdots\!80\)\( p^{78} T^{23} + \)\(25\!\cdots\!80\)\( p^{98} T^{24} - \)\(14\!\cdots\!24\)\( p^{118} T^{25} + \)\(51\!\cdots\!77\)\( p^{139} T^{26} - \)\(32\!\cdots\!29\)\( p^{158} T^{27} + \)\(33\!\cdots\!76\)\( p^{178} T^{28} - \)\(36\!\cdots\!25\)\( p^{198} T^{29} + \)\(97\!\cdots\!09\)\( p^{218} T^{30} + \)\(58\!\cdots\!52\)\( p^{238} T^{31} + \)\(23\!\cdots\!55\)\( p^{258} T^{32} + \)\(40\!\cdots\!83\)\( p^{278} T^{33} + \)\(47\!\cdots\!48\)\( p^{298} T^{34} + \)\(10\!\cdots\!99\)\( p^{318} T^{35} + \)\(70\!\cdots\!79\)\( p^{338} T^{36} + \)\(14\!\cdots\!62\)\( p^{358} T^{37} + \)\(70\!\cdots\!37\)\( p^{378} T^{38} + 4884366861977 p^{399} T^{39} + p^{420} T^{40} \) | |
| 23 | \( 1 + 15498282386265 p T + \)\(45\!\cdots\!32\)\( T^{2} + \)\(13\!\cdots\!89\)\( T^{3} + \)\(98\!\cdots\!93\)\( T^{4} + \)\(25\!\cdots\!84\)\( T^{5} + \)\(13\!\cdots\!66\)\( T^{6} + \)\(29\!\cdots\!58\)\( T^{7} + \)\(13\!\cdots\!24\)\( T^{8} + \)\(25\!\cdots\!82\)\( T^{9} + \)\(98\!\cdots\!70\)\( T^{10} + \)\(16\!\cdots\!08\)\( T^{11} + \)\(60\!\cdots\!87\)\( T^{12} + \)\(85\!\cdots\!33\)\( T^{13} + \)\(31\!\cdots\!24\)\( T^{14} + \)\(37\!\cdots\!07\)\( T^{15} + \)\(14\!\cdots\!79\)\( T^{16} + \)\(15\!\cdots\!48\)\( T^{17} + \)\(62\!\cdots\!04\)\( T^{18} + \)\(58\!\cdots\!80\)\( T^{19} + \)\(25\!\cdots\!72\)\( T^{20} + \)\(58\!\cdots\!80\)\( p^{21} T^{21} + \)\(62\!\cdots\!04\)\( p^{42} T^{22} + \)\(15\!\cdots\!48\)\( p^{63} T^{23} + \)\(14\!\cdots\!79\)\( p^{84} T^{24} + \)\(37\!\cdots\!07\)\( p^{105} T^{25} + \)\(31\!\cdots\!24\)\( p^{126} T^{26} + \)\(85\!\cdots\!33\)\( p^{147} T^{27} + \)\(60\!\cdots\!87\)\( p^{168} T^{28} + \)\(16\!\cdots\!08\)\( p^{189} T^{29} + \)\(98\!\cdots\!70\)\( p^{210} T^{30} + \)\(25\!\cdots\!82\)\( p^{231} T^{31} + \)\(13\!\cdots\!24\)\( p^{252} T^{32} + \)\(29\!\cdots\!58\)\( p^{273} T^{33} + \)\(13\!\cdots\!66\)\( p^{294} T^{34} + \)\(25\!\cdots\!84\)\( p^{315} T^{35} + \)\(98\!\cdots\!93\)\( p^{336} T^{36} + \)\(13\!\cdots\!89\)\( p^{357} T^{37} + \)\(45\!\cdots\!32\)\( p^{378} T^{38} + 15498282386265 p^{400} T^{39} + p^{420} T^{40} \) | |
| 29 | \( 1 - 2772828270755157 T + \)\(51\!\cdots\!80\)\( T^{2} - \)\(15\!\cdots\!61\)\( T^{3} + \)\(13\!\cdots\!09\)\( T^{4} - \)\(42\!\cdots\!96\)\( T^{5} + \)\(26\!\cdots\!54\)\( T^{6} - \)\(79\!\cdots\!58\)\( T^{7} + \)\(37\!\cdots\!64\)\( T^{8} - \)\(11\!\cdots\!14\)\( T^{9} + \)\(44\!\cdots\!06\)\( T^{10} - \)\(12\!\cdots\!36\)\( T^{11} + \)\(43\!\cdots\!19\)\( T^{12} - \)\(11\!\cdots\!43\)\( T^{13} + \)\(36\!\cdots\!72\)\( T^{14} - \)\(94\!\cdots\!15\)\( T^{15} + \)\(26\!\cdots\!35\)\( T^{16} - \)\(64\!\cdots\!28\)\( T^{17} + \)\(16\!\cdots\!68\)\( T^{18} - \)\(37\!\cdots\!92\)\( T^{19} + \)\(90\!\cdots\!40\)\( T^{20} - \)\(37\!\cdots\!92\)\( p^{21} T^{21} + \)\(16\!\cdots\!68\)\( p^{42} T^{22} - \)\(64\!\cdots\!28\)\( p^{63} T^{23} + \)\(26\!\cdots\!35\)\( p^{84} T^{24} - \)\(94\!\cdots\!15\)\( p^{105} T^{25} + \)\(36\!\cdots\!72\)\( p^{126} T^{26} - \)\(11\!\cdots\!43\)\( p^{147} T^{27} + \)\(43\!\cdots\!19\)\( p^{168} T^{28} - \)\(12\!\cdots\!36\)\( p^{189} T^{29} + \)\(44\!\cdots\!06\)\( p^{210} T^{30} - \)\(11\!\cdots\!14\)\( p^{231} T^{31} + \)\(37\!\cdots\!64\)\( p^{252} T^{32} - \)\(79\!\cdots\!58\)\( p^{273} T^{33} + \)\(26\!\cdots\!54\)\( p^{294} T^{34} - \)\(42\!\cdots\!96\)\( p^{315} T^{35} + \)\(13\!\cdots\!09\)\( p^{336} T^{36} - \)\(15\!\cdots\!61\)\( p^{357} T^{37} + \)\(51\!\cdots\!80\)\( p^{378} T^{38} - 2772828270755157 p^{399} T^{39} + p^{420} T^{40} \) | |
| 31 | \( 1 + 3064842508546901 T + \)\(23\!\cdots\!34\)\( T^{2} + \)\(63\!\cdots\!11\)\( T^{3} + \)\(28\!\cdots\!61\)\( T^{4} + \)\(67\!\cdots\!92\)\( T^{5} + \)\(22\!\cdots\!82\)\( T^{6} + \)\(48\!\cdots\!74\)\( T^{7} + \)\(13\!\cdots\!56\)\( T^{8} + \)\(26\!\cdots\!02\)\( T^{9} + \)\(62\!\cdots\!74\)\( T^{10} + \)\(11\!\cdots\!56\)\( T^{11} + \)\(24\!\cdots\!67\)\( T^{12} + \)\(42\!\cdots\!83\)\( T^{13} + \)\(81\!\cdots\!46\)\( T^{14} + \)\(13\!\cdots\!53\)\( T^{15} + \)\(23\!\cdots\!39\)\( T^{16} + \)\(35\!\cdots\!20\)\( T^{17} + \)\(59\!\cdots\!80\)\( T^{18} + \)\(83\!\cdots\!24\)\( T^{19} + \)\(13\!\cdots\!64\)\( T^{20} + \)\(83\!\cdots\!24\)\( p^{21} T^{21} + \)\(59\!\cdots\!80\)\( p^{42} T^{22} + \)\(35\!\cdots\!20\)\( p^{63} T^{23} + \)\(23\!\cdots\!39\)\( p^{84} T^{24} + \)\(13\!\cdots\!53\)\( p^{105} T^{25} + \)\(81\!\cdots\!46\)\( p^{126} T^{26} + \)\(42\!\cdots\!83\)\( p^{147} T^{27} + \)\(24\!\cdots\!67\)\( p^{168} T^{28} + \)\(11\!\cdots\!56\)\( p^{189} T^{29} + \)\(62\!\cdots\!74\)\( p^{210} T^{30} + \)\(26\!\cdots\!02\)\( p^{231} T^{31} + \)\(13\!\cdots\!56\)\( p^{252} T^{32} + \)\(48\!\cdots\!74\)\( p^{273} T^{33} + \)\(22\!\cdots\!82\)\( p^{294} T^{34} + \)\(67\!\cdots\!92\)\( p^{315} T^{35} + \)\(28\!\cdots\!61\)\( p^{336} T^{36} + \)\(63\!\cdots\!11\)\( p^{357} T^{37} + \)\(23\!\cdots\!34\)\( p^{378} T^{38} + 3064842508546901 p^{399} T^{39} + p^{420} T^{40} \) | |
| 37 | \( 1 - 15041349243939694 T + \)\(79\!\cdots\!36\)\( T^{2} - \)\(18\!\cdots\!90\)\( T^{3} + \)\(32\!\cdots\!58\)\( T^{4} - \)\(98\!\cdots\!54\)\( T^{5} + \)\(95\!\cdots\!08\)\( T^{6} - \)\(32\!\cdots\!74\)\( T^{7} + \)\(21\!\cdots\!85\)\( T^{8} - \)\(77\!\cdots\!56\)\( T^{9} + \)\(40\!\cdots\!32\)\( T^{10} - \)\(14\!\cdots\!44\)\( T^{11} + \)\(65\!\cdots\!44\)\( T^{12} - \)\(22\!\cdots\!76\)\( T^{13} + \)\(89\!\cdots\!36\)\( T^{14} - \)\(29\!\cdots\!52\)\( T^{15} + \)\(10\!\cdots\!90\)\( T^{16} - \)\(33\!\cdots\!68\)\( T^{17} + \)\(11\!\cdots\!32\)\( T^{18} - \)\(32\!\cdots\!16\)\( T^{19} + \)\(10\!\cdots\!44\)\( T^{20} - \)\(32\!\cdots\!16\)\( p^{21} T^{21} + \)\(11\!\cdots\!32\)\( p^{42} T^{22} - \)\(33\!\cdots\!68\)\( p^{63} T^{23} + \)\(10\!\cdots\!90\)\( p^{84} T^{24} - \)\(29\!\cdots\!52\)\( p^{105} T^{25} + \)\(89\!\cdots\!36\)\( p^{126} T^{26} - \)\(22\!\cdots\!76\)\( p^{147} T^{27} + \)\(65\!\cdots\!44\)\( p^{168} T^{28} - \)\(14\!\cdots\!44\)\( p^{189} T^{29} + \)\(40\!\cdots\!32\)\( p^{210} T^{30} - \)\(77\!\cdots\!56\)\( p^{231} T^{31} + \)\(21\!\cdots\!85\)\( p^{252} T^{32} - \)\(32\!\cdots\!74\)\( p^{273} T^{33} + \)\(95\!\cdots\!08\)\( p^{294} T^{34} - \)\(98\!\cdots\!54\)\( p^{315} T^{35} + \)\(32\!\cdots\!58\)\( p^{336} T^{36} - \)\(18\!\cdots\!90\)\( p^{357} T^{37} + \)\(79\!\cdots\!36\)\( p^{378} T^{38} - 15041349243939694 p^{399} T^{39} + p^{420} T^{40} \) | |
| 41 | \( 1 + 4985514195953106 T + \)\(73\!\cdots\!92\)\( T^{2} - \)\(25\!\cdots\!24\)\( T^{3} + \)\(27\!\cdots\!01\)\( T^{4} - \)\(75\!\cdots\!28\)\( p T^{5} + \)\(70\!\cdots\!16\)\( T^{6} - \)\(12\!\cdots\!66\)\( T^{7} + \)\(13\!\cdots\!51\)\( T^{8} - \)\(31\!\cdots\!76\)\( T^{9} + \)\(21\!\cdots\!12\)\( T^{10} - \)\(57\!\cdots\!44\)\( T^{11} + \)\(29\!\cdots\!46\)\( T^{12} - \)\(83\!\cdots\!44\)\( T^{13} + \)\(33\!\cdots\!00\)\( T^{14} - \)\(99\!\cdots\!40\)\( T^{15} + \)\(34\!\cdots\!41\)\( T^{16} - \)\(10\!\cdots\!06\)\( T^{17} + \)\(30\!\cdots\!92\)\( T^{18} - \)\(86\!\cdots\!16\)\( T^{19} + \)\(23\!\cdots\!79\)\( T^{20} - \)\(86\!\cdots\!16\)\( p^{21} T^{21} + \)\(30\!\cdots\!92\)\( p^{42} T^{22} - \)\(10\!\cdots\!06\)\( p^{63} T^{23} + \)\(34\!\cdots\!41\)\( p^{84} T^{24} - \)\(99\!\cdots\!40\)\( p^{105} T^{25} + \)\(33\!\cdots\!00\)\( p^{126} T^{26} - \)\(83\!\cdots\!44\)\( p^{147} T^{27} + \)\(29\!\cdots\!46\)\( p^{168} T^{28} - \)\(57\!\cdots\!44\)\( p^{189} T^{29} + \)\(21\!\cdots\!12\)\( p^{210} T^{30} - \)\(31\!\cdots\!76\)\( p^{231} T^{31} + \)\(13\!\cdots\!51\)\( p^{252} T^{32} - \)\(12\!\cdots\!66\)\( p^{273} T^{33} + \)\(70\!\cdots\!16\)\( p^{294} T^{34} - \)\(75\!\cdots\!28\)\( p^{316} T^{35} + \)\(27\!\cdots\!01\)\( p^{336} T^{36} - \)\(25\!\cdots\!24\)\( p^{357} T^{37} + \)\(73\!\cdots\!92\)\( p^{378} T^{38} + 4985514195953106 p^{399} T^{39} + p^{420} T^{40} \) | |
| 43 | \( 1 - 41986108659289660 T + \)\(22\!\cdots\!14\)\( T^{2} + \)\(15\!\cdots\!24\)\( T^{3} + \)\(24\!\cdots\!75\)\( T^{4} + \)\(12\!\cdots\!92\)\( T^{5} + \)\(17\!\cdots\!58\)\( T^{6} + \)\(16\!\cdots\!88\)\( T^{7} + \)\(99\!\cdots\!59\)\( T^{8} + \)\(12\!\cdots\!80\)\( T^{9} + \)\(45\!\cdots\!20\)\( T^{10} + \)\(14\!\cdots\!44\)\( p T^{11} + \)\(17\!\cdots\!78\)\( T^{12} + \)\(25\!\cdots\!40\)\( T^{13} + \)\(59\!\cdots\!84\)\( T^{14} + \)\(85\!\cdots\!08\)\( T^{15} + \)\(39\!\cdots\!07\)\( p T^{16} + \)\(23\!\cdots\!08\)\( T^{17} + \)\(42\!\cdots\!62\)\( T^{18} + \)\(55\!\cdots\!44\)\( T^{19} + \)\(92\!\cdots\!37\)\( T^{20} + \)\(55\!\cdots\!44\)\( p^{21} T^{21} + \)\(42\!\cdots\!62\)\( p^{42} T^{22} + \)\(23\!\cdots\!08\)\( p^{63} T^{23} + \)\(39\!\cdots\!07\)\( p^{85} T^{24} + \)\(85\!\cdots\!08\)\( p^{105} T^{25} + \)\(59\!\cdots\!84\)\( p^{126} T^{26} + \)\(25\!\cdots\!40\)\( p^{147} T^{27} + \)\(17\!\cdots\!78\)\( p^{168} T^{28} + \)\(14\!\cdots\!44\)\( p^{190} T^{29} + \)\(45\!\cdots\!20\)\( p^{210} T^{30} + \)\(12\!\cdots\!80\)\( p^{231} T^{31} + \)\(99\!\cdots\!59\)\( p^{252} T^{32} + \)\(16\!\cdots\!88\)\( p^{273} T^{33} + \)\(17\!\cdots\!58\)\( p^{294} T^{34} + \)\(12\!\cdots\!92\)\( p^{315} T^{35} + \)\(24\!\cdots\!75\)\( p^{336} T^{36} + \)\(15\!\cdots\!24\)\( p^{357} T^{37} + \)\(22\!\cdots\!14\)\( p^{378} T^{38} - 41986108659289660 p^{399} T^{39} + p^{420} T^{40} \) | |
| 47 | \( 1 + 971751895603333665 T + \)\(13\!\cdots\!36\)\( T^{2} + \)\(92\!\cdots\!75\)\( T^{3} + \)\(75\!\cdots\!89\)\( T^{4} + \)\(40\!\cdots\!80\)\( T^{5} + \)\(25\!\cdots\!82\)\( T^{6} + \)\(11\!\cdots\!50\)\( T^{7} + \)\(59\!\cdots\!20\)\( T^{8} + \)\(24\!\cdots\!50\)\( T^{9} + \)\(11\!\cdots\!34\)\( T^{10} + \)\(45\!\cdots\!00\)\( T^{11} + \)\(20\!\cdots\!67\)\( T^{12} + \)\(81\!\cdots\!15\)\( T^{13} + \)\(34\!\cdots\!68\)\( T^{14} + \)\(13\!\cdots\!65\)\( T^{15} + \)\(52\!\cdots\!23\)\( T^{16} + \)\(20\!\cdots\!80\)\( T^{17} + \)\(74\!\cdots\!32\)\( T^{18} + \)\(29\!\cdots\!80\)\( T^{19} + \)\(99\!\cdots\!60\)\( T^{20} + \)\(29\!\cdots\!80\)\( p^{21} T^{21} + \)\(74\!\cdots\!32\)\( p^{42} T^{22} + \)\(20\!\cdots\!80\)\( p^{63} T^{23} + \)\(52\!\cdots\!23\)\( p^{84} T^{24} + \)\(13\!\cdots\!65\)\( p^{105} T^{25} + \)\(34\!\cdots\!68\)\( p^{126} T^{26} + \)\(81\!\cdots\!15\)\( p^{147} T^{27} + \)\(20\!\cdots\!67\)\( p^{168} T^{28} + \)\(45\!\cdots\!00\)\( p^{189} T^{29} + \)\(11\!\cdots\!34\)\( p^{210} T^{30} + \)\(24\!\cdots\!50\)\( p^{231} T^{31} + \)\(59\!\cdots\!20\)\( p^{252} T^{32} + \)\(11\!\cdots\!50\)\( p^{273} T^{33} + \)\(25\!\cdots\!82\)\( p^{294} T^{34} + \)\(40\!\cdots\!80\)\( p^{315} T^{35} + \)\(75\!\cdots\!89\)\( p^{336} T^{36} + \)\(92\!\cdots\!75\)\( p^{357} T^{37} + \)\(13\!\cdots\!36\)\( p^{378} T^{38} + 971751895603333665 p^{399} T^{39} + p^{420} T^{40} \) | |
| 53 | \( 1 + 3200468347464244650 T + \)\(26\!\cdots\!40\)\( T^{2} + \)\(74\!\cdots\!14\)\( T^{3} + \)\(34\!\cdots\!38\)\( T^{4} + \)\(84\!\cdots\!30\)\( T^{5} + \)\(28\!\cdots\!64\)\( T^{6} + \)\(62\!\cdots\!66\)\( T^{7} + \)\(16\!\cdots\!21\)\( T^{8} + \)\(33\!\cdots\!36\)\( T^{9} + \)\(76\!\cdots\!80\)\( T^{10} + \)\(14\!\cdots\!56\)\( T^{11} + \)\(27\!\cdots\!04\)\( T^{12} + \)\(46\!\cdots\!88\)\( T^{13} + \)\(83\!\cdots\!04\)\( T^{14} + \)\(12\!\cdots\!40\)\( T^{15} + \)\(20\!\cdots\!06\)\( T^{16} + \)\(29\!\cdots\!84\)\( T^{17} + \)\(43\!\cdots\!60\)\( T^{18} + \)\(10\!\cdots\!16\)\( p T^{19} + \)\(75\!\cdots\!40\)\( T^{20} + \)\(10\!\cdots\!16\)\( p^{22} T^{21} + \)\(43\!\cdots\!60\)\( p^{42} T^{22} + \)\(29\!\cdots\!84\)\( p^{63} T^{23} + \)\(20\!\cdots\!06\)\( p^{84} T^{24} + \)\(12\!\cdots\!40\)\( p^{105} T^{25} + \)\(83\!\cdots\!04\)\( p^{126} T^{26} + \)\(46\!\cdots\!88\)\( p^{147} T^{27} + \)\(27\!\cdots\!04\)\( p^{168} T^{28} + \)\(14\!\cdots\!56\)\( p^{189} T^{29} + \)\(76\!\cdots\!80\)\( p^{210} T^{30} + \)\(33\!\cdots\!36\)\( p^{231} T^{31} + \)\(16\!\cdots\!21\)\( p^{252} T^{32} + \)\(62\!\cdots\!66\)\( p^{273} T^{33} + \)\(28\!\cdots\!64\)\( p^{294} T^{34} + \)\(84\!\cdots\!30\)\( p^{315} T^{35} + \)\(34\!\cdots\!38\)\( p^{336} T^{36} + \)\(74\!\cdots\!14\)\( p^{357} T^{37} + \)\(26\!\cdots\!40\)\( p^{378} T^{38} + 3200468347464244650 p^{399} T^{39} + p^{420} T^{40} \) | |
| 59 | \( 1 + 11335149505478767926 T + \)\(24\!\cdots\!68\)\( T^{2} + \)\(24\!\cdots\!68\)\( T^{3} + \)\(30\!\cdots\!77\)\( T^{4} + \)\(25\!\cdots\!12\)\( T^{5} + \)\(23\!\cdots\!08\)\( T^{6} + \)\(17\!\cdots\!94\)\( T^{7} + \)\(13\!\cdots\!51\)\( T^{8} + \)\(89\!\cdots\!40\)\( T^{9} + \)\(59\!\cdots\!00\)\( T^{10} + \)\(35\!\cdots\!68\)\( T^{11} + \)\(20\!\cdots\!42\)\( T^{12} + \)\(11\!\cdots\!92\)\( T^{13} + \)\(59\!\cdots\!56\)\( T^{14} + \)\(29\!\cdots\!36\)\( T^{15} + \)\(14\!\cdots\!37\)\( T^{16} + \)\(63\!\cdots\!14\)\( T^{17} + \)\(47\!\cdots\!44\)\( p T^{18} + \)\(11\!\cdots\!12\)\( T^{19} + \)\(46\!\cdots\!31\)\( T^{20} + \)\(11\!\cdots\!12\)\( p^{21} T^{21} + \)\(47\!\cdots\!44\)\( p^{43} T^{22} + \)\(63\!\cdots\!14\)\( p^{63} T^{23} + \)\(14\!\cdots\!37\)\( p^{84} T^{24} + \)\(29\!\cdots\!36\)\( p^{105} T^{25} + \)\(59\!\cdots\!56\)\( p^{126} T^{26} + \)\(11\!\cdots\!92\)\( p^{147} T^{27} + \)\(20\!\cdots\!42\)\( p^{168} T^{28} + \)\(35\!\cdots\!68\)\( p^{189} T^{29} + \)\(59\!\cdots\!00\)\( p^{210} T^{30} + \)\(89\!\cdots\!40\)\( p^{231} T^{31} + \)\(13\!\cdots\!51\)\( p^{252} T^{32} + \)\(17\!\cdots\!94\)\( p^{273} T^{33} + \)\(23\!\cdots\!08\)\( p^{294} T^{34} + \)\(25\!\cdots\!12\)\( p^{315} T^{35} + \)\(30\!\cdots\!77\)\( p^{336} T^{36} + \)\(24\!\cdots\!68\)\( p^{357} T^{37} + \)\(24\!\cdots\!68\)\( p^{378} T^{38} + 11335149505478767926 p^{399} T^{39} + p^{420} T^{40} \) | |
| 61 | \( 1 - 5471638213245247 p T + \)\(42\!\cdots\!78\)\( T^{2} - \)\(13\!\cdots\!39\)\( T^{3} + \)\(90\!\cdots\!41\)\( T^{4} - \)\(29\!\cdots\!20\)\( T^{5} + \)\(12\!\cdots\!82\)\( T^{6} - \)\(44\!\cdots\!50\)\( T^{7} + \)\(12\!\cdots\!12\)\( T^{8} - \)\(49\!\cdots\!90\)\( T^{9} + \)\(10\!\cdots\!50\)\( T^{10} - \)\(41\!\cdots\!64\)\( T^{11} + \)\(66\!\cdots\!95\)\( T^{12} - \)\(28\!\cdots\!25\)\( T^{13} + \)\(59\!\cdots\!74\)\( p T^{14} - \)\(15\!\cdots\!61\)\( T^{15} + \)\(16\!\cdots\!27\)\( T^{16} - \)\(68\!\cdots\!80\)\( T^{17} + \)\(64\!\cdots\!44\)\( T^{18} - \)\(25\!\cdots\!88\)\( T^{19} + \)\(21\!\cdots\!60\)\( T^{20} - \)\(25\!\cdots\!88\)\( p^{21} T^{21} + \)\(64\!\cdots\!44\)\( p^{42} T^{22} - \)\(68\!\cdots\!80\)\( p^{63} T^{23} + \)\(16\!\cdots\!27\)\( p^{84} T^{24} - \)\(15\!\cdots\!61\)\( p^{105} T^{25} + \)\(59\!\cdots\!74\)\( p^{127} T^{26} - \)\(28\!\cdots\!25\)\( p^{147} T^{27} + \)\(66\!\cdots\!95\)\( p^{168} T^{28} - \)\(41\!\cdots\!64\)\( p^{189} T^{29} + \)\(10\!\cdots\!50\)\( p^{210} T^{30} - \)\(49\!\cdots\!90\)\( p^{231} T^{31} + \)\(12\!\cdots\!12\)\( p^{252} T^{32} - \)\(44\!\cdots\!50\)\( p^{273} T^{33} + \)\(12\!\cdots\!82\)\( p^{294} T^{34} - \)\(29\!\cdots\!20\)\( p^{315} T^{35} + \)\(90\!\cdots\!41\)\( p^{336} T^{36} - \)\(13\!\cdots\!39\)\( p^{357} T^{37} + \)\(42\!\cdots\!78\)\( p^{378} T^{38} - 5471638213245247 p^{400} T^{39} + p^{420} T^{40} \) | |
| 67 | \( 1 + 14325054629523996728 T + \)\(24\!\cdots\!26\)\( T^{2} + \)\(26\!\cdots\!64\)\( T^{3} + \)\(28\!\cdots\!43\)\( T^{4} + \)\(24\!\cdots\!24\)\( T^{5} + \)\(21\!\cdots\!34\)\( T^{6} + \)\(14\!\cdots\!68\)\( T^{7} + \)\(12\!\cdots\!95\)\( T^{8} + \)\(69\!\cdots\!16\)\( T^{9} + \)\(58\!\cdots\!80\)\( T^{10} + \)\(26\!\cdots\!36\)\( T^{11} + \)\(22\!\cdots\!66\)\( T^{12} + \)\(84\!\cdots\!92\)\( T^{13} + \)\(76\!\cdots\!92\)\( T^{14} + \)\(23\!\cdots\!76\)\( T^{15} + \)\(22\!\cdots\!45\)\( T^{16} + \)\(61\!\cdots\!24\)\( T^{17} + \)\(60\!\cdots\!26\)\( T^{18} + \)\(14\!\cdots\!44\)\( T^{19} + \)\(14\!\cdots\!85\)\( T^{20} + \)\(14\!\cdots\!44\)\( p^{21} T^{21} + \)\(60\!\cdots\!26\)\( p^{42} T^{22} + \)\(61\!\cdots\!24\)\( p^{63} T^{23} + \)\(22\!\cdots\!45\)\( p^{84} T^{24} + \)\(23\!\cdots\!76\)\( p^{105} T^{25} + \)\(76\!\cdots\!92\)\( p^{126} T^{26} + \)\(84\!\cdots\!92\)\( p^{147} T^{27} + \)\(22\!\cdots\!66\)\( p^{168} T^{28} + \)\(26\!\cdots\!36\)\( p^{189} T^{29} + \)\(58\!\cdots\!80\)\( p^{210} T^{30} + \)\(69\!\cdots\!16\)\( p^{231} T^{31} + \)\(12\!\cdots\!95\)\( p^{252} T^{32} + \)\(14\!\cdots\!68\)\( p^{273} T^{33} + \)\(21\!\cdots\!34\)\( p^{294} T^{34} + \)\(24\!\cdots\!24\)\( p^{315} T^{35} + \)\(28\!\cdots\!43\)\( p^{336} T^{36} + \)\(26\!\cdots\!64\)\( p^{357} T^{37} + \)\(24\!\cdots\!26\)\( p^{378} T^{38} + 14325054629523996728 p^{399} T^{39} + p^{420} T^{40} \) | |
| 71 | \( 1 + \)\(10\!\cdots\!60\)\( T + \)\(11\!\cdots\!56\)\( T^{2} + \)\(71\!\cdots\!20\)\( T^{3} + \)\(43\!\cdots\!22\)\( T^{4} + \)\(19\!\cdots\!56\)\( T^{5} + \)\(87\!\cdots\!12\)\( T^{6} + \)\(31\!\cdots\!92\)\( T^{7} + \)\(10\!\cdots\!25\)\( T^{8} + \)\(30\!\cdots\!28\)\( T^{9} + \)\(88\!\cdots\!04\)\( T^{10} + \)\(20\!\cdots\!00\)\( T^{11} + \)\(49\!\cdots\!04\)\( T^{12} + \)\(80\!\cdots\!64\)\( T^{13} + \)\(12\!\cdots\!48\)\( T^{14} - \)\(21\!\cdots\!36\)\( T^{15} - \)\(15\!\cdots\!06\)\( T^{16} - \)\(92\!\cdots\!24\)\( T^{17} - \)\(32\!\cdots\!92\)\( T^{18} - \)\(16\!\cdots\!64\)\( p T^{19} - \)\(31\!\cdots\!00\)\( T^{20} - \)\(16\!\cdots\!64\)\( p^{22} T^{21} - \)\(32\!\cdots\!92\)\( p^{42} T^{22} - \)\(92\!\cdots\!24\)\( p^{63} T^{23} - \)\(15\!\cdots\!06\)\( p^{84} T^{24} - \)\(21\!\cdots\!36\)\( p^{105} T^{25} + \)\(12\!\cdots\!48\)\( p^{126} T^{26} + \)\(80\!\cdots\!64\)\( p^{147} T^{27} + \)\(49\!\cdots\!04\)\( p^{168} T^{28} + \)\(20\!\cdots\!00\)\( p^{189} T^{29} + \)\(88\!\cdots\!04\)\( p^{210} T^{30} + \)\(30\!\cdots\!28\)\( p^{231} T^{31} + \)\(10\!\cdots\!25\)\( p^{252} T^{32} + \)\(31\!\cdots\!92\)\( p^{273} T^{33} + \)\(87\!\cdots\!12\)\( p^{294} T^{34} + \)\(19\!\cdots\!56\)\( p^{315} T^{35} + \)\(43\!\cdots\!22\)\( p^{336} T^{36} + \)\(71\!\cdots\!20\)\( p^{357} T^{37} + \)\(11\!\cdots\!56\)\( p^{378} T^{38} + \)\(10\!\cdots\!60\)\( p^{399} T^{39} + p^{420} T^{40} \) | |
| 73 | \( 1 + 2415544298256170813 T + \)\(12\!\cdots\!95\)\( T^{2} + \)\(10\!\cdots\!88\)\( T^{3} + \)\(81\!\cdots\!09\)\( T^{4} - \)\(11\!\cdots\!01\)\( T^{5} + \)\(34\!\cdots\!08\)\( T^{6} - \)\(15\!\cdots\!75\)\( T^{7} + \)\(10\!\cdots\!93\)\( T^{8} - \)\(89\!\cdots\!30\)\( T^{9} + \)\(27\!\cdots\!75\)\( T^{10} - \)\(34\!\cdots\!33\)\( T^{11} + \)\(56\!\cdots\!16\)\( T^{12} - \)\(10\!\cdots\!71\)\( T^{13} + \)\(10\!\cdots\!45\)\( T^{14} - \)\(24\!\cdots\!62\)\( T^{15} + \)\(16\!\cdots\!70\)\( T^{16} - \)\(46\!\cdots\!22\)\( T^{17} + \)\(23\!\cdots\!69\)\( T^{18} - \)\(74\!\cdots\!71\)\( T^{19} + \)\(33\!\cdots\!42\)\( T^{20} - \)\(74\!\cdots\!71\)\( p^{21} T^{21} + \)\(23\!\cdots\!69\)\( p^{42} T^{22} - \)\(46\!\cdots\!22\)\( p^{63} T^{23} + \)\(16\!\cdots\!70\)\( p^{84} T^{24} - \)\(24\!\cdots\!62\)\( p^{105} T^{25} + \)\(10\!\cdots\!45\)\( p^{126} T^{26} - \)\(10\!\cdots\!71\)\( p^{147} T^{27} + \)\(56\!\cdots\!16\)\( p^{168} T^{28} - \)\(34\!\cdots\!33\)\( p^{189} T^{29} + \)\(27\!\cdots\!75\)\( p^{210} T^{30} - \)\(89\!\cdots\!30\)\( p^{231} T^{31} + \)\(10\!\cdots\!93\)\( p^{252} T^{32} - \)\(15\!\cdots\!75\)\( p^{273} T^{33} + \)\(34\!\cdots\!08\)\( p^{294} T^{34} - \)\(11\!\cdots\!01\)\( p^{315} T^{35} + \)\(81\!\cdots\!09\)\( p^{336} T^{36} + \)\(10\!\cdots\!88\)\( p^{357} T^{37} + \)\(12\!\cdots\!95\)\( p^{378} T^{38} + 2415544298256170813 p^{399} T^{39} + p^{420} T^{40} \) | |
| 79 | \( 1 + 45386959469213650259 T + \)\(82\!\cdots\!90\)\( T^{2} + \)\(27\!\cdots\!73\)\( T^{3} + \)\(33\!\cdots\!01\)\( T^{4} + \)\(72\!\cdots\!16\)\( T^{5} + \)\(88\!\cdots\!82\)\( T^{6} + \)\(98\!\cdots\!70\)\( T^{7} + \)\(17\!\cdots\!04\)\( T^{8} + \)\(32\!\cdots\!42\)\( T^{9} + \)\(26\!\cdots\!22\)\( T^{10} - \)\(15\!\cdots\!32\)\( T^{11} + \)\(35\!\cdots\!71\)\( T^{12} - \)\(38\!\cdots\!31\)\( T^{13} + \)\(39\!\cdots\!86\)\( T^{14} - \)\(56\!\cdots\!53\)\( T^{15} + \)\(38\!\cdots\!83\)\( T^{16} - \)\(61\!\cdots\!88\)\( T^{17} + \)\(32\!\cdots\!36\)\( T^{18} - \)\(53\!\cdots\!36\)\( T^{19} + \)\(24\!\cdots\!16\)\( T^{20} - \)\(53\!\cdots\!36\)\( p^{21} T^{21} + \)\(32\!\cdots\!36\)\( p^{42} T^{22} - \)\(61\!\cdots\!88\)\( p^{63} T^{23} + \)\(38\!\cdots\!83\)\( p^{84} T^{24} - \)\(56\!\cdots\!53\)\( p^{105} T^{25} + \)\(39\!\cdots\!86\)\( p^{126} T^{26} - \)\(38\!\cdots\!31\)\( p^{147} T^{27} + \)\(35\!\cdots\!71\)\( p^{168} T^{28} - \)\(15\!\cdots\!32\)\( p^{189} T^{29} + \)\(26\!\cdots\!22\)\( p^{210} T^{30} + \)\(32\!\cdots\!42\)\( p^{231} T^{31} + \)\(17\!\cdots\!04\)\( p^{252} T^{32} + \)\(98\!\cdots\!70\)\( p^{273} T^{33} + \)\(88\!\cdots\!82\)\( p^{294} T^{34} + \)\(72\!\cdots\!16\)\( p^{315} T^{35} + \)\(33\!\cdots\!01\)\( p^{336} T^{36} + \)\(27\!\cdots\!73\)\( p^{357} T^{37} + \)\(82\!\cdots\!90\)\( p^{378} T^{38} + 45386959469213650259 p^{399} T^{39} + p^{420} T^{40} \) | |
| 83 | \( 1 + \)\(22\!\cdots\!15\)\( T + \)\(25\!\cdots\!62\)\( T^{2} + \)\(58\!\cdots\!85\)\( T^{3} + \)\(32\!\cdots\!49\)\( T^{4} + \)\(73\!\cdots\!60\)\( T^{5} + \)\(28\!\cdots\!18\)\( T^{6} + \)\(60\!\cdots\!50\)\( T^{7} + \)\(18\!\cdots\!00\)\( T^{8} + \)\(37\!\cdots\!70\)\( T^{9} + \)\(96\!\cdots\!46\)\( T^{10} + \)\(17\!\cdots\!80\)\( T^{11} + \)\(40\!\cdots\!07\)\( T^{12} + \)\(68\!\cdots\!45\)\( T^{13} + \)\(13\!\cdots\!18\)\( T^{14} + \)\(22\!\cdots\!95\)\( T^{15} + \)\(41\!\cdots\!83\)\( T^{16} + \)\(60\!\cdots\!00\)\( T^{17} + \)\(10\!\cdots\!08\)\( T^{18} + \)\(14\!\cdots\!60\)\( T^{19} + \)\(22\!\cdots\!20\)\( T^{20} + \)\(14\!\cdots\!60\)\( p^{21} T^{21} + \)\(10\!\cdots\!08\)\( p^{42} T^{22} + \)\(60\!\cdots\!00\)\( p^{63} T^{23} + \)\(41\!\cdots\!83\)\( p^{84} T^{24} + \)\(22\!\cdots\!95\)\( p^{105} T^{25} + \)\(13\!\cdots\!18\)\( p^{126} T^{26} + \)\(68\!\cdots\!45\)\( p^{147} T^{27} + \)\(40\!\cdots\!07\)\( p^{168} T^{28} + \)\(17\!\cdots\!80\)\( p^{189} T^{29} + \)\(96\!\cdots\!46\)\( p^{210} T^{30} + \)\(37\!\cdots\!70\)\( p^{231} T^{31} + \)\(18\!\cdots\!00\)\( p^{252} T^{32} + \)\(60\!\cdots\!50\)\( p^{273} T^{33} + \)\(28\!\cdots\!18\)\( p^{294} T^{34} + \)\(73\!\cdots\!60\)\( p^{315} T^{35} + \)\(32\!\cdots\!49\)\( p^{336} T^{36} + \)\(58\!\cdots\!85\)\( p^{357} T^{37} + \)\(25\!\cdots\!62\)\( p^{378} T^{38} + \)\(22\!\cdots\!15\)\( p^{399} T^{39} + p^{420} T^{40} \) | |
| 89 | \( 1 - \)\(14\!\cdots\!60\)\( T + \)\(91\!\cdots\!32\)\( T^{2} - \)\(78\!\cdots\!04\)\( T^{3} + \)\(40\!\cdots\!74\)\( T^{4} - \)\(13\!\cdots\!24\)\( T^{5} + \)\(11\!\cdots\!68\)\( T^{6} + \)\(22\!\cdots\!60\)\( T^{7} + \)\(26\!\cdots\!13\)\( T^{8} + \)\(16\!\cdots\!28\)\( T^{9} + \)\(47\!\cdots\!92\)\( T^{10} + \)\(45\!\cdots\!88\)\( T^{11} + \)\(73\!\cdots\!08\)\( T^{12} + \)\(85\!\cdots\!24\)\( T^{13} + \)\(98\!\cdots\!28\)\( T^{14} + \)\(12\!\cdots\!28\)\( T^{15} + \)\(11\!\cdots\!90\)\( T^{16} + \)\(15\!\cdots\!60\)\( T^{17} + \)\(11\!\cdots\!52\)\( T^{18} + \)\(15\!\cdots\!20\)\( T^{19} + \)\(11\!\cdots\!44\)\( T^{20} + \)\(15\!\cdots\!20\)\( p^{21} T^{21} + \)\(11\!\cdots\!52\)\( p^{42} T^{22} + \)\(15\!\cdots\!60\)\( p^{63} T^{23} + \)\(11\!\cdots\!90\)\( p^{84} T^{24} + \)\(12\!\cdots\!28\)\( p^{105} T^{25} + \)\(98\!\cdots\!28\)\( p^{126} T^{26} + \)\(85\!\cdots\!24\)\( p^{147} T^{27} + \)\(73\!\cdots\!08\)\( p^{168} T^{28} + \)\(45\!\cdots\!88\)\( p^{189} T^{29} + \)\(47\!\cdots\!92\)\( p^{210} T^{30} + \)\(16\!\cdots\!28\)\( p^{231} T^{31} + \)\(26\!\cdots\!13\)\( p^{252} T^{32} + \)\(22\!\cdots\!60\)\( p^{273} T^{33} + \)\(11\!\cdots\!68\)\( p^{294} T^{34} - \)\(13\!\cdots\!24\)\( p^{315} T^{35} + \)\(40\!\cdots\!74\)\( p^{336} T^{36} - \)\(78\!\cdots\!04\)\( p^{357} T^{37} + \)\(91\!\cdots\!32\)\( p^{378} T^{38} - \)\(14\!\cdots\!60\)\( p^{399} T^{39} + p^{420} T^{40} \) | |
| 97 | \( 1 - \)\(65\!\cdots\!46\)\( T + \)\(46\!\cdots\!12\)\( T^{2} - \)\(23\!\cdots\!00\)\( T^{3} + \)\(11\!\cdots\!77\)\( T^{4} - \)\(42\!\cdots\!20\)\( T^{5} + \)\(18\!\cdots\!64\)\( T^{6} - \)\(55\!\cdots\!58\)\( T^{7} + \)\(22\!\cdots\!15\)\( T^{8} - \)\(55\!\cdots\!04\)\( T^{9} + \)\(24\!\cdots\!88\)\( T^{10} - \)\(46\!\cdots\!04\)\( T^{11} + \)\(21\!\cdots\!14\)\( T^{12} - \)\(33\!\cdots\!48\)\( T^{13} + \)\(16\!\cdots\!72\)\( T^{14} - \)\(21\!\cdots\!08\)\( T^{15} + \)\(11\!\cdots\!93\)\( T^{16} - \)\(12\!\cdots\!82\)\( T^{17} + \)\(72\!\cdots\!64\)\( T^{18} - \)\(69\!\cdots\!00\)\( T^{19} + \)\(40\!\cdots\!35\)\( T^{20} - \)\(69\!\cdots\!00\)\( p^{21} T^{21} + \)\(72\!\cdots\!64\)\( p^{42} T^{22} - \)\(12\!\cdots\!82\)\( p^{63} T^{23} + \)\(11\!\cdots\!93\)\( p^{84} T^{24} - \)\(21\!\cdots\!08\)\( p^{105} T^{25} + \)\(16\!\cdots\!72\)\( p^{126} T^{26} - \)\(33\!\cdots\!48\)\( p^{147} T^{27} + \)\(21\!\cdots\!14\)\( p^{168} T^{28} - \)\(46\!\cdots\!04\)\( p^{189} T^{29} + \)\(24\!\cdots\!88\)\( p^{210} T^{30} - \)\(55\!\cdots\!04\)\( p^{231} T^{31} + \)\(22\!\cdots\!15\)\( p^{252} T^{32} - \)\(55\!\cdots\!58\)\( p^{273} T^{33} + \)\(18\!\cdots\!64\)\( p^{294} T^{34} - \)\(42\!\cdots\!20\)\( p^{315} T^{35} + \)\(11\!\cdots\!77\)\( p^{336} T^{36} - \)\(23\!\cdots\!00\)\( p^{357} T^{37} + \)\(46\!\cdots\!12\)\( p^{378} T^{38} - \)\(65\!\cdots\!46\)\( p^{399} T^{39} + p^{420} T^{40} \) | |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{40} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−1.94665195705960214675213322609, −1.94075292217226474265432738686, −1.82920487330190709269865925739, −1.80969808928522212705011222664, −1.75187627643294300691468797640, −1.74123645291855586074572067577, −1.71646170070627317676808133271, −1.56517441648243970620988824663, −1.55680798576073407106540172177, −1.54178961824202162628802405902, −1.32071540762433458785290689702, −1.24612075471883602871414657207, −1.20545986399593910924881034455, −1.09909537047823872291568971789, −1.05041060419604603105899433618, −1.04044718785937523000294119943, −1.03178882719172204537282408162, −1.02441310318948375277773484037, −1.00531330890919917642355584497, −0.880415163710535721454422587404, −0.864974456706238540213095520037, −0.836339020300147156954176015966, −0.816113697917234417576000267272, −0.801818681266095791954270353962, −0.51572616219828689024118862713, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0.51572616219828689024118862713, 0.801818681266095791954270353962, 0.816113697917234417576000267272, 0.836339020300147156954176015966, 0.864974456706238540213095520037, 0.880415163710535721454422587404, 1.00531330890919917642355584497, 1.02441310318948375277773484037, 1.03178882719172204537282408162, 1.04044718785937523000294119943, 1.05041060419604603105899433618, 1.09909537047823872291568971789, 1.20545986399593910924881034455, 1.24612075471883602871414657207, 1.32071540762433458785290689702, 1.54178961824202162628802405902, 1.55680798576073407106540172177, 1.56517441648243970620988824663, 1.71646170070627317676808133271, 1.74123645291855586074572067577, 1.75187627643294300691468797640, 1.80969808928522212705011222664, 1.82920487330190709269865925739, 1.94075292217226474265432738686, 1.94665195705960214675213322609
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.