Dirichlet series
| L(s) = 1 | + 1.43e4·2-s − 1.16e5·3-s + 1.10e8·4-s − 3.62e7·5-s − 1.66e9·6-s − 1.99e9·7-s + 6.01e11·8-s − 4.46e10·9-s − 5.20e11·10-s + 3.04e9·11-s − 1.28e13·12-s − 2.89e11·13-s − 2.85e13·14-s + 4.21e12·15-s + 2.61e15·16-s − 8.65e12·17-s − 6.40e14·18-s + 4.96e12·19-s − 3.99e15·20-s + 2.31e14·21-s + 4.37e13·22-s + 7.64e13·23-s − 6.99e16·24-s − 2.09e15·25-s − 4.14e15·26-s + 5.55e15·27-s − 2.19e17·28-s + ⋯ |
| L(s) = 1 | + 9.89·2-s − 1.13·3-s + 52.5·4-s − 1.66·5-s − 11.2·6-s − 2.66·7-s + 197.·8-s − 4.26·9-s − 16.4·10-s + 0.0354·11-s − 59.6·12-s − 0.581·13-s − 26.3·14-s + 1.88·15-s + 595·16-s − 1.04·17-s − 42.2·18-s + 0.185·19-s − 87.2·20-s + 3.03·21-s + 0.350·22-s + 0.384·23-s − 225.·24-s − 4.38·25-s − 5.75·26-s + 5.19·27-s − 139.·28-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{14} \cdot 37^{14}\right)^{s/2} \, \Gamma_{\C}(s)^{14} \, L(s)\cr=\mathstrut & \,\Lambda(22-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{14} \cdot 37^{14}\right)^{s/2} \, \Gamma_{\C}(s+21/2)^{14} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(28\) |
| Conductor: | \(2^{14} \cdot 37^{14}\) |
| Sign: | $1$ |
| Analytic conductor: | \(2.61876\times 10^{32}\) |
| Root analytic conductor: | \(14.3810\) |
| Motivic weight: | \(21\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(14\) |
| Selberg data: | \((28,\ 2^{14} \cdot 37^{14} ,\ ( \ : [21/2]^{14} ),\ 1 )\) |
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 | 2 | \( ( 1 - p^{10} T )^{14} \) |
| 37 | \( ( 1 - p^{10} T )^{14} \) | |
| good | 3 | \( 1 + 116288 T + 19389224042 p T^{2} + 236895654064949 p^{3} T^{3} + 67757171611154023600 p^{3} T^{4} + \)\(24\!\cdots\!09\)\( p^{4} T^{5} + \)\(19\!\cdots\!77\)\( p^{7} T^{6} + \)\(73\!\cdots\!38\)\( p^{10} T^{7} + \)\(48\!\cdots\!74\)\( p^{13} T^{8} + \)\(17\!\cdots\!25\)\( p^{16} T^{9} + \)\(10\!\cdots\!95\)\( p^{19} T^{10} + \)\(11\!\cdots\!30\)\( p^{23} T^{11} + \)\(69\!\cdots\!99\)\( p^{28} T^{12} + \)\(22\!\cdots\!14\)\( p^{31} T^{13} + \)\(35\!\cdots\!00\)\( p^{35} T^{14} + \)\(22\!\cdots\!14\)\( p^{52} T^{15} + \)\(69\!\cdots\!99\)\( p^{70} T^{16} + \)\(11\!\cdots\!30\)\( p^{86} T^{17} + \)\(10\!\cdots\!95\)\( p^{103} T^{18} + \)\(17\!\cdots\!25\)\( p^{121} T^{19} + \)\(48\!\cdots\!74\)\( p^{139} T^{20} + \)\(73\!\cdots\!38\)\( p^{157} T^{21} + \)\(19\!\cdots\!77\)\( p^{175} T^{22} + \)\(24\!\cdots\!09\)\( p^{193} T^{23} + 67757171611154023600 p^{213} T^{24} + 236895654064949 p^{234} T^{25} + 19389224042 p^{253} T^{26} + 116288 p^{273} T^{27} + p^{294} T^{28} \) |
| 5 | \( 1 + 7255269 p T + 3406798590943474 T^{2} + \)\(23\!\cdots\!36\)\( p T^{3} + \)\(23\!\cdots\!99\)\( p^{2} T^{4} + \)\(15\!\cdots\!34\)\( p^{3} T^{5} + \)\(22\!\cdots\!11\)\( p^{5} T^{6} + \)\(12\!\cdots\!44\)\( p^{6} T^{7} + \)\(16\!\cdots\!89\)\( p^{8} T^{8} + \)\(32\!\cdots\!56\)\( p^{11} T^{9} + \)\(36\!\cdots\!58\)\( p^{13} T^{10} + \)\(33\!\cdots\!67\)\( p^{15} T^{11} + \)\(16\!\cdots\!83\)\( p^{16} T^{12} + \)\(14\!\cdots\!58\)\( p^{18} T^{13} + \)\(27\!\cdots\!66\)\( p^{21} T^{14} + \)\(14\!\cdots\!58\)\( p^{39} T^{15} + \)\(16\!\cdots\!83\)\( p^{58} T^{16} + \)\(33\!\cdots\!67\)\( p^{78} T^{17} + \)\(36\!\cdots\!58\)\( p^{97} T^{18} + \)\(32\!\cdots\!56\)\( p^{116} T^{19} + \)\(16\!\cdots\!89\)\( p^{134} T^{20} + \)\(12\!\cdots\!44\)\( p^{153} T^{21} + \)\(22\!\cdots\!11\)\( p^{173} T^{22} + \)\(15\!\cdots\!34\)\( p^{192} T^{23} + \)\(23\!\cdots\!99\)\( p^{212} T^{24} + \)\(23\!\cdots\!36\)\( p^{232} T^{25} + 3406798590943474 p^{252} T^{26} + 7255269 p^{274} T^{27} + p^{294} T^{28} \) | |
| 7 | \( 1 + 1992761657 T + 5801082226280431389 T^{2} + \)\(85\!\cdots\!88\)\( T^{3} + \)\(14\!\cdots\!68\)\( T^{4} + \)\(26\!\cdots\!90\)\( p T^{5} + \)\(50\!\cdots\!29\)\( p^{2} T^{6} + \)\(77\!\cdots\!79\)\( p^{3} T^{7} + \)\(17\!\cdots\!40\)\( p^{5} T^{8} + \)\(24\!\cdots\!09\)\( p^{6} T^{9} + \)\(34\!\cdots\!17\)\( p^{7} T^{10} + \)\(29\!\cdots\!94\)\( p^{7} T^{11} + \)\(37\!\cdots\!63\)\( p^{8} T^{12} + \)\(41\!\cdots\!33\)\( p^{9} T^{13} + \)\(46\!\cdots\!02\)\( p^{10} T^{14} + \)\(41\!\cdots\!33\)\( p^{30} T^{15} + \)\(37\!\cdots\!63\)\( p^{50} T^{16} + \)\(29\!\cdots\!94\)\( p^{70} T^{17} + \)\(34\!\cdots\!17\)\( p^{91} T^{18} + \)\(24\!\cdots\!09\)\( p^{111} T^{19} + \)\(17\!\cdots\!40\)\( p^{131} T^{20} + \)\(77\!\cdots\!79\)\( p^{150} T^{21} + \)\(50\!\cdots\!29\)\( p^{170} T^{22} + \)\(26\!\cdots\!90\)\( p^{190} T^{23} + \)\(14\!\cdots\!68\)\( p^{210} T^{24} + \)\(85\!\cdots\!88\)\( p^{231} T^{25} + 5801082226280431389 p^{252} T^{26} + 1992761657 p^{273} T^{27} + p^{294} T^{28} \) | |
| 11 | \( 1 - 3048981618 T + \)\(66\!\cdots\!86\)\( T^{2} + \)\(53\!\cdots\!29\)\( T^{3} + \)\(22\!\cdots\!60\)\( T^{4} + \)\(33\!\cdots\!51\)\( T^{5} + \)\(48\!\cdots\!27\)\( T^{6} + \)\(96\!\cdots\!38\)\( T^{7} + \)\(71\!\cdots\!58\)\( p T^{8} + \)\(14\!\cdots\!41\)\( p^{2} T^{9} + \)\(74\!\cdots\!51\)\( p^{3} T^{10} + \)\(15\!\cdots\!44\)\( p^{4} T^{11} + \)\(61\!\cdots\!81\)\( p^{5} T^{12} + \)\(12\!\cdots\!10\)\( p^{6} T^{13} + \)\(41\!\cdots\!72\)\( p^{7} T^{14} + \)\(12\!\cdots\!10\)\( p^{27} T^{15} + \)\(61\!\cdots\!81\)\( p^{47} T^{16} + \)\(15\!\cdots\!44\)\( p^{67} T^{17} + \)\(74\!\cdots\!51\)\( p^{87} T^{18} + \)\(14\!\cdots\!41\)\( p^{107} T^{19} + \)\(71\!\cdots\!58\)\( p^{127} T^{20} + \)\(96\!\cdots\!38\)\( p^{147} T^{21} + \)\(48\!\cdots\!27\)\( p^{168} T^{22} + \)\(33\!\cdots\!51\)\( p^{189} T^{23} + \)\(22\!\cdots\!60\)\( p^{210} T^{24} + \)\(53\!\cdots\!29\)\( p^{231} T^{25} + \)\(66\!\cdots\!86\)\( p^{252} T^{26} - 3048981618 p^{273} T^{27} + p^{294} T^{28} \) | |
| 13 | \( 1 + 289023666401 T + \)\(18\!\cdots\!14\)\( T^{2} + \)\(64\!\cdots\!00\)\( T^{3} + \)\(17\!\cdots\!35\)\( T^{4} + \)\(65\!\cdots\!22\)\( T^{5} + \)\(12\!\cdots\!63\)\( T^{6} + \)\(33\!\cdots\!56\)\( p T^{7} + \)\(27\!\cdots\!49\)\( p^{3} T^{8} + \)\(96\!\cdots\!28\)\( p^{3} T^{9} + \)\(86\!\cdots\!82\)\( p^{4} T^{10} + \)\(21\!\cdots\!71\)\( p^{5} T^{11} + \)\(16\!\cdots\!47\)\( p^{6} T^{12} + \)\(38\!\cdots\!78\)\( p^{7} T^{13} + \)\(26\!\cdots\!10\)\( p^{8} T^{14} + \)\(38\!\cdots\!78\)\( p^{28} T^{15} + \)\(16\!\cdots\!47\)\( p^{48} T^{16} + \)\(21\!\cdots\!71\)\( p^{68} T^{17} + \)\(86\!\cdots\!82\)\( p^{88} T^{18} + \)\(96\!\cdots\!28\)\( p^{108} T^{19} + \)\(27\!\cdots\!49\)\( p^{129} T^{20} + \)\(33\!\cdots\!56\)\( p^{148} T^{21} + \)\(12\!\cdots\!63\)\( p^{168} T^{22} + \)\(65\!\cdots\!22\)\( p^{189} T^{23} + \)\(17\!\cdots\!35\)\( p^{210} T^{24} + \)\(64\!\cdots\!00\)\( p^{231} T^{25} + \)\(18\!\cdots\!14\)\( p^{252} T^{26} + 289023666401 p^{273} T^{27} + p^{294} T^{28} \) | |
| 17 | \( 1 + 8653439514264 T + \)\(67\!\cdots\!22\)\( T^{2} + \)\(47\!\cdots\!04\)\( T^{3} + \)\(21\!\cdots\!75\)\( T^{4} + \)\(12\!\cdots\!44\)\( T^{5} + \)\(24\!\cdots\!16\)\( p T^{6} + \)\(71\!\cdots\!16\)\( p^{2} T^{7} + \)\(12\!\cdots\!37\)\( p^{3} T^{8} + \)\(30\!\cdots\!96\)\( p^{4} T^{9} + \)\(28\!\cdots\!86\)\( p^{6} T^{10} + \)\(10\!\cdots\!68\)\( p^{6} T^{11} + \)\(15\!\cdots\!95\)\( p^{7} T^{12} + \)\(29\!\cdots\!56\)\( p^{8} T^{13} + \)\(40\!\cdots\!12\)\( p^{9} T^{14} + \)\(29\!\cdots\!56\)\( p^{29} T^{15} + \)\(15\!\cdots\!95\)\( p^{49} T^{16} + \)\(10\!\cdots\!68\)\( p^{69} T^{17} + \)\(28\!\cdots\!86\)\( p^{90} T^{18} + \)\(30\!\cdots\!96\)\( p^{109} T^{19} + \)\(12\!\cdots\!37\)\( p^{129} T^{20} + \)\(71\!\cdots\!16\)\( p^{149} T^{21} + \)\(24\!\cdots\!16\)\( p^{169} T^{22} + \)\(12\!\cdots\!44\)\( p^{189} T^{23} + \)\(21\!\cdots\!75\)\( p^{210} T^{24} + \)\(47\!\cdots\!04\)\( p^{231} T^{25} + \)\(67\!\cdots\!22\)\( p^{252} T^{26} + 8653439514264 p^{273} T^{27} + p^{294} T^{28} \) | |
| 19 | \( 1 - 261167153014 p T + \)\(38\!\cdots\!50\)\( T^{2} - \)\(17\!\cdots\!82\)\( T^{3} + \)\(36\!\cdots\!97\)\( p T^{4} - \)\(66\!\cdots\!76\)\( p^{2} T^{5} + \)\(11\!\cdots\!56\)\( p^{3} T^{6} - \)\(11\!\cdots\!52\)\( p^{4} T^{7} + \)\(29\!\cdots\!27\)\( p^{5} T^{8} - \)\(52\!\cdots\!54\)\( p^{6} T^{9} + \)\(59\!\cdots\!62\)\( p^{7} T^{10} + \)\(27\!\cdots\!54\)\( p^{8} T^{11} + \)\(11\!\cdots\!97\)\( p^{9} T^{12} + \)\(11\!\cdots\!76\)\( p^{10} T^{13} + \)\(20\!\cdots\!44\)\( p^{11} T^{14} + \)\(11\!\cdots\!76\)\( p^{31} T^{15} + \)\(11\!\cdots\!97\)\( p^{51} T^{16} + \)\(27\!\cdots\!54\)\( p^{71} T^{17} + \)\(59\!\cdots\!62\)\( p^{91} T^{18} - \)\(52\!\cdots\!54\)\( p^{111} T^{19} + \)\(29\!\cdots\!27\)\( p^{131} T^{20} - \)\(11\!\cdots\!52\)\( p^{151} T^{21} + \)\(11\!\cdots\!56\)\( p^{171} T^{22} - \)\(66\!\cdots\!76\)\( p^{191} T^{23} + \)\(36\!\cdots\!97\)\( p^{211} T^{24} - \)\(17\!\cdots\!82\)\( p^{231} T^{25} + \)\(38\!\cdots\!50\)\( p^{252} T^{26} - 261167153014 p^{274} T^{27} + p^{294} T^{28} \) | |
| 23 | \( 1 - 76444090561653 T + \)\(13\!\cdots\!30\)\( p T^{2} - \)\(33\!\cdots\!10\)\( T^{3} + \)\(45\!\cdots\!79\)\( T^{4} - \)\(65\!\cdots\!32\)\( T^{5} + \)\(19\!\cdots\!37\)\( p T^{6} - \)\(76\!\cdots\!66\)\( T^{7} + \)\(35\!\cdots\!37\)\( T^{8} - \)\(61\!\cdots\!50\)\( T^{9} + \)\(22\!\cdots\!02\)\( T^{10} - \)\(37\!\cdots\!53\)\( T^{11} + \)\(11\!\cdots\!91\)\( T^{12} - \)\(18\!\cdots\!22\)\( T^{13} + \)\(49\!\cdots\!14\)\( T^{14} - \)\(18\!\cdots\!22\)\( p^{21} T^{15} + \)\(11\!\cdots\!91\)\( p^{42} T^{16} - \)\(37\!\cdots\!53\)\( p^{63} T^{17} + \)\(22\!\cdots\!02\)\( p^{84} T^{18} - \)\(61\!\cdots\!50\)\( p^{105} T^{19} + \)\(35\!\cdots\!37\)\( p^{126} T^{20} - \)\(76\!\cdots\!66\)\( p^{147} T^{21} + \)\(19\!\cdots\!37\)\( p^{169} T^{22} - \)\(65\!\cdots\!32\)\( p^{189} T^{23} + \)\(45\!\cdots\!79\)\( p^{210} T^{24} - \)\(33\!\cdots\!10\)\( p^{231} T^{25} + \)\(13\!\cdots\!30\)\( p^{253} T^{26} - 76444090561653 p^{273} T^{27} + p^{294} T^{28} \) | |
| 29 | \( 1 + 8455814168873703 T + \)\(65\!\cdots\!96\)\( T^{2} + \)\(32\!\cdots\!82\)\( T^{3} + \)\(15\!\cdots\!25\)\( T^{4} + \)\(58\!\cdots\!66\)\( T^{5} + \)\(21\!\cdots\!49\)\( T^{6} + \)\(67\!\cdots\!78\)\( T^{7} + \)\(71\!\cdots\!91\)\( p T^{8} + \)\(69\!\cdots\!72\)\( p^{2} T^{9} + \)\(16\!\cdots\!40\)\( T^{10} + \)\(42\!\cdots\!55\)\( T^{11} + \)\(10\!\cdots\!15\)\( T^{12} + \)\(25\!\cdots\!84\)\( T^{13} + \)\(60\!\cdots\!30\)\( T^{14} + \)\(25\!\cdots\!84\)\( p^{21} T^{15} + \)\(10\!\cdots\!15\)\( p^{42} T^{16} + \)\(42\!\cdots\!55\)\( p^{63} T^{17} + \)\(16\!\cdots\!40\)\( p^{84} T^{18} + \)\(69\!\cdots\!72\)\( p^{107} T^{19} + \)\(71\!\cdots\!91\)\( p^{127} T^{20} + \)\(67\!\cdots\!78\)\( p^{147} T^{21} + \)\(21\!\cdots\!49\)\( p^{168} T^{22} + \)\(58\!\cdots\!66\)\( p^{189} T^{23} + \)\(15\!\cdots\!25\)\( p^{210} T^{24} + \)\(32\!\cdots\!82\)\( p^{231} T^{25} + \)\(65\!\cdots\!96\)\( p^{252} T^{26} + 8455814168873703 p^{273} T^{27} + p^{294} T^{28} \) | |
| 31 | \( 1 + 20256434803157351 T + \)\(32\!\cdots\!86\)\( T^{2} + \)\(34\!\cdots\!96\)\( T^{3} + \)\(33\!\cdots\!55\)\( T^{4} + \)\(25\!\cdots\!24\)\( T^{5} + \)\(17\!\cdots\!91\)\( T^{6} + \)\(99\!\cdots\!40\)\( T^{7} + \)\(51\!\cdots\!97\)\( T^{8} + \)\(70\!\cdots\!06\)\( p T^{9} + \)\(81\!\cdots\!98\)\( T^{10} + \)\(19\!\cdots\!09\)\( T^{11} + \)\(55\!\cdots\!57\)\( p T^{12} - \)\(79\!\cdots\!16\)\( p T^{13} - \)\(14\!\cdots\!70\)\( T^{14} - \)\(79\!\cdots\!16\)\( p^{22} T^{15} + \)\(55\!\cdots\!57\)\( p^{43} T^{16} + \)\(19\!\cdots\!09\)\( p^{63} T^{17} + \)\(81\!\cdots\!98\)\( p^{84} T^{18} + \)\(70\!\cdots\!06\)\( p^{106} T^{19} + \)\(51\!\cdots\!97\)\( p^{126} T^{20} + \)\(99\!\cdots\!40\)\( p^{147} T^{21} + \)\(17\!\cdots\!91\)\( p^{168} T^{22} + \)\(25\!\cdots\!24\)\( p^{189} T^{23} + \)\(33\!\cdots\!55\)\( p^{210} T^{24} + \)\(34\!\cdots\!96\)\( p^{231} T^{25} + \)\(32\!\cdots\!86\)\( p^{252} T^{26} + 20256434803157351 p^{273} T^{27} + p^{294} T^{28} \) | |
| 41 | \( 1 - 141749959306768422 T + \)\(79\!\cdots\!02\)\( T^{2} - \)\(10\!\cdots\!17\)\( T^{3} + \)\(29\!\cdots\!30\)\( T^{4} - \)\(33\!\cdots\!13\)\( T^{5} + \)\(70\!\cdots\!59\)\( T^{6} - \)\(72\!\cdots\!10\)\( T^{7} + \)\(11\!\cdots\!86\)\( T^{8} - \)\(11\!\cdots\!93\)\( T^{9} + \)\(15\!\cdots\!15\)\( T^{10} - \)\(13\!\cdots\!18\)\( T^{11} + \)\(15\!\cdots\!09\)\( T^{12} - \)\(11\!\cdots\!52\)\( T^{13} + \)\(12\!\cdots\!26\)\( T^{14} - \)\(11\!\cdots\!52\)\( p^{21} T^{15} + \)\(15\!\cdots\!09\)\( p^{42} T^{16} - \)\(13\!\cdots\!18\)\( p^{63} T^{17} + \)\(15\!\cdots\!15\)\( p^{84} T^{18} - \)\(11\!\cdots\!93\)\( p^{105} T^{19} + \)\(11\!\cdots\!86\)\( p^{126} T^{20} - \)\(72\!\cdots\!10\)\( p^{147} T^{21} + \)\(70\!\cdots\!59\)\( p^{168} T^{22} - \)\(33\!\cdots\!13\)\( p^{189} T^{23} + \)\(29\!\cdots\!30\)\( p^{210} T^{24} - \)\(10\!\cdots\!17\)\( p^{231} T^{25} + \)\(79\!\cdots\!02\)\( p^{252} T^{26} - 141749959306768422 p^{273} T^{27} + p^{294} T^{28} \) | |
| 43 | \( 1 - 207178824983915962 T + \)\(20\!\cdots\!62\)\( T^{2} - \)\(44\!\cdots\!42\)\( T^{3} + \)\(19\!\cdots\!47\)\( T^{4} - \)\(44\!\cdots\!52\)\( T^{5} + \)\(12\!\cdots\!40\)\( T^{6} - \)\(27\!\cdots\!16\)\( T^{7} + \)\(61\!\cdots\!57\)\( T^{8} - \)\(12\!\cdots\!58\)\( T^{9} + \)\(22\!\cdots\!02\)\( T^{10} - \)\(39\!\cdots\!06\)\( T^{11} + \)\(65\!\cdots\!87\)\( T^{12} - \)\(99\!\cdots\!80\)\( T^{13} + \)\(14\!\cdots\!92\)\( T^{14} - \)\(99\!\cdots\!80\)\( p^{21} T^{15} + \)\(65\!\cdots\!87\)\( p^{42} T^{16} - \)\(39\!\cdots\!06\)\( p^{63} T^{17} + \)\(22\!\cdots\!02\)\( p^{84} T^{18} - \)\(12\!\cdots\!58\)\( p^{105} T^{19} + \)\(61\!\cdots\!57\)\( p^{126} T^{20} - \)\(27\!\cdots\!16\)\( p^{147} T^{21} + \)\(12\!\cdots\!40\)\( p^{168} T^{22} - \)\(44\!\cdots\!52\)\( p^{189} T^{23} + \)\(19\!\cdots\!47\)\( p^{210} T^{24} - \)\(44\!\cdots\!42\)\( p^{231} T^{25} + \)\(20\!\cdots\!62\)\( p^{252} T^{26} - 207178824983915962 p^{273} T^{27} + p^{294} T^{28} \) | |
| 47 | \( 1 + 227924735787051405 T + \)\(12\!\cdots\!01\)\( T^{2} + \)\(18\!\cdots\!12\)\( T^{3} + \)\(69\!\cdots\!56\)\( T^{4} + \)\(60\!\cdots\!38\)\( T^{5} + \)\(25\!\cdots\!05\)\( T^{6} + \)\(86\!\cdots\!25\)\( T^{7} + \)\(70\!\cdots\!04\)\( T^{8} - \)\(45\!\cdots\!07\)\( T^{9} + \)\(14\!\cdots\!63\)\( T^{10} - \)\(47\!\cdots\!14\)\( T^{11} + \)\(25\!\cdots\!31\)\( T^{12} - \)\(10\!\cdots\!97\)\( T^{13} + \)\(36\!\cdots\!06\)\( T^{14} - \)\(10\!\cdots\!97\)\( p^{21} T^{15} + \)\(25\!\cdots\!31\)\( p^{42} T^{16} - \)\(47\!\cdots\!14\)\( p^{63} T^{17} + \)\(14\!\cdots\!63\)\( p^{84} T^{18} - \)\(45\!\cdots\!07\)\( p^{105} T^{19} + \)\(70\!\cdots\!04\)\( p^{126} T^{20} + \)\(86\!\cdots\!25\)\( p^{147} T^{21} + \)\(25\!\cdots\!05\)\( p^{168} T^{22} + \)\(60\!\cdots\!38\)\( p^{189} T^{23} + \)\(69\!\cdots\!56\)\( p^{210} T^{24} + \)\(18\!\cdots\!12\)\( p^{231} T^{25} + \)\(12\!\cdots\!01\)\( p^{252} T^{26} + 227924735787051405 p^{273} T^{27} + p^{294} T^{28} \) | |
| 53 | \( 1 + 136202641020546003 T + \)\(52\!\cdots\!61\)\( T^{2} - \)\(35\!\cdots\!86\)\( T^{3} + \)\(19\!\cdots\!52\)\( T^{4} - \)\(13\!\cdots\!20\)\( T^{5} + \)\(62\!\cdots\!49\)\( T^{6} - \)\(46\!\cdots\!51\)\( T^{7} + \)\(14\!\cdots\!72\)\( T^{8} - \)\(11\!\cdots\!53\)\( T^{9} + \)\(31\!\cdots\!71\)\( T^{10} - \)\(22\!\cdots\!72\)\( T^{11} + \)\(57\!\cdots\!63\)\( T^{12} - \)\(41\!\cdots\!17\)\( T^{13} + \)\(99\!\cdots\!18\)\( T^{14} - \)\(41\!\cdots\!17\)\( p^{21} T^{15} + \)\(57\!\cdots\!63\)\( p^{42} T^{16} - \)\(22\!\cdots\!72\)\( p^{63} T^{17} + \)\(31\!\cdots\!71\)\( p^{84} T^{18} - \)\(11\!\cdots\!53\)\( p^{105} T^{19} + \)\(14\!\cdots\!72\)\( p^{126} T^{20} - \)\(46\!\cdots\!51\)\( p^{147} T^{21} + \)\(62\!\cdots\!49\)\( p^{168} T^{22} - \)\(13\!\cdots\!20\)\( p^{189} T^{23} + \)\(19\!\cdots\!52\)\( p^{210} T^{24} - \)\(35\!\cdots\!86\)\( p^{231} T^{25} + \)\(52\!\cdots\!61\)\( p^{252} T^{26} + 136202641020546003 p^{273} T^{27} + p^{294} T^{28} \) | |
| 59 | \( 1 + 8182105818231312258 T + \)\(71\!\cdots\!30\)\( T^{2} + \)\(31\!\cdots\!70\)\( T^{3} + \)\(17\!\cdots\!51\)\( T^{4} + \)\(53\!\cdots\!64\)\( T^{5} + \)\(31\!\cdots\!00\)\( T^{6} + \)\(93\!\cdots\!36\)\( T^{7} + \)\(72\!\cdots\!65\)\( T^{8} + \)\(24\!\cdots\!62\)\( T^{9} + \)\(16\!\cdots\!02\)\( T^{10} + \)\(46\!\cdots\!10\)\( T^{11} + \)\(25\!\cdots\!59\)\( T^{12} + \)\(58\!\cdots\!88\)\( T^{13} + \)\(36\!\cdots\!84\)\( T^{14} + \)\(58\!\cdots\!88\)\( p^{21} T^{15} + \)\(25\!\cdots\!59\)\( p^{42} T^{16} + \)\(46\!\cdots\!10\)\( p^{63} T^{17} + \)\(16\!\cdots\!02\)\( p^{84} T^{18} + \)\(24\!\cdots\!62\)\( p^{105} T^{19} + \)\(72\!\cdots\!65\)\( p^{126} T^{20} + \)\(93\!\cdots\!36\)\( p^{147} T^{21} + \)\(31\!\cdots\!00\)\( p^{168} T^{22} + \)\(53\!\cdots\!64\)\( p^{189} T^{23} + \)\(17\!\cdots\!51\)\( p^{210} T^{24} + \)\(31\!\cdots\!70\)\( p^{231} T^{25} + \)\(71\!\cdots\!30\)\( p^{252} T^{26} + 8182105818231312258 p^{273} T^{27} + p^{294} T^{28} \) | |
| 61 | \( 1 + 17403349115903587511 T + \)\(38\!\cdots\!78\)\( T^{2} + \)\(41\!\cdots\!40\)\( T^{3} + \)\(50\!\cdots\!51\)\( T^{4} + \)\(37\!\cdots\!10\)\( T^{5} + \)\(32\!\cdots\!83\)\( T^{6} + \)\(15\!\cdots\!68\)\( T^{7} + \)\(11\!\cdots\!97\)\( T^{8} + \)\(30\!\cdots\!20\)\( T^{9} + \)\(28\!\cdots\!14\)\( T^{10} + \)\(44\!\cdots\!85\)\( T^{11} + \)\(10\!\cdots\!55\)\( T^{12} + \)\(22\!\cdots\!38\)\( T^{13} + \)\(37\!\cdots\!58\)\( T^{14} + \)\(22\!\cdots\!38\)\( p^{21} T^{15} + \)\(10\!\cdots\!55\)\( p^{42} T^{16} + \)\(44\!\cdots\!85\)\( p^{63} T^{17} + \)\(28\!\cdots\!14\)\( p^{84} T^{18} + \)\(30\!\cdots\!20\)\( p^{105} T^{19} + \)\(11\!\cdots\!97\)\( p^{126} T^{20} + \)\(15\!\cdots\!68\)\( p^{147} T^{21} + \)\(32\!\cdots\!83\)\( p^{168} T^{22} + \)\(37\!\cdots\!10\)\( p^{189} T^{23} + \)\(50\!\cdots\!51\)\( p^{210} T^{24} + \)\(41\!\cdots\!40\)\( p^{231} T^{25} + \)\(38\!\cdots\!78\)\( p^{252} T^{26} + 17403349115903587511 p^{273} T^{27} + p^{294} T^{28} \) | |
| 67 | \( 1 + 32274698588879719067 T + \)\(22\!\cdots\!76\)\( T^{2} + \)\(65\!\cdots\!64\)\( T^{3} + \)\(25\!\cdots\!73\)\( T^{4} + \)\(63\!\cdots\!24\)\( T^{5} + \)\(18\!\cdots\!17\)\( T^{6} + \)\(40\!\cdots\!52\)\( T^{7} + \)\(93\!\cdots\!87\)\( T^{8} + \)\(18\!\cdots\!46\)\( T^{9} + \)\(36\!\cdots\!12\)\( T^{10} + \)\(64\!\cdots\!29\)\( T^{11} + \)\(11\!\cdots\!27\)\( T^{12} + \)\(17\!\cdots\!28\)\( T^{13} + \)\(28\!\cdots\!10\)\( T^{14} + \)\(17\!\cdots\!28\)\( p^{21} T^{15} + \)\(11\!\cdots\!27\)\( p^{42} T^{16} + \)\(64\!\cdots\!29\)\( p^{63} T^{17} + \)\(36\!\cdots\!12\)\( p^{84} T^{18} + \)\(18\!\cdots\!46\)\( p^{105} T^{19} + \)\(93\!\cdots\!87\)\( p^{126} T^{20} + \)\(40\!\cdots\!52\)\( p^{147} T^{21} + \)\(18\!\cdots\!17\)\( p^{168} T^{22} + \)\(63\!\cdots\!24\)\( p^{189} T^{23} + \)\(25\!\cdots\!73\)\( p^{210} T^{24} + \)\(65\!\cdots\!64\)\( p^{231} T^{25} + \)\(22\!\cdots\!76\)\( p^{252} T^{26} + 32274698588879719067 p^{273} T^{27} + p^{294} T^{28} \) | |
| 71 | \( 1 + 47331816180146780301 T + \)\(59\!\cdots\!15\)\( T^{2} + \)\(21\!\cdots\!32\)\( T^{3} + \)\(15\!\cdots\!02\)\( T^{4} + \)\(44\!\cdots\!94\)\( T^{5} + \)\(25\!\cdots\!27\)\( T^{6} + \)\(52\!\cdots\!57\)\( T^{7} + \)\(27\!\cdots\!28\)\( T^{8} + \)\(35\!\cdots\!97\)\( T^{9} + \)\(20\!\cdots\!13\)\( T^{10} + \)\(63\!\cdots\!98\)\( T^{11} + \)\(12\!\cdots\!09\)\( T^{12} - \)\(10\!\cdots\!49\)\( T^{13} + \)\(75\!\cdots\!10\)\( T^{14} - \)\(10\!\cdots\!49\)\( p^{21} T^{15} + \)\(12\!\cdots\!09\)\( p^{42} T^{16} + \)\(63\!\cdots\!98\)\( p^{63} T^{17} + \)\(20\!\cdots\!13\)\( p^{84} T^{18} + \)\(35\!\cdots\!97\)\( p^{105} T^{19} + \)\(27\!\cdots\!28\)\( p^{126} T^{20} + \)\(52\!\cdots\!57\)\( p^{147} T^{21} + \)\(25\!\cdots\!27\)\( p^{168} T^{22} + \)\(44\!\cdots\!94\)\( p^{189} T^{23} + \)\(15\!\cdots\!02\)\( p^{210} T^{24} + \)\(21\!\cdots\!32\)\( p^{231} T^{25} + \)\(59\!\cdots\!15\)\( p^{252} T^{26} + 47331816180146780301 p^{273} T^{27} + p^{294} T^{28} \) | |
| 73 | \( 1 + 76503702763464317552 T + \)\(16\!\cdots\!28\)\( p T^{2} + \)\(67\!\cdots\!41\)\( T^{3} + \)\(61\!\cdots\!50\)\( T^{4} + \)\(24\!\cdots\!01\)\( T^{5} + \)\(16\!\cdots\!49\)\( T^{6} + \)\(45\!\cdots\!26\)\( T^{7} + \)\(27\!\cdots\!00\)\( T^{8} + \)\(28\!\cdots\!15\)\( T^{9} + \)\(24\!\cdots\!61\)\( T^{10} - \)\(59\!\cdots\!72\)\( T^{11} + \)\(64\!\cdots\!77\)\( T^{12} - \)\(18\!\cdots\!24\)\( T^{13} - \)\(83\!\cdots\!02\)\( T^{14} - \)\(18\!\cdots\!24\)\( p^{21} T^{15} + \)\(64\!\cdots\!77\)\( p^{42} T^{16} - \)\(59\!\cdots\!72\)\( p^{63} T^{17} + \)\(24\!\cdots\!61\)\( p^{84} T^{18} + \)\(28\!\cdots\!15\)\( p^{105} T^{19} + \)\(27\!\cdots\!00\)\( p^{126} T^{20} + \)\(45\!\cdots\!26\)\( p^{147} T^{21} + \)\(16\!\cdots\!49\)\( p^{168} T^{22} + \)\(24\!\cdots\!01\)\( p^{189} T^{23} + \)\(61\!\cdots\!50\)\( p^{210} T^{24} + \)\(67\!\cdots\!41\)\( p^{231} T^{25} + \)\(16\!\cdots\!28\)\( p^{253} T^{26} + 76503702763464317552 p^{273} T^{27} + p^{294} T^{28} \) | |
| 79 | \( 1 + 23285518628800253735 T + \)\(52\!\cdots\!42\)\( T^{2} + \)\(18\!\cdots\!66\)\( T^{3} + \)\(13\!\cdots\!35\)\( T^{4} + \)\(64\!\cdots\!68\)\( T^{5} + \)\(25\!\cdots\!47\)\( T^{6} + \)\(13\!\cdots\!30\)\( T^{7} + \)\(36\!\cdots\!61\)\( T^{8} + \)\(19\!\cdots\!98\)\( T^{9} + \)\(41\!\cdots\!62\)\( T^{10} + \)\(21\!\cdots\!63\)\( T^{11} + \)\(38\!\cdots\!43\)\( T^{12} + \)\(24\!\cdots\!02\)\( p T^{13} + \)\(29\!\cdots\!38\)\( T^{14} + \)\(24\!\cdots\!02\)\( p^{22} T^{15} + \)\(38\!\cdots\!43\)\( p^{42} T^{16} + \)\(21\!\cdots\!63\)\( p^{63} T^{17} + \)\(41\!\cdots\!62\)\( p^{84} T^{18} + \)\(19\!\cdots\!98\)\( p^{105} T^{19} + \)\(36\!\cdots\!61\)\( p^{126} T^{20} + \)\(13\!\cdots\!30\)\( p^{147} T^{21} + \)\(25\!\cdots\!47\)\( p^{168} T^{22} + \)\(64\!\cdots\!68\)\( p^{189} T^{23} + \)\(13\!\cdots\!35\)\( p^{210} T^{24} + \)\(18\!\cdots\!66\)\( p^{231} T^{25} + \)\(52\!\cdots\!42\)\( p^{252} T^{26} + 23285518628800253735 p^{273} T^{27} + p^{294} T^{28} \) | |
| 83 | \( 1 + \)\(29\!\cdots\!21\)\( T + \)\(24\!\cdots\!35\)\( T^{2} + \)\(60\!\cdots\!72\)\( T^{3} + \)\(27\!\cdots\!18\)\( T^{4} + \)\(56\!\cdots\!42\)\( T^{5} + \)\(18\!\cdots\!23\)\( T^{6} + \)\(33\!\cdots\!25\)\( T^{7} + \)\(84\!\cdots\!00\)\( T^{8} + \)\(13\!\cdots\!89\)\( T^{9} + \)\(29\!\cdots\!93\)\( T^{10} + \)\(42\!\cdots\!38\)\( T^{11} + \)\(78\!\cdots\!73\)\( T^{12} + \)\(10\!\cdots\!51\)\( T^{13} + \)\(17\!\cdots\!70\)\( T^{14} + \)\(10\!\cdots\!51\)\( p^{21} T^{15} + \)\(78\!\cdots\!73\)\( p^{42} T^{16} + \)\(42\!\cdots\!38\)\( p^{63} T^{17} + \)\(29\!\cdots\!93\)\( p^{84} T^{18} + \)\(13\!\cdots\!89\)\( p^{105} T^{19} + \)\(84\!\cdots\!00\)\( p^{126} T^{20} + \)\(33\!\cdots\!25\)\( p^{147} T^{21} + \)\(18\!\cdots\!23\)\( p^{168} T^{22} + \)\(56\!\cdots\!42\)\( p^{189} T^{23} + \)\(27\!\cdots\!18\)\( p^{210} T^{24} + \)\(60\!\cdots\!72\)\( p^{231} T^{25} + \)\(24\!\cdots\!35\)\( p^{252} T^{26} + \)\(29\!\cdots\!21\)\( p^{273} T^{27} + p^{294} T^{28} \) | |
| 89 | \( 1 + \)\(27\!\cdots\!08\)\( T + \)\(50\!\cdots\!74\)\( T^{2} + \)\(17\!\cdots\!08\)\( T^{3} + \)\(14\!\cdots\!79\)\( T^{4} + \)\(54\!\cdots\!16\)\( T^{5} + \)\(30\!\cdots\!48\)\( T^{6} + \)\(11\!\cdots\!16\)\( T^{7} + \)\(48\!\cdots\!97\)\( T^{8} + \)\(17\!\cdots\!40\)\( T^{9} + \)\(65\!\cdots\!98\)\( T^{10} + \)\(21\!\cdots\!64\)\( T^{11} + \)\(72\!\cdots\!79\)\( T^{12} + \)\(22\!\cdots\!80\)\( T^{13} + \)\(68\!\cdots\!68\)\( T^{14} + \)\(22\!\cdots\!80\)\( p^{21} T^{15} + \)\(72\!\cdots\!79\)\( p^{42} T^{16} + \)\(21\!\cdots\!64\)\( p^{63} T^{17} + \)\(65\!\cdots\!98\)\( p^{84} T^{18} + \)\(17\!\cdots\!40\)\( p^{105} T^{19} + \)\(48\!\cdots\!97\)\( p^{126} T^{20} + \)\(11\!\cdots\!16\)\( p^{147} T^{21} + \)\(30\!\cdots\!48\)\( p^{168} T^{22} + \)\(54\!\cdots\!16\)\( p^{189} T^{23} + \)\(14\!\cdots\!79\)\( p^{210} T^{24} + \)\(17\!\cdots\!08\)\( p^{231} T^{25} + \)\(50\!\cdots\!74\)\( p^{252} T^{26} + \)\(27\!\cdots\!08\)\( p^{273} T^{27} + p^{294} T^{28} \) | |
| 97 | \( 1 + \)\(14\!\cdots\!14\)\( T + \)\(56\!\cdots\!46\)\( T^{2} + \)\(73\!\cdots\!14\)\( T^{3} + \)\(15\!\cdots\!55\)\( T^{4} + \)\(18\!\cdots\!68\)\( T^{5} + \)\(28\!\cdots\!32\)\( T^{6} + \)\(29\!\cdots\!04\)\( T^{7} + \)\(36\!\cdots\!65\)\( T^{8} + \)\(33\!\cdots\!34\)\( T^{9} + \)\(34\!\cdots\!78\)\( T^{10} + \)\(29\!\cdots\!34\)\( T^{11} + \)\(26\!\cdots\!83\)\( T^{12} + \)\(19\!\cdots\!08\)\( T^{13} + \)\(15\!\cdots\!88\)\( T^{14} + \)\(19\!\cdots\!08\)\( p^{21} T^{15} + \)\(26\!\cdots\!83\)\( p^{42} T^{16} + \)\(29\!\cdots\!34\)\( p^{63} T^{17} + \)\(34\!\cdots\!78\)\( p^{84} T^{18} + \)\(33\!\cdots\!34\)\( p^{105} T^{19} + \)\(36\!\cdots\!65\)\( p^{126} T^{20} + \)\(29\!\cdots\!04\)\( p^{147} T^{21} + \)\(28\!\cdots\!32\)\( p^{168} T^{22} + \)\(18\!\cdots\!68\)\( p^{189} T^{23} + \)\(15\!\cdots\!55\)\( p^{210} T^{24} + \)\(73\!\cdots\!14\)\( p^{231} T^{25} + \)\(56\!\cdots\!46\)\( p^{252} T^{26} + \)\(14\!\cdots\!14\)\( p^{273} T^{27} + p^{294} T^{28} \) | |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{28} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−2.72037602720778235915062300616, −2.63685665686266974470498337417, −2.59227941570638031249915716014, −2.54629672282052341081552737328, −2.51195206477955285173795358676, −2.47077223224172450536515690502, −2.39744451384979875558250603069, −2.39619769654157445588166137383, −2.35655120085127306355010833025, −2.32318673690282771886258068468, −1.90939565573207832695141329300, −1.90930898601944234917016835212, −1.75438275793762387750407829171, −1.67166811130227907618813303134, −1.63395409168180353073816352394, −1.45208007973781681070303298511, −1.43561085174042585594823066691, −1.40944110251080518480071538564, −1.36220153592782154675731291793, −1.31069924360238216369636297958, −1.22118802289487408946470973322, −0.971153662886435301971412667299, −0.958470684420376839004406571844, −0.949210923710613049961493307804, −0.904619292122774957046816574234, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0.904619292122774957046816574234, 0.949210923710613049961493307804, 0.958470684420376839004406571844, 0.971153662886435301971412667299, 1.22118802289487408946470973322, 1.31069924360238216369636297958, 1.36220153592782154675731291793, 1.40944110251080518480071538564, 1.43561085174042585594823066691, 1.45208007973781681070303298511, 1.63395409168180353073816352394, 1.67166811130227907618813303134, 1.75438275793762387750407829171, 1.90930898601944234917016835212, 1.90939565573207832695141329300, 2.32318673690282771886258068468, 2.35655120085127306355010833025, 2.39619769654157445588166137383, 2.39744451384979875558250603069, 2.47077223224172450536515690502, 2.51195206477955285173795358676, 2.54629672282052341081552737328, 2.59227941570638031249915716014, 2.63685665686266974470498337417, 2.72037602720778235915062300616
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.