Dirichlet series
| L(s) = 1 | + 27·2-s + 163·3-s − 2.02e3·4-s + 2.96e3·5-s + 4.40e3·6-s + 3.36e4·7-s − 5.14e4·8-s − 5.24e4·9-s + 8.00e4·10-s + 8.18e4·11-s − 3.30e5·12-s − 3.99e5·13-s + 9.07e5·14-s + 4.83e5·15-s + 2.00e6·16-s − 4.49e4·17-s − 1.41e6·18-s + 1.71e5·19-s − 6.00e6·20-s + 5.47e6·21-s + 2.20e6·22-s + 1.93e6·23-s − 8.38e6·24-s − 5.16e6·25-s − 1.07e7·26-s − 9.67e6·27-s − 6.80e7·28-s + ⋯ |
| L(s) = 1 | + 1.19·2-s + 1.16·3-s − 3.95·4-s + 2.12·5-s + 1.38·6-s + 5.29·7-s − 4.43·8-s − 2.66·9-s + 2.53·10-s + 1.68·11-s − 4.59·12-s − 3.88·13-s + 6.31·14-s + 2.46·15-s + 7.65·16-s − 0.130·17-s − 3.17·18-s + 0.302·19-s − 8.38·20-s + 6.14·21-s + 2.01·22-s + 1.43·23-s − 5.15·24-s − 2.64·25-s − 4.63·26-s − 3.50·27-s − 20.9·28-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{14} \cdot 13^{14}\right)^{s/2} \, \Gamma_{\C}(s)^{14} \, L(s)\cr=\mathstrut & \,\Lambda(10-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{14} \cdot 13^{14}\right)^{s/2} \, \Gamma_{\C}(s+9/2)^{14} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(28\) |
| Conductor: | \(7^{14} \cdot 13^{14}\) |
| Sign: | $1$ |
| Analytic conductor: | \(2.46776\times 10^{23}\) |
| Root analytic conductor: | \(6.84603\) |
| Motivic weight: | \(9\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((28,\ 7^{14} \cdot 13^{14} ,\ ( \ : [9/2]^{14} ),\ 1 )\) |
Particular Values
| \(L(5)\) | \(\approx\) | \(15.09856434\) |
| \(L(\frac12)\) | \(\approx\) | \(15.09856434\) |
| \(L(\frac{11}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 7 | \( ( 1 - p^{4} T )^{14} \) |
| 13 | \( ( 1 + p^{4} T )^{14} \) | |
| good | 2 | \( 1 - 27 T + 1377 p T^{2} - 38805 p T^{3} + 1069281 p^{2} T^{4} - 28848717 p^{2} T^{5} + 73175451 p^{6} T^{6} - 7409915853 p^{4} T^{7} + 31144636437 p^{7} T^{8} - 741359351943 p^{7} T^{9} + 5474016847337 p^{9} T^{10} - 7721871998535 p^{13} T^{11} + 103866285371265 p^{14} T^{12} - 1115630113590351 p^{15} T^{13} + 7014783057588309 p^{17} T^{14} - 1115630113590351 p^{24} T^{15} + 103866285371265 p^{32} T^{16} - 7721871998535 p^{40} T^{17} + 5474016847337 p^{45} T^{18} - 741359351943 p^{52} T^{19} + 31144636437 p^{61} T^{20} - 7409915853 p^{67} T^{21} + 73175451 p^{78} T^{22} - 28848717 p^{83} T^{23} + 1069281 p^{92} T^{24} - 38805 p^{100} T^{25} + 1377 p^{109} T^{26} - 27 p^{117} T^{27} + p^{126} T^{28} \) |
| 3 | \( 1 - 163 T + 79001 T^{2} - 11743846 T^{3} + 3660497167 T^{4} - 172350814733 p T^{5} + 4601034182254 p^{3} T^{6} - 564335298695467 p^{3} T^{7} + 39426949915856885 p^{4} T^{8} - 148238475926086256 p^{7} T^{9} + 90926416539403733807 p^{6} T^{10} - \)\(25\!\cdots\!31\)\( p^{7} T^{11} + \)\(20\!\cdots\!31\)\( p^{10} T^{12} - \)\(45\!\cdots\!38\)\( p^{9} T^{13} + \)\(37\!\cdots\!68\)\( p^{10} T^{14} - \)\(45\!\cdots\!38\)\( p^{18} T^{15} + \)\(20\!\cdots\!31\)\( p^{28} T^{16} - \)\(25\!\cdots\!31\)\( p^{34} T^{17} + 90926416539403733807 p^{42} T^{18} - 148238475926086256 p^{52} T^{19} + 39426949915856885 p^{58} T^{20} - 564335298695467 p^{66} T^{21} + 4601034182254 p^{75} T^{22} - 172350814733 p^{82} T^{23} + 3660497167 p^{90} T^{24} - 11743846 p^{99} T^{25} + 79001 p^{108} T^{26} - 163 p^{117} T^{27} + p^{126} T^{28} \) | |
| 5 | \( 1 - 2964 T + 13953351 T^{2} - 31433062842 T^{3} + 91547325518177 T^{4} - 180392840875004034 T^{5} + \)\(41\!\cdots\!36\)\( T^{6} - \)\(73\!\cdots\!54\)\( T^{7} + \)\(14\!\cdots\!68\)\( T^{8} - \)\(23\!\cdots\!38\)\( T^{9} + \)\(82\!\cdots\!12\)\( p T^{10} - \)\(25\!\cdots\!22\)\( p^{2} T^{11} + \)\(79\!\cdots\!54\)\( p^{3} T^{12} - \)\(22\!\cdots\!26\)\( p^{4} T^{13} + \)\(66\!\cdots\!22\)\( p^{5} T^{14} - \)\(22\!\cdots\!26\)\( p^{13} T^{15} + \)\(79\!\cdots\!54\)\( p^{21} T^{16} - \)\(25\!\cdots\!22\)\( p^{29} T^{17} + \)\(82\!\cdots\!12\)\( p^{37} T^{18} - \)\(23\!\cdots\!38\)\( p^{45} T^{19} + \)\(14\!\cdots\!68\)\( p^{54} T^{20} - \)\(73\!\cdots\!54\)\( p^{63} T^{21} + \)\(41\!\cdots\!36\)\( p^{72} T^{22} - 180392840875004034 p^{81} T^{23} + 91547325518177 p^{90} T^{24} - 31433062842 p^{99} T^{25} + 13953351 p^{108} T^{26} - 2964 p^{117} T^{27} + p^{126} T^{28} \) | |
| 11 | \( 1 - 81825 T + 25097754017 T^{2} - 1688109128916810 T^{3} + \)\(29\!\cdots\!15\)\( T^{4} - \)\(16\!\cdots\!17\)\( T^{5} + \)\(21\!\cdots\!74\)\( T^{6} - \)\(10\!\cdots\!11\)\( T^{7} + \)\(11\!\cdots\!45\)\( T^{8} - \)\(49\!\cdots\!36\)\( T^{9} + \)\(45\!\cdots\!63\)\( T^{10} - \)\(17\!\cdots\!11\)\( T^{11} + \)\(14\!\cdots\!23\)\( T^{12} - \)\(50\!\cdots\!62\)\( T^{13} + \)\(37\!\cdots\!24\)\( T^{14} - \)\(50\!\cdots\!62\)\( p^{9} T^{15} + \)\(14\!\cdots\!23\)\( p^{18} T^{16} - \)\(17\!\cdots\!11\)\( p^{27} T^{17} + \)\(45\!\cdots\!63\)\( p^{36} T^{18} - \)\(49\!\cdots\!36\)\( p^{45} T^{19} + \)\(11\!\cdots\!45\)\( p^{54} T^{20} - \)\(10\!\cdots\!11\)\( p^{63} T^{21} + \)\(21\!\cdots\!74\)\( p^{72} T^{22} - \)\(16\!\cdots\!17\)\( p^{81} T^{23} + \)\(29\!\cdots\!15\)\( p^{90} T^{24} - 1688109128916810 p^{99} T^{25} + 25097754017 p^{108} T^{26} - 81825 p^{117} T^{27} + p^{126} T^{28} \) | |
| 17 | \( 1 + 44922 T + 806688183834 T^{2} + 99444772394050602 T^{3} + \)\(32\!\cdots\!75\)\( T^{4} + \)\(64\!\cdots\!52\)\( T^{5} + \)\(90\!\cdots\!48\)\( T^{6} + \)\(23\!\cdots\!08\)\( T^{7} + \)\(19\!\cdots\!53\)\( T^{8} + \)\(58\!\cdots\!70\)\( T^{9} + \)\(33\!\cdots\!50\)\( T^{10} + \)\(10\!\cdots\!66\)\( T^{11} + \)\(49\!\cdots\!71\)\( T^{12} + \)\(16\!\cdots\!12\)\( T^{13} + \)\(36\!\cdots\!84\)\( p T^{14} + \)\(16\!\cdots\!12\)\( p^{9} T^{15} + \)\(49\!\cdots\!71\)\( p^{18} T^{16} + \)\(10\!\cdots\!66\)\( p^{27} T^{17} + \)\(33\!\cdots\!50\)\( p^{36} T^{18} + \)\(58\!\cdots\!70\)\( p^{45} T^{19} + \)\(19\!\cdots\!53\)\( p^{54} T^{20} + \)\(23\!\cdots\!08\)\( p^{63} T^{21} + \)\(90\!\cdots\!48\)\( p^{72} T^{22} + \)\(64\!\cdots\!52\)\( p^{81} T^{23} + \)\(32\!\cdots\!75\)\( p^{90} T^{24} + 99444772394050602 p^{99} T^{25} + 806688183834 p^{108} T^{26} + 44922 p^{117} T^{27} + p^{126} T^{28} \) | |
| 19 | \( 1 - 171756 T + 1859405118175 T^{2} - 397639084389480270 T^{3} + \)\(18\!\cdots\!49\)\( T^{4} - \)\(42\!\cdots\!42\)\( T^{5} + \)\(13\!\cdots\!64\)\( T^{6} - \)\(29\!\cdots\!70\)\( T^{7} + \)\(76\!\cdots\!56\)\( T^{8} - \)\(84\!\cdots\!06\)\( p T^{9} + \)\(98\!\cdots\!80\)\( p^{2} T^{10} - \)\(36\!\cdots\!30\)\( p T^{11} + \)\(14\!\cdots\!06\)\( T^{12} - \)\(25\!\cdots\!82\)\( T^{13} + \)\(49\!\cdots\!30\)\( T^{14} - \)\(25\!\cdots\!82\)\( p^{9} T^{15} + \)\(14\!\cdots\!06\)\( p^{18} T^{16} - \)\(36\!\cdots\!30\)\( p^{28} T^{17} + \)\(98\!\cdots\!80\)\( p^{38} T^{18} - \)\(84\!\cdots\!06\)\( p^{46} T^{19} + \)\(76\!\cdots\!56\)\( p^{54} T^{20} - \)\(29\!\cdots\!70\)\( p^{63} T^{21} + \)\(13\!\cdots\!64\)\( p^{72} T^{22} - \)\(42\!\cdots\!42\)\( p^{81} T^{23} + \)\(18\!\cdots\!49\)\( p^{90} T^{24} - 397639084389480270 p^{99} T^{25} + 1859405118175 p^{108} T^{26} - 171756 p^{117} T^{27} + p^{126} T^{28} \) | |
| 23 | \( 1 - 1930479 T + 15826080814344 T^{2} - 31330957167012015231 T^{3} + \)\(12\!\cdots\!70\)\( T^{4} - \)\(24\!\cdots\!37\)\( T^{5} + \)\(71\!\cdots\!18\)\( T^{6} - \)\(12\!\cdots\!55\)\( T^{7} + \)\(28\!\cdots\!71\)\( T^{8} - \)\(46\!\cdots\!80\)\( T^{9} + \)\(90\!\cdots\!04\)\( T^{10} - \)\(13\!\cdots\!48\)\( T^{11} + \)\(22\!\cdots\!35\)\( T^{12} - \)\(29\!\cdots\!49\)\( T^{13} + \)\(45\!\cdots\!98\)\( T^{14} - \)\(29\!\cdots\!49\)\( p^{9} T^{15} + \)\(22\!\cdots\!35\)\( p^{18} T^{16} - \)\(13\!\cdots\!48\)\( p^{27} T^{17} + \)\(90\!\cdots\!04\)\( p^{36} T^{18} - \)\(46\!\cdots\!80\)\( p^{45} T^{19} + \)\(28\!\cdots\!71\)\( p^{54} T^{20} - \)\(12\!\cdots\!55\)\( p^{63} T^{21} + \)\(71\!\cdots\!18\)\( p^{72} T^{22} - \)\(24\!\cdots\!37\)\( p^{81} T^{23} + \)\(12\!\cdots\!70\)\( p^{90} T^{24} - 31330957167012015231 p^{99} T^{25} + 15826080814344 p^{108} T^{26} - 1930479 p^{117} T^{27} + p^{126} T^{28} \) | |
| 29 | \( 1 + 3799608 T + 3621785345355 p T^{2} + \)\(39\!\cdots\!82\)\( T^{3} + \)\(53\!\cdots\!69\)\( T^{4} + \)\(20\!\cdots\!18\)\( T^{5} + \)\(17\!\cdots\!68\)\( T^{6} + \)\(24\!\cdots\!38\)\( p T^{7} + \)\(42\!\cdots\!68\)\( T^{8} + \)\(18\!\cdots\!34\)\( T^{9} + \)\(85\!\cdots\!16\)\( T^{10} + \)\(36\!\cdots\!78\)\( T^{11} + \)\(14\!\cdots\!22\)\( T^{12} + \)\(60\!\cdots\!98\)\( T^{13} + \)\(22\!\cdots\!10\)\( T^{14} + \)\(60\!\cdots\!98\)\( p^{9} T^{15} + \)\(14\!\cdots\!22\)\( p^{18} T^{16} + \)\(36\!\cdots\!78\)\( p^{27} T^{17} + \)\(85\!\cdots\!16\)\( p^{36} T^{18} + \)\(18\!\cdots\!34\)\( p^{45} T^{19} + \)\(42\!\cdots\!68\)\( p^{54} T^{20} + \)\(24\!\cdots\!38\)\( p^{64} T^{21} + \)\(17\!\cdots\!68\)\( p^{72} T^{22} + \)\(20\!\cdots\!18\)\( p^{81} T^{23} + \)\(53\!\cdots\!69\)\( p^{90} T^{24} + \)\(39\!\cdots\!82\)\( p^{99} T^{25} + 3621785345355 p^{109} T^{26} + 3799608 p^{117} T^{27} + p^{126} T^{28} \) | |
| 31 | \( 1 + 4392203 T + 89558033027720 T^{2} + \)\(48\!\cdots\!71\)\( T^{3} + \)\(54\!\cdots\!22\)\( T^{4} + \)\(23\!\cdots\!61\)\( T^{5} + \)\(24\!\cdots\!70\)\( T^{6} + \)\(90\!\cdots\!95\)\( T^{7} + \)\(86\!\cdots\!39\)\( T^{8} + \)\(30\!\cdots\!68\)\( T^{9} + \)\(29\!\cdots\!40\)\( T^{10} + \)\(10\!\cdots\!40\)\( T^{11} + \)\(93\!\cdots\!03\)\( T^{12} + \)\(31\!\cdots\!09\)\( T^{13} + \)\(26\!\cdots\!42\)\( T^{14} + \)\(31\!\cdots\!09\)\( p^{9} T^{15} + \)\(93\!\cdots\!03\)\( p^{18} T^{16} + \)\(10\!\cdots\!40\)\( p^{27} T^{17} + \)\(29\!\cdots\!40\)\( p^{36} T^{18} + \)\(30\!\cdots\!68\)\( p^{45} T^{19} + \)\(86\!\cdots\!39\)\( p^{54} T^{20} + \)\(90\!\cdots\!95\)\( p^{63} T^{21} + \)\(24\!\cdots\!70\)\( p^{72} T^{22} + \)\(23\!\cdots\!61\)\( p^{81} T^{23} + \)\(54\!\cdots\!22\)\( p^{90} T^{24} + \)\(48\!\cdots\!71\)\( p^{99} T^{25} + 89558033027720 p^{108} T^{26} + 4392203 p^{117} T^{27} + p^{126} T^{28} \) | |
| 37 | \( 1 - 29198909 T + 1147458784003631 T^{2} - \)\(24\!\cdots\!44\)\( T^{3} + \)\(16\!\cdots\!29\)\( p T^{4} - \)\(11\!\cdots\!99\)\( T^{5} + \)\(59\!\cdots\!18\)\( p T^{6} - \)\(34\!\cdots\!91\)\( T^{7} + \)\(58\!\cdots\!91\)\( T^{8} - \)\(81\!\cdots\!06\)\( T^{9} + \)\(12\!\cdots\!13\)\( T^{10} - \)\(15\!\cdots\!81\)\( T^{11} + \)\(20\!\cdots\!67\)\( T^{12} - \)\(23\!\cdots\!06\)\( T^{13} + \)\(29\!\cdots\!68\)\( T^{14} - \)\(23\!\cdots\!06\)\( p^{9} T^{15} + \)\(20\!\cdots\!67\)\( p^{18} T^{16} - \)\(15\!\cdots\!81\)\( p^{27} T^{17} + \)\(12\!\cdots\!13\)\( p^{36} T^{18} - \)\(81\!\cdots\!06\)\( p^{45} T^{19} + \)\(58\!\cdots\!91\)\( p^{54} T^{20} - \)\(34\!\cdots\!91\)\( p^{63} T^{21} + \)\(59\!\cdots\!18\)\( p^{73} T^{22} - \)\(11\!\cdots\!99\)\( p^{81} T^{23} + \)\(16\!\cdots\!29\)\( p^{91} T^{24} - \)\(24\!\cdots\!44\)\( p^{99} T^{25} + 1147458784003631 p^{108} T^{26} - 29198909 p^{117} T^{27} + p^{126} T^{28} \) | |
| 41 | \( 1 - 48410973 T + 3181898094770747 T^{2} - \)\(12\!\cdots\!52\)\( T^{3} + \)\(49\!\cdots\!69\)\( T^{4} - \)\(15\!\cdots\!91\)\( T^{5} + \)\(49\!\cdots\!58\)\( T^{6} - \)\(13\!\cdots\!19\)\( T^{7} + \)\(36\!\cdots\!99\)\( T^{8} - \)\(86\!\cdots\!98\)\( T^{9} + \)\(20\!\cdots\!97\)\( T^{10} - \)\(43\!\cdots\!09\)\( T^{11} + \)\(91\!\cdots\!39\)\( T^{12} - \)\(17\!\cdots\!90\)\( T^{13} + \)\(33\!\cdots\!00\)\( T^{14} - \)\(17\!\cdots\!90\)\( p^{9} T^{15} + \)\(91\!\cdots\!39\)\( p^{18} T^{16} - \)\(43\!\cdots\!09\)\( p^{27} T^{17} + \)\(20\!\cdots\!97\)\( p^{36} T^{18} - \)\(86\!\cdots\!98\)\( p^{45} T^{19} + \)\(36\!\cdots\!99\)\( p^{54} T^{20} - \)\(13\!\cdots\!19\)\( p^{63} T^{21} + \)\(49\!\cdots\!58\)\( p^{72} T^{22} - \)\(15\!\cdots\!91\)\( p^{81} T^{23} + \)\(49\!\cdots\!69\)\( p^{90} T^{24} - \)\(12\!\cdots\!52\)\( p^{99} T^{25} + 3181898094770747 p^{108} T^{26} - 48410973 p^{117} T^{27} + p^{126} T^{28} \) | |
| 43 | \( 1 - 52650242 T + 5406876927964247 T^{2} - \)\(22\!\cdots\!18\)\( T^{3} + \)\(13\!\cdots\!09\)\( T^{4} - \)\(48\!\cdots\!96\)\( T^{5} + \)\(21\!\cdots\!88\)\( T^{6} - \)\(66\!\cdots\!00\)\( T^{7} + \)\(24\!\cdots\!12\)\( T^{8} - \)\(66\!\cdots\!84\)\( T^{9} + \)\(21\!\cdots\!24\)\( T^{10} - \)\(52\!\cdots\!76\)\( T^{11} + \)\(14\!\cdots\!22\)\( T^{12} - \)\(32\!\cdots\!20\)\( T^{13} + \)\(81\!\cdots\!14\)\( T^{14} - \)\(32\!\cdots\!20\)\( p^{9} T^{15} + \)\(14\!\cdots\!22\)\( p^{18} T^{16} - \)\(52\!\cdots\!76\)\( p^{27} T^{17} + \)\(21\!\cdots\!24\)\( p^{36} T^{18} - \)\(66\!\cdots\!84\)\( p^{45} T^{19} + \)\(24\!\cdots\!12\)\( p^{54} T^{20} - \)\(66\!\cdots\!00\)\( p^{63} T^{21} + \)\(21\!\cdots\!88\)\( p^{72} T^{22} - \)\(48\!\cdots\!96\)\( p^{81} T^{23} + \)\(13\!\cdots\!09\)\( p^{90} T^{24} - \)\(22\!\cdots\!18\)\( p^{99} T^{25} + 5406876927964247 p^{108} T^{26} - 52650242 p^{117} T^{27} + p^{126} T^{28} \) | |
| 47 | \( 1 - 160580841 T + 21409825633626140 T^{2} - \)\(20\!\cdots\!01\)\( T^{3} + \)\(17\!\cdots\!26\)\( T^{4} - \)\(12\!\cdots\!99\)\( T^{5} + \)\(79\!\cdots\!70\)\( T^{6} - \)\(46\!\cdots\!85\)\( T^{7} + \)\(24\!\cdots\!15\)\( T^{8} - \)\(12\!\cdots\!88\)\( T^{9} + \)\(54\!\cdots\!76\)\( T^{10} - \)\(23\!\cdots\!32\)\( T^{11} + \)\(90\!\cdots\!51\)\( T^{12} - \)\(33\!\cdots\!07\)\( T^{13} + \)\(11\!\cdots\!30\)\( T^{14} - \)\(33\!\cdots\!07\)\( p^{9} T^{15} + \)\(90\!\cdots\!51\)\( p^{18} T^{16} - \)\(23\!\cdots\!32\)\( p^{27} T^{17} + \)\(54\!\cdots\!76\)\( p^{36} T^{18} - \)\(12\!\cdots\!88\)\( p^{45} T^{19} + \)\(24\!\cdots\!15\)\( p^{54} T^{20} - \)\(46\!\cdots\!85\)\( p^{63} T^{21} + \)\(79\!\cdots\!70\)\( p^{72} T^{22} - \)\(12\!\cdots\!99\)\( p^{81} T^{23} + \)\(17\!\cdots\!26\)\( p^{90} T^{24} - \)\(20\!\cdots\!01\)\( p^{99} T^{25} + 21409825633626140 p^{108} T^{26} - 160580841 p^{117} T^{27} + p^{126} T^{28} \) | |
| 53 | \( 1 - 80753796 T + 30677229399252111 T^{2} - \)\(37\!\cdots\!86\)\( p T^{3} + \)\(43\!\cdots\!97\)\( T^{4} - \)\(23\!\cdots\!30\)\( T^{5} + \)\(38\!\cdots\!96\)\( T^{6} - \)\(17\!\cdots\!82\)\( T^{7} + \)\(24\!\cdots\!68\)\( T^{8} - \)\(94\!\cdots\!66\)\( T^{9} + \)\(12\!\cdots\!56\)\( T^{10} - \)\(41\!\cdots\!62\)\( T^{11} + \)\(51\!\cdots\!90\)\( T^{12} - \)\(15\!\cdots\!30\)\( T^{13} + \)\(18\!\cdots\!66\)\( T^{14} - \)\(15\!\cdots\!30\)\( p^{9} T^{15} + \)\(51\!\cdots\!90\)\( p^{18} T^{16} - \)\(41\!\cdots\!62\)\( p^{27} T^{17} + \)\(12\!\cdots\!56\)\( p^{36} T^{18} - \)\(94\!\cdots\!66\)\( p^{45} T^{19} + \)\(24\!\cdots\!68\)\( p^{54} T^{20} - \)\(17\!\cdots\!82\)\( p^{63} T^{21} + \)\(38\!\cdots\!96\)\( p^{72} T^{22} - \)\(23\!\cdots\!30\)\( p^{81} T^{23} + \)\(43\!\cdots\!97\)\( p^{90} T^{24} - \)\(37\!\cdots\!86\)\( p^{100} T^{25} + 30677229399252111 p^{108} T^{26} - 80753796 p^{117} T^{27} + p^{126} T^{28} \) | |
| 59 | \( 1 - 442445502 T + 129501752167165894 T^{2} - \)\(27\!\cdots\!94\)\( T^{3} + \)\(50\!\cdots\!91\)\( T^{4} - \)\(79\!\cdots\!40\)\( T^{5} + \)\(11\!\cdots\!40\)\( T^{6} - \)\(15\!\cdots\!36\)\( T^{7} + \)\(19\!\cdots\!97\)\( T^{8} - \)\(23\!\cdots\!70\)\( T^{9} + \)\(26\!\cdots\!38\)\( T^{10} - \)\(28\!\cdots\!38\)\( T^{11} + \)\(29\!\cdots\!39\)\( T^{12} - \)\(28\!\cdots\!64\)\( T^{13} + \)\(27\!\cdots\!52\)\( T^{14} - \)\(28\!\cdots\!64\)\( p^{9} T^{15} + \)\(29\!\cdots\!39\)\( p^{18} T^{16} - \)\(28\!\cdots\!38\)\( p^{27} T^{17} + \)\(26\!\cdots\!38\)\( p^{36} T^{18} - \)\(23\!\cdots\!70\)\( p^{45} T^{19} + \)\(19\!\cdots\!97\)\( p^{54} T^{20} - \)\(15\!\cdots\!36\)\( p^{63} T^{21} + \)\(11\!\cdots\!40\)\( p^{72} T^{22} - \)\(79\!\cdots\!40\)\( p^{81} T^{23} + \)\(50\!\cdots\!91\)\( p^{90} T^{24} - \)\(27\!\cdots\!94\)\( p^{99} T^{25} + 129501752167165894 p^{108} T^{26} - 442445502 p^{117} T^{27} + p^{126} T^{28} \) | |
| 61 | \( 1 - 270199089 T + 113884524810665767 T^{2} - \)\(22\!\cdots\!64\)\( T^{3} + \)\(54\!\cdots\!61\)\( T^{4} - \)\(82\!\cdots\!59\)\( T^{5} + \)\(15\!\cdots\!38\)\( T^{6} - \)\(19\!\cdots\!35\)\( T^{7} + \)\(29\!\cdots\!07\)\( T^{8} - \)\(33\!\cdots\!38\)\( T^{9} + \)\(49\!\cdots\!81\)\( T^{10} - \)\(52\!\cdots\!37\)\( T^{11} + \)\(71\!\cdots\!19\)\( T^{12} - \)\(72\!\cdots\!54\)\( T^{13} + \)\(91\!\cdots\!20\)\( T^{14} - \)\(72\!\cdots\!54\)\( p^{9} T^{15} + \)\(71\!\cdots\!19\)\( p^{18} T^{16} - \)\(52\!\cdots\!37\)\( p^{27} T^{17} + \)\(49\!\cdots\!81\)\( p^{36} T^{18} - \)\(33\!\cdots\!38\)\( p^{45} T^{19} + \)\(29\!\cdots\!07\)\( p^{54} T^{20} - \)\(19\!\cdots\!35\)\( p^{63} T^{21} + \)\(15\!\cdots\!38\)\( p^{72} T^{22} - \)\(82\!\cdots\!59\)\( p^{81} T^{23} + \)\(54\!\cdots\!61\)\( p^{90} T^{24} - \)\(22\!\cdots\!64\)\( p^{99} T^{25} + 113884524810665767 p^{108} T^{26} - 270199089 p^{117} T^{27} + p^{126} T^{28} \) | |
| 67 | \( 1 - 92500909 T + 232124556425133785 T^{2} - \)\(20\!\cdots\!42\)\( T^{3} + \)\(26\!\cdots\!91\)\( T^{4} - \)\(23\!\cdots\!21\)\( T^{5} + \)\(20\!\cdots\!26\)\( T^{6} - \)\(16\!\cdots\!71\)\( T^{7} + \)\(11\!\cdots\!21\)\( T^{8} - \)\(91\!\cdots\!88\)\( T^{9} + \)\(49\!\cdots\!15\)\( T^{10} - \)\(38\!\cdots\!27\)\( T^{11} + \)\(17\!\cdots\!39\)\( T^{12} - \)\(12\!\cdots\!14\)\( T^{13} + \)\(53\!\cdots\!88\)\( T^{14} - \)\(12\!\cdots\!14\)\( p^{9} T^{15} + \)\(17\!\cdots\!39\)\( p^{18} T^{16} - \)\(38\!\cdots\!27\)\( p^{27} T^{17} + \)\(49\!\cdots\!15\)\( p^{36} T^{18} - \)\(91\!\cdots\!88\)\( p^{45} T^{19} + \)\(11\!\cdots\!21\)\( p^{54} T^{20} - \)\(16\!\cdots\!71\)\( p^{63} T^{21} + \)\(20\!\cdots\!26\)\( p^{72} T^{22} - \)\(23\!\cdots\!21\)\( p^{81} T^{23} + \)\(26\!\cdots\!91\)\( p^{90} T^{24} - \)\(20\!\cdots\!42\)\( p^{99} T^{25} + 232124556425133785 p^{108} T^{26} - 92500909 p^{117} T^{27} + p^{126} T^{28} \) | |
| 71 | \( 1 - 84383796 T + 363035507792751910 T^{2} - \)\(30\!\cdots\!40\)\( T^{3} + \)\(59\!\cdots\!39\)\( T^{4} + \)\(49\!\cdots\!32\)\( T^{5} + \)\(63\!\cdots\!92\)\( T^{6} + \)\(10\!\cdots\!76\)\( T^{7} + \)\(52\!\cdots\!85\)\( T^{8} + \)\(10\!\cdots\!48\)\( T^{9} + \)\(37\!\cdots\!46\)\( T^{10} + \)\(79\!\cdots\!44\)\( T^{11} + \)\(23\!\cdots\!79\)\( T^{12} + \)\(44\!\cdots\!60\)\( T^{13} + \)\(11\!\cdots\!28\)\( T^{14} + \)\(44\!\cdots\!60\)\( p^{9} T^{15} + \)\(23\!\cdots\!79\)\( p^{18} T^{16} + \)\(79\!\cdots\!44\)\( p^{27} T^{17} + \)\(37\!\cdots\!46\)\( p^{36} T^{18} + \)\(10\!\cdots\!48\)\( p^{45} T^{19} + \)\(52\!\cdots\!85\)\( p^{54} T^{20} + \)\(10\!\cdots\!76\)\( p^{63} T^{21} + \)\(63\!\cdots\!92\)\( p^{72} T^{22} + \)\(49\!\cdots\!32\)\( p^{81} T^{23} + \)\(59\!\cdots\!39\)\( p^{90} T^{24} - \)\(30\!\cdots\!40\)\( p^{99} T^{25} + 363035507792751910 p^{108} T^{26} - 84383796 p^{117} T^{27} + p^{126} T^{28} \) | |
| 73 | \( 1 - 5031155 p T + 344491973633637818 T^{2} - \)\(64\!\cdots\!37\)\( T^{3} + \)\(39\!\cdots\!78\)\( T^{4} - \)\(76\!\cdots\!09\)\( T^{5} + \)\(20\!\cdots\!08\)\( T^{6} + \)\(80\!\cdots\!31\)\( p T^{7} + \)\(82\!\cdots\!75\)\( T^{8} + \)\(61\!\cdots\!48\)\( T^{9} + \)\(87\!\cdots\!36\)\( T^{10} + \)\(34\!\cdots\!32\)\( T^{11} + \)\(95\!\cdots\!59\)\( T^{12} + \)\(16\!\cdots\!87\)\( T^{13} + \)\(67\!\cdots\!36\)\( T^{14} + \)\(16\!\cdots\!87\)\( p^{9} T^{15} + \)\(95\!\cdots\!59\)\( p^{18} T^{16} + \)\(34\!\cdots\!32\)\( p^{27} T^{17} + \)\(87\!\cdots\!36\)\( p^{36} T^{18} + \)\(61\!\cdots\!48\)\( p^{45} T^{19} + \)\(82\!\cdots\!75\)\( p^{54} T^{20} + \)\(80\!\cdots\!31\)\( p^{64} T^{21} + \)\(20\!\cdots\!08\)\( p^{72} T^{22} - \)\(76\!\cdots\!09\)\( p^{81} T^{23} + \)\(39\!\cdots\!78\)\( p^{90} T^{24} - \)\(64\!\cdots\!37\)\( p^{99} T^{25} + 344491973633637818 p^{108} T^{26} - 5031155 p^{118} T^{27} + p^{126} T^{28} \) | |
| 79 | \( 1 - 434861545 T + 834393196482950884 T^{2} - \)\(38\!\cdots\!61\)\( T^{3} + \)\(34\!\cdots\!38\)\( T^{4} - \)\(15\!\cdots\!55\)\( T^{5} + \)\(94\!\cdots\!62\)\( T^{6} - \)\(39\!\cdots\!69\)\( T^{7} + \)\(18\!\cdots\!15\)\( T^{8} - \)\(69\!\cdots\!84\)\( T^{9} + \)\(26\!\cdots\!80\)\( T^{10} - \)\(92\!\cdots\!40\)\( T^{11} + \)\(32\!\cdots\!35\)\( T^{12} - \)\(11\!\cdots\!55\)\( T^{13} + \)\(37\!\cdots\!18\)\( T^{14} - \)\(11\!\cdots\!55\)\( p^{9} T^{15} + \)\(32\!\cdots\!35\)\( p^{18} T^{16} - \)\(92\!\cdots\!40\)\( p^{27} T^{17} + \)\(26\!\cdots\!80\)\( p^{36} T^{18} - \)\(69\!\cdots\!84\)\( p^{45} T^{19} + \)\(18\!\cdots\!15\)\( p^{54} T^{20} - \)\(39\!\cdots\!69\)\( p^{63} T^{21} + \)\(94\!\cdots\!62\)\( p^{72} T^{22} - \)\(15\!\cdots\!55\)\( p^{81} T^{23} + \)\(34\!\cdots\!38\)\( p^{90} T^{24} - \)\(38\!\cdots\!61\)\( p^{99} T^{25} + 834393196482950884 p^{108} T^{26} - 434861545 p^{117} T^{27} + p^{126} T^{28} \) | |
| 83 | \( 1 - 1013603934 T + 2134247149602395047 T^{2} - \)\(17\!\cdots\!54\)\( T^{3} + \)\(21\!\cdots\!85\)\( T^{4} - \)\(15\!\cdots\!28\)\( T^{5} + \)\(13\!\cdots\!00\)\( T^{6} - \)\(82\!\cdots\!72\)\( T^{7} + \)\(57\!\cdots\!64\)\( T^{8} - \)\(32\!\cdots\!04\)\( T^{9} + \)\(19\!\cdots\!72\)\( T^{10} - \)\(97\!\cdots\!24\)\( T^{11} + \)\(49\!\cdots\!58\)\( T^{12} - \)\(22\!\cdots\!68\)\( T^{13} + \)\(10\!\cdots\!58\)\( T^{14} - \)\(22\!\cdots\!68\)\( p^{9} T^{15} + \)\(49\!\cdots\!58\)\( p^{18} T^{16} - \)\(97\!\cdots\!24\)\( p^{27} T^{17} + \)\(19\!\cdots\!72\)\( p^{36} T^{18} - \)\(32\!\cdots\!04\)\( p^{45} T^{19} + \)\(57\!\cdots\!64\)\( p^{54} T^{20} - \)\(82\!\cdots\!72\)\( p^{63} T^{21} + \)\(13\!\cdots\!00\)\( p^{72} T^{22} - \)\(15\!\cdots\!28\)\( p^{81} T^{23} + \)\(21\!\cdots\!85\)\( p^{90} T^{24} - \)\(17\!\cdots\!54\)\( p^{99} T^{25} + 2134247149602395047 p^{108} T^{26} - 1013603934 p^{117} T^{27} + p^{126} T^{28} \) | |
| 89 | \( 1 - 1069739706 T + 3907904573617261371 T^{2} - \)\(36\!\cdots\!02\)\( T^{3} + \)\(73\!\cdots\!45\)\( T^{4} - \)\(67\!\cdots\!24\)\( p T^{5} + \)\(88\!\cdots\!36\)\( T^{6} - \)\(64\!\cdots\!04\)\( T^{7} + \)\(75\!\cdots\!36\)\( T^{8} - \)\(49\!\cdots\!68\)\( T^{9} + \)\(48\!\cdots\!32\)\( T^{10} - \)\(28\!\cdots\!84\)\( T^{11} + \)\(24\!\cdots\!58\)\( T^{12} - \)\(12\!\cdots\!88\)\( T^{13} + \)\(94\!\cdots\!18\)\( T^{14} - \)\(12\!\cdots\!88\)\( p^{9} T^{15} + \)\(24\!\cdots\!58\)\( p^{18} T^{16} - \)\(28\!\cdots\!84\)\( p^{27} T^{17} + \)\(48\!\cdots\!32\)\( p^{36} T^{18} - \)\(49\!\cdots\!68\)\( p^{45} T^{19} + \)\(75\!\cdots\!36\)\( p^{54} T^{20} - \)\(64\!\cdots\!04\)\( p^{63} T^{21} + \)\(88\!\cdots\!36\)\( p^{72} T^{22} - \)\(67\!\cdots\!24\)\( p^{82} T^{23} + \)\(73\!\cdots\!45\)\( p^{90} T^{24} - \)\(36\!\cdots\!02\)\( p^{99} T^{25} + 3907904573617261371 p^{108} T^{26} - 1069739706 p^{117} T^{27} + p^{126} T^{28} \) | |
| 97 | \( 1 - 2839636281 T + 5564930172861095674 T^{2} - \)\(91\!\cdots\!51\)\( T^{3} + \)\(14\!\cdots\!34\)\( T^{4} - \)\(21\!\cdots\!51\)\( T^{5} + \)\(27\!\cdots\!64\)\( T^{6} - \)\(33\!\cdots\!87\)\( T^{7} + \)\(39\!\cdots\!87\)\( T^{8} - \)\(43\!\cdots\!40\)\( T^{9} + \)\(45\!\cdots\!48\)\( T^{10} - \)\(45\!\cdots\!92\)\( T^{11} + \)\(43\!\cdots\!19\)\( T^{12} - \)\(40\!\cdots\!43\)\( T^{13} + \)\(35\!\cdots\!44\)\( T^{14} - \)\(40\!\cdots\!43\)\( p^{9} T^{15} + \)\(43\!\cdots\!19\)\( p^{18} T^{16} - \)\(45\!\cdots\!92\)\( p^{27} T^{17} + \)\(45\!\cdots\!48\)\( p^{36} T^{18} - \)\(43\!\cdots\!40\)\( p^{45} T^{19} + \)\(39\!\cdots\!87\)\( p^{54} T^{20} - \)\(33\!\cdots\!87\)\( p^{63} T^{21} + \)\(27\!\cdots\!64\)\( p^{72} T^{22} - \)\(21\!\cdots\!51\)\( p^{81} T^{23} + \)\(14\!\cdots\!34\)\( p^{90} T^{24} - \)\(91\!\cdots\!51\)\( p^{99} T^{25} + 5564930172861095674 p^{108} T^{26} - 2839636281 p^{117} T^{27} + p^{126} T^{28} \) | |
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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
−2.65743615732589159037877200561, −2.64620203620183362337787860214, −2.59651457854491211998585178637, −2.29944392306766621666166347430, −2.25357487095976324275316674942, −2.22339962827378184836564788266, −2.18335528961099792398081017143, −2.09880612604627666476616484851, −2.05686895660745202905172783682, −1.96854342225959871955601401414, −1.66059908064621886440916643509, −1.65713004771621425858433958945, −1.49651914945754100230405644580, −1.42182065475238225539366024584, −1.18839382110036102510958961076, −0.994983507711497648269126682747, −0.871451186995177830713079223319, −0.860316376261049036105543361877, −0.71565627793444312550521160865, −0.59147610864647509971680316337, −0.56222213535582695671047548599, −0.53458370226179092235280719196, −0.42412711234879009744327072273, −0.40500036780705057180151778646, −0.03700038668019936861904113773, 0.03700038668019936861904113773, 0.40500036780705057180151778646, 0.42412711234879009744327072273, 0.53458370226179092235280719196, 0.56222213535582695671047548599, 0.59147610864647509971680316337, 0.71565627793444312550521160865, 0.860316376261049036105543361877, 0.871451186995177830713079223319, 0.994983507711497648269126682747, 1.18839382110036102510958961076, 1.42182065475238225539366024584, 1.49651914945754100230405644580, 1.65713004771621425858433958945, 1.66059908064621886440916643509, 1.96854342225959871955601401414, 2.05686895660745202905172783682, 2.09880612604627666476616484851, 2.18335528961099792398081017143, 2.22339962827378184836564788266, 2.25357487095976324275316674942, 2.29944392306766621666166347430, 2.59651457854491211998585178637, 2.64620203620183362337787860214, 2.65743615732589159037877200561
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.