Dirichlet series
| L(s) = 1 | − 1.22e4·2-s + 4.10e4·3-s + 8.17e7·4-s + 3.20e7·5-s − 5.04e8·6-s + 4.42e8·7-s − 3.90e11·8-s − 3.90e10·9-s − 3.93e11·10-s − 7.56e10·11-s + 3.35e12·12-s + 1.10e12·13-s − 5.43e12·14-s + 1.31e12·15-s + 1.50e15·16-s + 2.13e13·17-s + 4.79e14·18-s − 6.41e13·19-s + 2.61e15·20-s + 1.81e13·21-s + 9.29e14·22-s − 9.24e13·23-s − 1.60e16·24-s − 1.82e15·25-s − 1.35e16·26-s − 1.63e15·27-s + 3.61e16·28-s + ⋯ |
| L(s) = 1 | − 8.48·2-s + 0.401·3-s + 39·4-s + 1.46·5-s − 3.40·6-s + 0.591·7-s − 128.·8-s − 3.72·9-s − 12.4·10-s − 0.878·11-s + 15.6·12-s + 2.22·13-s − 5.02·14-s + 0.588·15-s + 341.·16-s + 2.56·17-s + 31.6·18-s − 2.39·19-s + 57.1·20-s + 0.237·21-s + 7.45·22-s − 0.465·23-s − 51.6·24-s − 3.82·25-s − 18.8·26-s − 1.52·27-s + 23.0·28-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 29^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(22-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 29^{12}\right)^{s/2} \, \Gamma_{\C}(s+21/2)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(24\) |
| Conductor: | \(2^{12} \cdot 29^{12}\) |
| Sign: | $1$ |
| Analytic conductor: | \(3.29075\times 10^{26}\) |
| Root analytic conductor: | \(12.7317\) |
| Motivic weight: | \(21\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((24,\ 2^{12} \cdot 29^{12} ,\ ( \ : [21/2]^{12} ),\ 1 )\) |
Particular Values
| \(L(11)\) | \(\approx\) | \(0.1330337749\) |
| \(L(\frac12)\) | \(\approx\) | \(0.1330337749\) |
| \(L(\frac{23}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( ( 1 + p^{10} T )^{12} \) |
| 29 | \( ( 1 - p^{10} T )^{12} \) | |
| good | 3 | \( 1 - 41066 T + 13563477733 p T^{2} - 60599116894114 p^{3} T^{3} + 39072598498982837594 p^{3} T^{4} - \)\(50\!\cdots\!82\)\( p^{4} T^{5} + \)\(10\!\cdots\!99\)\( p^{9} T^{6} - \)\(40\!\cdots\!38\)\( p^{11} T^{7} + \)\(20\!\cdots\!96\)\( p^{13} T^{8} - \)\(82\!\cdots\!02\)\( p^{17} T^{9} + \)\(13\!\cdots\!99\)\( p^{22} T^{10} - \)\(15\!\cdots\!22\)\( p^{25} T^{11} + \)\(70\!\cdots\!00\)\( p^{29} T^{12} - \)\(15\!\cdots\!22\)\( p^{46} T^{13} + \)\(13\!\cdots\!99\)\( p^{64} T^{14} - \)\(82\!\cdots\!02\)\( p^{80} T^{15} + \)\(20\!\cdots\!96\)\( p^{97} T^{16} - \)\(40\!\cdots\!38\)\( p^{116} T^{17} + \)\(10\!\cdots\!99\)\( p^{135} T^{18} - \)\(50\!\cdots\!82\)\( p^{151} T^{19} + 39072598498982837594 p^{171} T^{20} - 60599116894114 p^{192} T^{21} + 13563477733 p^{211} T^{22} - 41066 p^{231} T^{23} + p^{252} T^{24} \) |
| 5 | \( 1 - 32000498 T + 2847278091220413 T^{2} - \)\(79\!\cdots\!86\)\( T^{3} + \)\(76\!\cdots\!34\)\( p T^{4} - \)\(69\!\cdots\!74\)\( p^{3} T^{5} + \)\(97\!\cdots\!27\)\( p^{5} T^{6} - \)\(67\!\cdots\!14\)\( p^{7} T^{7} + \)\(79\!\cdots\!88\)\( p^{9} T^{8} - \)\(35\!\cdots\!98\)\( p^{11} T^{9} + \)\(43\!\cdots\!17\)\( p^{13} T^{10} - \)\(90\!\cdots\!26\)\( p^{15} T^{11} + \)\(24\!\cdots\!08\)\( p^{17} T^{12} - \)\(90\!\cdots\!26\)\( p^{36} T^{13} + \)\(43\!\cdots\!17\)\( p^{55} T^{14} - \)\(35\!\cdots\!98\)\( p^{74} T^{15} + \)\(79\!\cdots\!88\)\( p^{93} T^{16} - \)\(67\!\cdots\!14\)\( p^{112} T^{17} + \)\(97\!\cdots\!27\)\( p^{131} T^{18} - \)\(69\!\cdots\!74\)\( p^{150} T^{19} + \)\(76\!\cdots\!34\)\( p^{169} T^{20} - \)\(79\!\cdots\!86\)\( p^{189} T^{21} + 2847278091220413 p^{210} T^{22} - 32000498 p^{231} T^{23} + p^{252} T^{24} \) | |
| 7 | \( 1 - 442380608 T + 2580137941900504484 T^{2} - \)\(19\!\cdots\!24\)\( p T^{3} + \)\(59\!\cdots\!26\)\( p^{2} T^{4} - \)\(54\!\cdots\!04\)\( p^{3} T^{5} + \)\(86\!\cdots\!20\)\( p^{4} T^{6} - \)\(99\!\cdots\!48\)\( p^{5} T^{7} + \)\(10\!\cdots\!67\)\( p^{6} T^{8} - \)\(12\!\cdots\!96\)\( p^{7} T^{9} + \)\(14\!\cdots\!60\)\( p^{8} T^{10} - \)\(20\!\cdots\!28\)\( p^{10} T^{11} + \)\(18\!\cdots\!24\)\( p^{10} T^{12} - \)\(20\!\cdots\!28\)\( p^{31} T^{13} + \)\(14\!\cdots\!60\)\( p^{50} T^{14} - \)\(12\!\cdots\!96\)\( p^{70} T^{15} + \)\(10\!\cdots\!67\)\( p^{90} T^{16} - \)\(99\!\cdots\!48\)\( p^{110} T^{17} + \)\(86\!\cdots\!20\)\( p^{130} T^{18} - \)\(54\!\cdots\!04\)\( p^{150} T^{19} + \)\(59\!\cdots\!26\)\( p^{170} T^{20} - \)\(19\!\cdots\!24\)\( p^{190} T^{21} + 2580137941900504484 p^{210} T^{22} - 442380608 p^{231} T^{23} + p^{252} T^{24} \) | |
| 11 | \( 1 + 75612722022 T + \)\(51\!\cdots\!75\)\( T^{2} + \)\(27\!\cdots\!42\)\( T^{3} + \)\(12\!\cdots\!98\)\( T^{4} + \)\(47\!\cdots\!86\)\( T^{5} + \)\(20\!\cdots\!13\)\( T^{6} + \)\(50\!\cdots\!18\)\( p T^{7} + \)\(20\!\cdots\!60\)\( p^{2} T^{8} + \)\(38\!\cdots\!58\)\( p^{3} T^{9} + \)\(17\!\cdots\!15\)\( p^{4} T^{10} + \)\(24\!\cdots\!26\)\( p^{5} T^{11} + \)\(11\!\cdots\!60\)\( p^{6} T^{12} + \)\(24\!\cdots\!26\)\( p^{26} T^{13} + \)\(17\!\cdots\!15\)\( p^{46} T^{14} + \)\(38\!\cdots\!58\)\( p^{66} T^{15} + \)\(20\!\cdots\!60\)\( p^{86} T^{16} + \)\(50\!\cdots\!18\)\( p^{106} T^{17} + \)\(20\!\cdots\!13\)\( p^{126} T^{18} + \)\(47\!\cdots\!86\)\( p^{147} T^{19} + \)\(12\!\cdots\!98\)\( p^{168} T^{20} + \)\(27\!\cdots\!42\)\( p^{189} T^{21} + \)\(51\!\cdots\!75\)\( p^{210} T^{22} + 75612722022 p^{231} T^{23} + p^{252} T^{24} \) | |
| 13 | \( 1 - 1106340906306 T + \)\(19\!\cdots\!33\)\( T^{2} - \)\(16\!\cdots\!82\)\( T^{3} + \)\(17\!\cdots\!46\)\( T^{4} - \)\(98\!\cdots\!58\)\( p T^{5} + \)\(61\!\cdots\!31\)\( p^{2} T^{6} - \)\(29\!\cdots\!42\)\( p^{3} T^{7} + \)\(15\!\cdots\!24\)\( p^{4} T^{8} - \)\(67\!\cdots\!62\)\( p^{5} T^{9} + \)\(31\!\cdots\!57\)\( p^{6} T^{10} - \)\(12\!\cdots\!30\)\( p^{7} T^{11} + \)\(50\!\cdots\!36\)\( p^{8} T^{12} - \)\(12\!\cdots\!30\)\( p^{28} T^{13} + \)\(31\!\cdots\!57\)\( p^{48} T^{14} - \)\(67\!\cdots\!62\)\( p^{68} T^{15} + \)\(15\!\cdots\!24\)\( p^{88} T^{16} - \)\(29\!\cdots\!42\)\( p^{108} T^{17} + \)\(61\!\cdots\!31\)\( p^{128} T^{18} - \)\(98\!\cdots\!58\)\( p^{148} T^{19} + \)\(17\!\cdots\!46\)\( p^{168} T^{20} - \)\(16\!\cdots\!82\)\( p^{189} T^{21} + \)\(19\!\cdots\!33\)\( p^{210} T^{22} - 1106340906306 p^{231} T^{23} + p^{252} T^{24} \) | |
| 17 | \( 1 - 21318462894400 T + \)\(68\!\cdots\!96\)\( T^{2} - \)\(10\!\cdots\!84\)\( T^{3} + \)\(20\!\cdots\!34\)\( T^{4} - \)\(15\!\cdots\!20\)\( p T^{5} + \)\(12\!\cdots\!88\)\( p^{2} T^{6} - \)\(80\!\cdots\!40\)\( p^{3} T^{7} + \)\(55\!\cdots\!95\)\( p^{4} T^{8} - \)\(31\!\cdots\!16\)\( p^{5} T^{9} + \)\(18\!\cdots\!72\)\( p^{6} T^{10} - \)\(93\!\cdots\!08\)\( p^{7} T^{11} + \)\(49\!\cdots\!96\)\( p^{8} T^{12} - \)\(93\!\cdots\!08\)\( p^{28} T^{13} + \)\(18\!\cdots\!72\)\( p^{48} T^{14} - \)\(31\!\cdots\!16\)\( p^{68} T^{15} + \)\(55\!\cdots\!95\)\( p^{88} T^{16} - \)\(80\!\cdots\!40\)\( p^{108} T^{17} + \)\(12\!\cdots\!88\)\( p^{128} T^{18} - \)\(15\!\cdots\!20\)\( p^{148} T^{19} + \)\(20\!\cdots\!34\)\( p^{168} T^{20} - \)\(10\!\cdots\!84\)\( p^{189} T^{21} + \)\(68\!\cdots\!96\)\( p^{210} T^{22} - 21318462894400 p^{231} T^{23} + p^{252} T^{24} \) | |
| 19 | \( 1 + 3375679354448 p T + \)\(29\!\cdots\!08\)\( p T^{2} + \)\(69\!\cdots\!12\)\( p^{2} T^{3} + \)\(20\!\cdots\!50\)\( p^{3} T^{4} + \)\(39\!\cdots\!92\)\( p^{4} T^{5} + \)\(92\!\cdots\!92\)\( p^{5} T^{6} + \)\(15\!\cdots\!28\)\( p^{6} T^{7} + \)\(31\!\cdots\!01\)\( p^{7} T^{8} + \)\(45\!\cdots\!76\)\( p^{8} T^{9} + \)\(83\!\cdots\!32\)\( p^{9} T^{10} + \)\(11\!\cdots\!92\)\( p^{10} T^{11} + \)\(95\!\cdots\!40\)\( p^{12} T^{12} + \)\(11\!\cdots\!92\)\( p^{31} T^{13} + \)\(83\!\cdots\!32\)\( p^{51} T^{14} + \)\(45\!\cdots\!76\)\( p^{71} T^{15} + \)\(31\!\cdots\!01\)\( p^{91} T^{16} + \)\(15\!\cdots\!28\)\( p^{111} T^{17} + \)\(92\!\cdots\!92\)\( p^{131} T^{18} + \)\(39\!\cdots\!92\)\( p^{151} T^{19} + \)\(20\!\cdots\!50\)\( p^{171} T^{20} + \)\(69\!\cdots\!12\)\( p^{191} T^{21} + \)\(29\!\cdots\!08\)\( p^{211} T^{22} + 3375679354448 p^{232} T^{23} + p^{252} T^{24} \) | |
| 23 | \( 1 + 92442371463100 T + \)\(34\!\cdots\!04\)\( T^{2} + \)\(45\!\cdots\!60\)\( T^{3} + \)\(55\!\cdots\!06\)\( T^{4} + \)\(90\!\cdots\!68\)\( T^{5} + \)\(57\!\cdots\!28\)\( T^{6} + \)\(10\!\cdots\!28\)\( T^{7} + \)\(44\!\cdots\!79\)\( T^{8} + \)\(76\!\cdots\!84\)\( T^{9} + \)\(25\!\cdots\!72\)\( T^{10} + \)\(41\!\cdots\!40\)\( T^{11} + \)\(11\!\cdots\!44\)\( T^{12} + \)\(41\!\cdots\!40\)\( p^{21} T^{13} + \)\(25\!\cdots\!72\)\( p^{42} T^{14} + \)\(76\!\cdots\!84\)\( p^{63} T^{15} + \)\(44\!\cdots\!79\)\( p^{84} T^{16} + \)\(10\!\cdots\!28\)\( p^{105} T^{17} + \)\(57\!\cdots\!28\)\( p^{126} T^{18} + \)\(90\!\cdots\!68\)\( p^{147} T^{19} + \)\(55\!\cdots\!06\)\( p^{168} T^{20} + \)\(45\!\cdots\!60\)\( p^{189} T^{21} + \)\(34\!\cdots\!04\)\( p^{210} T^{22} + 92442371463100 p^{231} T^{23} + p^{252} T^{24} \) | |
| 31 | \( 1 - 667598339910262 T + \)\(16\!\cdots\!67\)\( T^{2} - \)\(24\!\cdots\!14\)\( T^{3} + \)\(13\!\cdots\!18\)\( T^{4} - \)\(29\!\cdots\!98\)\( T^{5} + \)\(68\!\cdots\!25\)\( T^{6} - \)\(18\!\cdots\!82\)\( T^{7} + \)\(25\!\cdots\!80\)\( T^{8} - \)\(78\!\cdots\!50\)\( T^{9} + \)\(73\!\cdots\!79\)\( T^{10} - \)\(22\!\cdots\!38\)\( T^{11} + \)\(17\!\cdots\!60\)\( T^{12} - \)\(22\!\cdots\!38\)\( p^{21} T^{13} + \)\(73\!\cdots\!79\)\( p^{42} T^{14} - \)\(78\!\cdots\!50\)\( p^{63} T^{15} + \)\(25\!\cdots\!80\)\( p^{84} T^{16} - \)\(18\!\cdots\!82\)\( p^{105} T^{17} + \)\(68\!\cdots\!25\)\( p^{126} T^{18} - \)\(29\!\cdots\!98\)\( p^{147} T^{19} + \)\(13\!\cdots\!18\)\( p^{168} T^{20} - \)\(24\!\cdots\!14\)\( p^{189} T^{21} + \)\(16\!\cdots\!67\)\( p^{210} T^{22} - 667598339910262 p^{231} T^{23} + p^{252} T^{24} \) | |
| 37 | \( 1 - 28910323697160228 T + \)\(46\!\cdots\!88\)\( T^{2} - \)\(17\!\cdots\!36\)\( T^{3} + \)\(12\!\cdots\!26\)\( T^{4} - \)\(46\!\cdots\!88\)\( T^{5} + \)\(23\!\cdots\!56\)\( T^{6} - \)\(83\!\cdots\!80\)\( T^{7} + \)\(34\!\cdots\!39\)\( T^{8} - \)\(11\!\cdots\!80\)\( T^{9} + \)\(39\!\cdots\!36\)\( T^{10} - \)\(12\!\cdots\!60\)\( T^{11} + \)\(37\!\cdots\!76\)\( T^{12} - \)\(12\!\cdots\!60\)\( p^{21} T^{13} + \)\(39\!\cdots\!36\)\( p^{42} T^{14} - \)\(11\!\cdots\!80\)\( p^{63} T^{15} + \)\(34\!\cdots\!39\)\( p^{84} T^{16} - \)\(83\!\cdots\!80\)\( p^{105} T^{17} + \)\(23\!\cdots\!56\)\( p^{126} T^{18} - \)\(46\!\cdots\!88\)\( p^{147} T^{19} + \)\(12\!\cdots\!26\)\( p^{168} T^{20} - \)\(17\!\cdots\!36\)\( p^{189} T^{21} + \)\(46\!\cdots\!88\)\( p^{210} T^{22} - 28910323697160228 p^{231} T^{23} + p^{252} T^{24} \) | |
| 41 | \( 1 - 114917975551005484 T + \)\(52\!\cdots\!88\)\( T^{2} - \)\(60\!\cdots\!84\)\( T^{3} + \)\(14\!\cdots\!02\)\( T^{4} - \)\(15\!\cdots\!84\)\( T^{5} + \)\(25\!\cdots\!84\)\( T^{6} - \)\(25\!\cdots\!56\)\( T^{7} + \)\(32\!\cdots\!99\)\( T^{8} - \)\(30\!\cdots\!52\)\( T^{9} + \)\(33\!\cdots\!40\)\( T^{10} - \)\(28\!\cdots\!36\)\( T^{11} + \)\(27\!\cdots\!44\)\( T^{12} - \)\(28\!\cdots\!36\)\( p^{21} T^{13} + \)\(33\!\cdots\!40\)\( p^{42} T^{14} - \)\(30\!\cdots\!52\)\( p^{63} T^{15} + \)\(32\!\cdots\!99\)\( p^{84} T^{16} - \)\(25\!\cdots\!56\)\( p^{105} T^{17} + \)\(25\!\cdots\!84\)\( p^{126} T^{18} - \)\(15\!\cdots\!84\)\( p^{147} T^{19} + \)\(14\!\cdots\!02\)\( p^{168} T^{20} - \)\(60\!\cdots\!84\)\( p^{189} T^{21} + \)\(52\!\cdots\!88\)\( p^{210} T^{22} - 114917975551005484 p^{231} T^{23} + p^{252} T^{24} \) | |
| 43 | \( 1 - 203268895640575250 T + \)\(19\!\cdots\!87\)\( T^{2} - \)\(32\!\cdots\!98\)\( T^{3} + \)\(17\!\cdots\!54\)\( T^{4} - \)\(58\!\cdots\!78\)\( p T^{5} + \)\(94\!\cdots\!53\)\( T^{6} - \)\(12\!\cdots\!78\)\( T^{7} + \)\(37\!\cdots\!52\)\( T^{8} - \)\(41\!\cdots\!90\)\( T^{9} + \)\(10\!\cdots\!87\)\( T^{10} - \)\(10\!\cdots\!14\)\( T^{11} + \)\(24\!\cdots\!04\)\( T^{12} - \)\(10\!\cdots\!14\)\( p^{21} T^{13} + \)\(10\!\cdots\!87\)\( p^{42} T^{14} - \)\(41\!\cdots\!90\)\( p^{63} T^{15} + \)\(37\!\cdots\!52\)\( p^{84} T^{16} - \)\(12\!\cdots\!78\)\( p^{105} T^{17} + \)\(94\!\cdots\!53\)\( p^{126} T^{18} - \)\(58\!\cdots\!78\)\( p^{148} T^{19} + \)\(17\!\cdots\!54\)\( p^{168} T^{20} - \)\(32\!\cdots\!98\)\( p^{189} T^{21} + \)\(19\!\cdots\!87\)\( p^{210} T^{22} - 203268895640575250 p^{231} T^{23} + p^{252} T^{24} \) | |
| 47 | \( 1 - 1050843994052953634 T + \)\(10\!\cdots\!59\)\( T^{2} - \)\(68\!\cdots\!34\)\( T^{3} + \)\(44\!\cdots\!26\)\( T^{4} - \)\(23\!\cdots\!66\)\( T^{5} + \)\(12\!\cdots\!57\)\( T^{6} - \)\(57\!\cdots\!90\)\( T^{7} + \)\(26\!\cdots\!32\)\( T^{8} - \)\(11\!\cdots\!58\)\( T^{9} + \)\(46\!\cdots\!67\)\( T^{10} - \)\(17\!\cdots\!46\)\( T^{11} + \)\(65\!\cdots\!00\)\( T^{12} - \)\(17\!\cdots\!46\)\( p^{21} T^{13} + \)\(46\!\cdots\!67\)\( p^{42} T^{14} - \)\(11\!\cdots\!58\)\( p^{63} T^{15} + \)\(26\!\cdots\!32\)\( p^{84} T^{16} - \)\(57\!\cdots\!90\)\( p^{105} T^{17} + \)\(12\!\cdots\!57\)\( p^{126} T^{18} - \)\(23\!\cdots\!66\)\( p^{147} T^{19} + \)\(44\!\cdots\!26\)\( p^{168} T^{20} - \)\(68\!\cdots\!34\)\( p^{189} T^{21} + \)\(10\!\cdots\!59\)\( p^{210} T^{22} - 1050843994052953634 p^{231} T^{23} + p^{252} T^{24} \) | |
| 53 | \( 1 - 1243305464914490870 T + \)\(73\!\cdots\!69\)\( T^{2} - \)\(84\!\cdots\!06\)\( T^{3} + \)\(24\!\cdots\!54\)\( T^{4} - \)\(21\!\cdots\!46\)\( T^{5} + \)\(43\!\cdots\!07\)\( T^{6} - \)\(16\!\cdots\!38\)\( T^{7} + \)\(28\!\cdots\!52\)\( T^{8} + \)\(55\!\cdots\!62\)\( T^{9} - \)\(47\!\cdots\!11\)\( T^{10} + \)\(20\!\cdots\!70\)\( T^{11} - \)\(14\!\cdots\!80\)\( T^{12} + \)\(20\!\cdots\!70\)\( p^{21} T^{13} - \)\(47\!\cdots\!11\)\( p^{42} T^{14} + \)\(55\!\cdots\!62\)\( p^{63} T^{15} + \)\(28\!\cdots\!52\)\( p^{84} T^{16} - \)\(16\!\cdots\!38\)\( p^{105} T^{17} + \)\(43\!\cdots\!07\)\( p^{126} T^{18} - \)\(21\!\cdots\!46\)\( p^{147} T^{19} + \)\(24\!\cdots\!54\)\( p^{168} T^{20} - \)\(84\!\cdots\!06\)\( p^{189} T^{21} + \)\(73\!\cdots\!69\)\( p^{210} T^{22} - 1243305464914490870 p^{231} T^{23} + p^{252} T^{24} \) | |
| 59 | \( 1 - 2566690023566975892 T + \)\(10\!\cdots\!32\)\( T^{2} - \)\(19\!\cdots\!12\)\( T^{3} + \)\(51\!\cdots\!26\)\( T^{4} - \)\(61\!\cdots\!88\)\( T^{5} + \)\(16\!\cdots\!92\)\( T^{6} - \)\(11\!\cdots\!08\)\( T^{7} + \)\(39\!\cdots\!27\)\( T^{8} - \)\(20\!\cdots\!84\)\( T^{9} + \)\(81\!\cdots\!64\)\( T^{10} - \)\(35\!\cdots\!28\)\( T^{11} + \)\(13\!\cdots\!96\)\( T^{12} - \)\(35\!\cdots\!28\)\( p^{21} T^{13} + \)\(81\!\cdots\!64\)\( p^{42} T^{14} - \)\(20\!\cdots\!84\)\( p^{63} T^{15} + \)\(39\!\cdots\!27\)\( p^{84} T^{16} - \)\(11\!\cdots\!08\)\( p^{105} T^{17} + \)\(16\!\cdots\!92\)\( p^{126} T^{18} - \)\(61\!\cdots\!88\)\( p^{147} T^{19} + \)\(51\!\cdots\!26\)\( p^{168} T^{20} - \)\(19\!\cdots\!12\)\( p^{189} T^{21} + \)\(10\!\cdots\!32\)\( p^{210} T^{22} - 2566690023566975892 p^{231} T^{23} + p^{252} T^{24} \) | |
| 61 | \( 1 + 11920971254026911592 T + \)\(23\!\cdots\!20\)\( T^{2} + \)\(19\!\cdots\!56\)\( T^{3} + \)\(24\!\cdots\!18\)\( T^{4} + \)\(16\!\cdots\!28\)\( T^{5} + \)\(16\!\cdots\!12\)\( T^{6} + \)\(95\!\cdots\!24\)\( T^{7} + \)\(79\!\cdots\!31\)\( T^{8} + \)\(41\!\cdots\!60\)\( T^{9} + \)\(31\!\cdots\!36\)\( T^{10} + \)\(14\!\cdots\!88\)\( T^{11} + \)\(10\!\cdots\!04\)\( T^{12} + \)\(14\!\cdots\!88\)\( p^{21} T^{13} + \)\(31\!\cdots\!36\)\( p^{42} T^{14} + \)\(41\!\cdots\!60\)\( p^{63} T^{15} + \)\(79\!\cdots\!31\)\( p^{84} T^{16} + \)\(95\!\cdots\!24\)\( p^{105} T^{17} + \)\(16\!\cdots\!12\)\( p^{126} T^{18} + \)\(16\!\cdots\!28\)\( p^{147} T^{19} + \)\(24\!\cdots\!18\)\( p^{168} T^{20} + \)\(19\!\cdots\!56\)\( p^{189} T^{21} + \)\(23\!\cdots\!20\)\( p^{210} T^{22} + 11920971254026911592 p^{231} T^{23} + p^{252} T^{24} \) | |
| 67 | \( 1 - 27467580398508962728 T + \)\(18\!\cdots\!72\)\( T^{2} - \)\(43\!\cdots\!48\)\( T^{3} + \)\(16\!\cdots\!74\)\( T^{4} - \)\(34\!\cdots\!20\)\( T^{5} + \)\(10\!\cdots\!12\)\( T^{6} - \)\(18\!\cdots\!08\)\( T^{7} + \)\(42\!\cdots\!95\)\( T^{8} - \)\(69\!\cdots\!40\)\( T^{9} + \)\(13\!\cdots\!52\)\( T^{10} - \)\(19\!\cdots\!76\)\( T^{11} + \)\(34\!\cdots\!28\)\( T^{12} - \)\(19\!\cdots\!76\)\( p^{21} T^{13} + \)\(13\!\cdots\!52\)\( p^{42} T^{14} - \)\(69\!\cdots\!40\)\( p^{63} T^{15} + \)\(42\!\cdots\!95\)\( p^{84} T^{16} - \)\(18\!\cdots\!08\)\( p^{105} T^{17} + \)\(10\!\cdots\!12\)\( p^{126} T^{18} - \)\(34\!\cdots\!20\)\( p^{147} T^{19} + \)\(16\!\cdots\!74\)\( p^{168} T^{20} - \)\(43\!\cdots\!48\)\( p^{189} T^{21} + \)\(18\!\cdots\!72\)\( p^{210} T^{22} - 27467580398508962728 p^{231} T^{23} + p^{252} T^{24} \) | |
| 71 | \( 1 - 39394008523302335116 T + \)\(57\!\cdots\!80\)\( T^{2} - \)\(17\!\cdots\!24\)\( T^{3} + \)\(14\!\cdots\!18\)\( T^{4} - \)\(34\!\cdots\!28\)\( T^{5} + \)\(23\!\cdots\!12\)\( T^{6} - \)\(46\!\cdots\!56\)\( T^{7} + \)\(28\!\cdots\!07\)\( T^{8} - \)\(48\!\cdots\!36\)\( T^{9} + \)\(28\!\cdots\!08\)\( T^{10} - \)\(43\!\cdots\!24\)\( T^{11} + \)\(23\!\cdots\!28\)\( T^{12} - \)\(43\!\cdots\!24\)\( p^{21} T^{13} + \)\(28\!\cdots\!08\)\( p^{42} T^{14} - \)\(48\!\cdots\!36\)\( p^{63} T^{15} + \)\(28\!\cdots\!07\)\( p^{84} T^{16} - \)\(46\!\cdots\!56\)\( p^{105} T^{17} + \)\(23\!\cdots\!12\)\( p^{126} T^{18} - \)\(34\!\cdots\!28\)\( p^{147} T^{19} + \)\(14\!\cdots\!18\)\( p^{168} T^{20} - \)\(17\!\cdots\!24\)\( p^{189} T^{21} + \)\(57\!\cdots\!80\)\( p^{210} T^{22} - 39394008523302335116 p^{231} T^{23} + p^{252} T^{24} \) | |
| 73 | \( 1 + \)\(10\!\cdots\!44\)\( T + \)\(11\!\cdots\!04\)\( T^{2} + \)\(77\!\cdots\!92\)\( T^{3} + \)\(50\!\cdots\!90\)\( T^{4} + \)\(24\!\cdots\!60\)\( T^{5} + \)\(11\!\cdots\!80\)\( T^{6} + \)\(46\!\cdots\!92\)\( T^{7} + \)\(18\!\cdots\!15\)\( T^{8} + \)\(63\!\cdots\!36\)\( T^{9} + \)\(23\!\cdots\!32\)\( T^{10} + \)\(11\!\cdots\!20\)\( p T^{11} + \)\(30\!\cdots\!72\)\( T^{12} + \)\(11\!\cdots\!20\)\( p^{22} T^{13} + \)\(23\!\cdots\!32\)\( p^{42} T^{14} + \)\(63\!\cdots\!36\)\( p^{63} T^{15} + \)\(18\!\cdots\!15\)\( p^{84} T^{16} + \)\(46\!\cdots\!92\)\( p^{105} T^{17} + \)\(11\!\cdots\!80\)\( p^{126} T^{18} + \)\(24\!\cdots\!60\)\( p^{147} T^{19} + \)\(50\!\cdots\!90\)\( p^{168} T^{20} + \)\(77\!\cdots\!92\)\( p^{189} T^{21} + \)\(11\!\cdots\!04\)\( p^{210} T^{22} + \)\(10\!\cdots\!44\)\( p^{231} T^{23} + p^{252} T^{24} \) | |
| 79 | \( 1 + \)\(43\!\cdots\!82\)\( T + \)\(12\!\cdots\!59\)\( T^{2} + \)\(26\!\cdots\!14\)\( T^{3} + \)\(47\!\cdots\!06\)\( T^{4} + \)\(73\!\cdots\!78\)\( T^{5} + \)\(10\!\cdots\!81\)\( T^{6} + \)\(12\!\cdots\!34\)\( T^{7} + \)\(14\!\cdots\!76\)\( T^{8} + \)\(15\!\cdots\!78\)\( T^{9} + \)\(15\!\cdots\!71\)\( T^{10} + \)\(14\!\cdots\!26\)\( T^{11} + \)\(12\!\cdots\!68\)\( T^{12} + \)\(14\!\cdots\!26\)\( p^{21} T^{13} + \)\(15\!\cdots\!71\)\( p^{42} T^{14} + \)\(15\!\cdots\!78\)\( p^{63} T^{15} + \)\(14\!\cdots\!76\)\( p^{84} T^{16} + \)\(12\!\cdots\!34\)\( p^{105} T^{17} + \)\(10\!\cdots\!81\)\( p^{126} T^{18} + \)\(73\!\cdots\!78\)\( p^{147} T^{19} + \)\(47\!\cdots\!06\)\( p^{168} T^{20} + \)\(26\!\cdots\!14\)\( p^{189} T^{21} + \)\(12\!\cdots\!59\)\( p^{210} T^{22} + \)\(43\!\cdots\!82\)\( p^{231} T^{23} + p^{252} T^{24} \) | |
| 83 | \( 1 + \)\(15\!\cdots\!56\)\( T + \)\(14\!\cdots\!28\)\( T^{2} + \)\(18\!\cdots\!04\)\( T^{3} + \)\(10\!\cdots\!66\)\( T^{4} + \)\(11\!\cdots\!72\)\( T^{5} + \)\(47\!\cdots\!48\)\( T^{6} + \)\(44\!\cdots\!12\)\( T^{7} + \)\(16\!\cdots\!51\)\( T^{8} + \)\(13\!\cdots\!44\)\( T^{9} + \)\(44\!\cdots\!80\)\( T^{10} + \)\(33\!\cdots\!12\)\( T^{11} + \)\(98\!\cdots\!44\)\( T^{12} + \)\(33\!\cdots\!12\)\( p^{21} T^{13} + \)\(44\!\cdots\!80\)\( p^{42} T^{14} + \)\(13\!\cdots\!44\)\( p^{63} T^{15} + \)\(16\!\cdots\!51\)\( p^{84} T^{16} + \)\(44\!\cdots\!12\)\( p^{105} T^{17} + \)\(47\!\cdots\!48\)\( p^{126} T^{18} + \)\(11\!\cdots\!72\)\( p^{147} T^{19} + \)\(10\!\cdots\!66\)\( p^{168} T^{20} + \)\(18\!\cdots\!04\)\( p^{189} T^{21} + \)\(14\!\cdots\!28\)\( p^{210} T^{22} + \)\(15\!\cdots\!56\)\( p^{231} T^{23} + p^{252} T^{24} \) | |
| 89 | \( 1 + \)\(37\!\cdots\!48\)\( T + \)\(70\!\cdots\!44\)\( T^{2} + \)\(26\!\cdots\!12\)\( T^{3} + \)\(23\!\cdots\!30\)\( T^{4} + \)\(87\!\cdots\!00\)\( T^{5} + \)\(50\!\cdots\!92\)\( T^{6} + \)\(17\!\cdots\!12\)\( T^{7} + \)\(79\!\cdots\!99\)\( T^{8} + \)\(24\!\cdots\!16\)\( T^{9} + \)\(95\!\cdots\!80\)\( T^{10} + \)\(26\!\cdots\!64\)\( T^{11} + \)\(92\!\cdots\!56\)\( T^{12} + \)\(26\!\cdots\!64\)\( p^{21} T^{13} + \)\(95\!\cdots\!80\)\( p^{42} T^{14} + \)\(24\!\cdots\!16\)\( p^{63} T^{15} + \)\(79\!\cdots\!99\)\( p^{84} T^{16} + \)\(17\!\cdots\!12\)\( p^{105} T^{17} + \)\(50\!\cdots\!92\)\( p^{126} T^{18} + \)\(87\!\cdots\!00\)\( p^{147} T^{19} + \)\(23\!\cdots\!30\)\( p^{168} T^{20} + \)\(26\!\cdots\!12\)\( p^{189} T^{21} + \)\(70\!\cdots\!44\)\( p^{210} T^{22} + \)\(37\!\cdots\!48\)\( p^{231} T^{23} + p^{252} T^{24} \) | |
| 97 | \( 1 + \)\(57\!\cdots\!36\)\( T + \)\(39\!\cdots\!72\)\( T^{2} + \)\(25\!\cdots\!48\)\( T^{3} + \)\(76\!\cdots\!06\)\( T^{4} + \)\(54\!\cdots\!32\)\( T^{5} + \)\(10\!\cdots\!68\)\( T^{6} + \)\(71\!\cdots\!44\)\( T^{7} + \)\(98\!\cdots\!43\)\( T^{8} + \)\(65\!\cdots\!60\)\( T^{9} + \)\(74\!\cdots\!80\)\( T^{10} + \)\(45\!\cdots\!36\)\( T^{11} + \)\(43\!\cdots\!04\)\( T^{12} + \)\(45\!\cdots\!36\)\( p^{21} T^{13} + \)\(74\!\cdots\!80\)\( p^{42} T^{14} + \)\(65\!\cdots\!60\)\( p^{63} T^{15} + \)\(98\!\cdots\!43\)\( p^{84} T^{16} + \)\(71\!\cdots\!44\)\( p^{105} T^{17} + \)\(10\!\cdots\!68\)\( p^{126} T^{18} + \)\(54\!\cdots\!32\)\( p^{147} T^{19} + \)\(76\!\cdots\!06\)\( p^{168} T^{20} + \)\(25\!\cdots\!48\)\( p^{189} T^{21} + \)\(39\!\cdots\!72\)\( p^{210} T^{22} + \)\(57\!\cdots\!36\)\( p^{231} T^{23} + p^{252} T^{24} \) | |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{24} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−2.56512052354162335728074629493, −2.23675310121980697903251431099, −2.02758526710333144596353140133, −2.01845687298556345311451312449, −1.79113389172273074792443610660, −1.72336430608867637708851674312, −1.70677491741781921429252223365, −1.70477371617383559617902293457, −1.64351263617284044985879204345, −1.58115682292653456752523359220, −1.56726293564722646002623789776, −1.35950277908891611357354183247, −1.20187419926749079055288077055, −1.15114274976963169136647108763, −1.04391620152906276493306099878, −0.835886786933630553692078509967, −0.70332012927020105611585911518, −0.56727013109762367082139995167, −0.48023484496649352311165104332, −0.46389279279466992497276011234, −0.46279738038379196943500935812, −0.45946960583092694331190826055, −0.40557169511916311443414394205, −0.15834907143693339187456793713, −0.07749873799918177491027165979, 0.07749873799918177491027165979, 0.15834907143693339187456793713, 0.40557169511916311443414394205, 0.45946960583092694331190826055, 0.46279738038379196943500935812, 0.46389279279466992497276011234, 0.48023484496649352311165104332, 0.56727013109762367082139995167, 0.70332012927020105611585911518, 0.835886786933630553692078509967, 1.04391620152906276493306099878, 1.15114274976963169136647108763, 1.20187419926749079055288077055, 1.35950277908891611357354183247, 1.56726293564722646002623789776, 1.58115682292653456752523359220, 1.64351263617284044985879204345, 1.70477371617383559617902293457, 1.70677491741781921429252223365, 1.72336430608867637708851674312, 1.79113389172273074792443610660, 2.01845687298556345311451312449, 2.02758526710333144596353140133, 2.23675310121980697903251431099, 2.56512052354162335728074629493
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.