Dirichlet series
| L(s) = 1 | + 6.14e3·2-s + 7.56e4·3-s + 2.04e7·4-s + 7.41e6·5-s + 4.64e8·6-s + 1.01e8·7-s + 4.88e10·8-s − 2.11e9·9-s + 4.55e10·10-s + 2.00e9·11-s + 1.54e12·12-s + 5.76e10·13-s + 6.24e11·14-s + 5.61e11·15-s + 9.38e13·16-s + 9.48e11·17-s − 1.29e13·18-s + 2.00e12·19-s + 1.51e14·20-s + 7.68e12·21-s + 1.23e13·22-s + 7.82e12·23-s + 3.69e15·24-s − 6.63e13·25-s + 3.54e14·26-s − 3.43e14·27-s + 2.07e15·28-s + ⋯ |
| L(s) = 1 | + 8.48·2-s + 2.21·3-s + 39·4-s + 1.69·5-s + 18.8·6-s + 0.951·7-s + 128.·8-s − 1.81·9-s + 14.4·10-s + 0.256·11-s + 86.5·12-s + 1.50·13-s + 8.07·14-s + 3.76·15-s + 341.·16-s + 1.94·17-s − 15.4·18-s + 1.42·19-s + 66.2·20-s + 2.11·21-s + 2.17·22-s + 0.905·23-s + 285.·24-s − 3.47·25-s + 12.7·26-s − 8.67·27-s + 37.1·28-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 29^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(20-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 29^{12}\right)^{s/2} \, \Gamma_{\C}(s+19/2)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(24\) |
| Conductor: | \(2^{12} \cdot 29^{12}\) |
| Sign: | $1$ |
| Analytic conductor: | \(2.98530\times 10^{25}\) |
| Root analytic conductor: | \(11.5201\) |
| Motivic weight: | \(19\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((24,\ 2^{12} \cdot 29^{12} ,\ ( \ : [19/2]^{12} ),\ 1 )\) |
Particular Values
| \(L(10)\) | \(\approx\) | \(1.974611502\times10^{6}\) |
| \(L(\frac12)\) | \(\approx\) | \(1.974611502\times10^{6}\) |
| \(L(\frac{21}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( ( 1 - p^{9} T )^{12} \) |
| 29 | \( ( 1 - p^{9} T )^{12} \) | |
| good | 3 | \( 1 - 75638 T + 7834583263 T^{2} - 136261318071566 p T^{3} + 8195535211862301754 p T^{4} - \)\(10\!\cdots\!22\)\( p^{2} T^{5} + \)\(18\!\cdots\!27\)\( p^{5} T^{6} - \)\(72\!\cdots\!50\)\( p^{7} T^{7} + \)\(11\!\cdots\!52\)\( p^{10} T^{8} - \)\(50\!\cdots\!34\)\( p^{14} T^{9} + \)\(23\!\cdots\!95\)\( p^{16} T^{10} - \)\(31\!\cdots\!54\)\( p^{19} T^{11} + \)\(15\!\cdots\!84\)\( p^{25} T^{12} - \)\(31\!\cdots\!54\)\( p^{38} T^{13} + \)\(23\!\cdots\!95\)\( p^{54} T^{14} - \)\(50\!\cdots\!34\)\( p^{71} T^{15} + \)\(11\!\cdots\!52\)\( p^{86} T^{16} - \)\(72\!\cdots\!50\)\( p^{102} T^{17} + \)\(18\!\cdots\!27\)\( p^{119} T^{18} - \)\(10\!\cdots\!22\)\( p^{135} T^{19} + 8195535211862301754 p^{153} T^{20} - 136261318071566 p^{172} T^{21} + 7834583263 p^{190} T^{22} - 75638 p^{209} T^{23} + p^{228} T^{24} \) |
| 5 | \( 1 - 1483426 p T + 121379435718373 T^{2} - \)\(73\!\cdots\!58\)\( T^{3} + \)\(26\!\cdots\!94\)\( p^{2} T^{4} - \)\(14\!\cdots\!66\)\( p^{2} T^{5} + \)\(18\!\cdots\!07\)\( p^{3} T^{6} - \)\(19\!\cdots\!74\)\( p^{4} T^{7} + \)\(20\!\cdots\!52\)\( p^{5} T^{8} - \)\(20\!\cdots\!66\)\( p^{6} T^{9} + \)\(74\!\cdots\!09\)\( p^{9} T^{10} - \)\(58\!\cdots\!62\)\( p^{13} T^{11} + \)\(96\!\cdots\!28\)\( p^{15} T^{12} - \)\(58\!\cdots\!62\)\( p^{32} T^{13} + \)\(74\!\cdots\!09\)\( p^{47} T^{14} - \)\(20\!\cdots\!66\)\( p^{63} T^{15} + \)\(20\!\cdots\!52\)\( p^{81} T^{16} - \)\(19\!\cdots\!74\)\( p^{99} T^{17} + \)\(18\!\cdots\!07\)\( p^{117} T^{18} - \)\(14\!\cdots\!66\)\( p^{135} T^{19} + \)\(26\!\cdots\!94\)\( p^{154} T^{20} - \)\(73\!\cdots\!58\)\( p^{171} T^{21} + 121379435718373 p^{190} T^{22} - 1483426 p^{210} T^{23} + p^{228} T^{24} \) | |
| 7 | \( 1 - 101600432 T + 58709124730002980 T^{2} - \)\(53\!\cdots\!28\)\( T^{3} + \)\(19\!\cdots\!62\)\( T^{4} - \)\(15\!\cdots\!12\)\( T^{5} + \)\(19\!\cdots\!88\)\( p^{4} T^{6} - \)\(98\!\cdots\!64\)\( p^{3} T^{7} + \)\(49\!\cdots\!21\)\( p^{5} T^{8} - \)\(67\!\cdots\!48\)\( p^{7} T^{9} + \)\(30\!\cdots\!40\)\( p^{9} T^{10} - \)\(37\!\cdots\!76\)\( p^{11} T^{11} + \)\(22\!\cdots\!36\)\( p^{14} T^{12} - \)\(37\!\cdots\!76\)\( p^{30} T^{13} + \)\(30\!\cdots\!40\)\( p^{47} T^{14} - \)\(67\!\cdots\!48\)\( p^{64} T^{15} + \)\(49\!\cdots\!21\)\( p^{81} T^{16} - \)\(98\!\cdots\!64\)\( p^{98} T^{17} + \)\(19\!\cdots\!88\)\( p^{118} T^{18} - \)\(15\!\cdots\!12\)\( p^{133} T^{19} + \)\(19\!\cdots\!62\)\( p^{152} T^{20} - \)\(53\!\cdots\!28\)\( p^{171} T^{21} + 58709124730002980 p^{190} T^{22} - 101600432 p^{209} T^{23} + p^{228} T^{24} \) | |
| 11 | \( 1 - 2007416390 T + \)\(30\!\cdots\!03\)\( T^{2} - \)\(90\!\cdots\!70\)\( T^{3} + \)\(52\!\cdots\!38\)\( T^{4} - \)\(20\!\cdots\!18\)\( T^{5} + \)\(64\!\cdots\!25\)\( T^{6} - \)\(26\!\cdots\!98\)\( p T^{7} + \)\(51\!\cdots\!36\)\( p^{2} T^{8} - \)\(22\!\cdots\!66\)\( p^{3} T^{9} + \)\(33\!\cdots\!31\)\( p^{4} T^{10} - \)\(13\!\cdots\!78\)\( p^{6} T^{11} + \)\(18\!\cdots\!12\)\( p^{6} T^{12} - \)\(13\!\cdots\!78\)\( p^{25} T^{13} + \)\(33\!\cdots\!31\)\( p^{42} T^{14} - \)\(22\!\cdots\!66\)\( p^{60} T^{15} + \)\(51\!\cdots\!36\)\( p^{78} T^{16} - \)\(26\!\cdots\!98\)\( p^{96} T^{17} + \)\(64\!\cdots\!25\)\( p^{114} T^{18} - \)\(20\!\cdots\!18\)\( p^{133} T^{19} + \)\(52\!\cdots\!38\)\( p^{152} T^{20} - \)\(90\!\cdots\!70\)\( p^{171} T^{21} + \)\(30\!\cdots\!03\)\( p^{190} T^{22} - 2007416390 p^{209} T^{23} + p^{228} T^{24} \) | |
| 13 | \( 1 - 57626609050 T + \)\(47\!\cdots\!65\)\( T^{2} - \)\(21\!\cdots\!58\)\( T^{3} + \)\(13\!\cdots\!82\)\( T^{4} - \)\(45\!\cdots\!74\)\( p T^{5} + \)\(17\!\cdots\!39\)\( p^{2} T^{6} - \)\(59\!\cdots\!54\)\( p^{3} T^{7} + \)\(21\!\cdots\!52\)\( p^{4} T^{8} - \)\(64\!\cdots\!90\)\( p^{5} T^{9} + \)\(21\!\cdots\!85\)\( p^{6} T^{10} - \)\(61\!\cdots\!54\)\( p^{7} T^{11} + \)\(19\!\cdots\!80\)\( p^{8} T^{12} - \)\(61\!\cdots\!54\)\( p^{26} T^{13} + \)\(21\!\cdots\!85\)\( p^{44} T^{14} - \)\(64\!\cdots\!90\)\( p^{62} T^{15} + \)\(21\!\cdots\!52\)\( p^{80} T^{16} - \)\(59\!\cdots\!54\)\( p^{98} T^{17} + \)\(17\!\cdots\!39\)\( p^{116} T^{18} - \)\(45\!\cdots\!74\)\( p^{134} T^{19} + \)\(13\!\cdots\!82\)\( p^{152} T^{20} - \)\(21\!\cdots\!58\)\( p^{171} T^{21} + \)\(47\!\cdots\!65\)\( p^{190} T^{22} - 57626609050 p^{209} T^{23} + p^{228} T^{24} \) | |
| 17 | \( 1 - 948593027440 T + \)\(14\!\cdots\!60\)\( p T^{2} - \)\(70\!\cdots\!92\)\( p^{2} T^{3} + \)\(56\!\cdots\!02\)\( p^{3} T^{4} - \)\(24\!\cdots\!24\)\( p^{4} T^{5} + \)\(13\!\cdots\!12\)\( p^{5} T^{6} - \)\(54\!\cdots\!24\)\( p^{6} T^{7} + \)\(24\!\cdots\!75\)\( p^{7} T^{8} - \)\(82\!\cdots\!08\)\( p^{8} T^{9} + \)\(30\!\cdots\!20\)\( p^{9} T^{10} - \)\(54\!\cdots\!96\)\( p^{11} T^{11} + \)\(29\!\cdots\!04\)\( p^{11} T^{12} - \)\(54\!\cdots\!96\)\( p^{30} T^{13} + \)\(30\!\cdots\!20\)\( p^{47} T^{14} - \)\(82\!\cdots\!08\)\( p^{65} T^{15} + \)\(24\!\cdots\!75\)\( p^{83} T^{16} - \)\(54\!\cdots\!24\)\( p^{101} T^{17} + \)\(13\!\cdots\!12\)\( p^{119} T^{18} - \)\(24\!\cdots\!24\)\( p^{137} T^{19} + \)\(56\!\cdots\!02\)\( p^{155} T^{20} - \)\(70\!\cdots\!92\)\( p^{173} T^{21} + \)\(14\!\cdots\!60\)\( p^{191} T^{22} - 948593027440 p^{209} T^{23} + p^{228} T^{24} \) | |
| 19 | \( 1 - 2003081583856 T + \)\(14\!\cdots\!28\)\( T^{2} - \)\(27\!\cdots\!24\)\( T^{3} + \)\(95\!\cdots\!02\)\( T^{4} - \)\(17\!\cdots\!36\)\( T^{5} + \)\(41\!\cdots\!36\)\( T^{6} - \)\(68\!\cdots\!76\)\( T^{7} + \)\(12\!\cdots\!95\)\( T^{8} - \)\(19\!\cdots\!64\)\( T^{9} + \)\(32\!\cdots\!64\)\( T^{10} - \)\(45\!\cdots\!12\)\( T^{11} + \)\(67\!\cdots\!08\)\( T^{12} - \)\(45\!\cdots\!12\)\( p^{19} T^{13} + \)\(32\!\cdots\!64\)\( p^{38} T^{14} - \)\(19\!\cdots\!64\)\( p^{57} T^{15} + \)\(12\!\cdots\!95\)\( p^{76} T^{16} - \)\(68\!\cdots\!76\)\( p^{95} T^{17} + \)\(41\!\cdots\!36\)\( p^{114} T^{18} - \)\(17\!\cdots\!36\)\( p^{133} T^{19} + \)\(95\!\cdots\!02\)\( p^{152} T^{20} - \)\(27\!\cdots\!24\)\( p^{171} T^{21} + \)\(14\!\cdots\!28\)\( p^{190} T^{22} - 2003081583856 p^{209} T^{23} + p^{228} T^{24} \) | |
| 23 | \( 1 - 7823442559484 T + \)\(42\!\cdots\!16\)\( T^{2} - \)\(37\!\cdots\!92\)\( T^{3} + \)\(10\!\cdots\!82\)\( T^{4} - \)\(88\!\cdots\!48\)\( T^{5} + \)\(16\!\cdots\!88\)\( T^{6} - \)\(14\!\cdots\!56\)\( T^{7} + \)\(21\!\cdots\!07\)\( T^{8} - \)\(16\!\cdots\!56\)\( T^{9} + \)\(21\!\cdots\!24\)\( T^{10} - \)\(15\!\cdots\!96\)\( T^{11} + \)\(17\!\cdots\!76\)\( T^{12} - \)\(15\!\cdots\!96\)\( p^{19} T^{13} + \)\(21\!\cdots\!24\)\( p^{38} T^{14} - \)\(16\!\cdots\!56\)\( p^{57} T^{15} + \)\(21\!\cdots\!07\)\( p^{76} T^{16} - \)\(14\!\cdots\!56\)\( p^{95} T^{17} + \)\(16\!\cdots\!88\)\( p^{114} T^{18} - \)\(88\!\cdots\!48\)\( p^{133} T^{19} + \)\(10\!\cdots\!82\)\( p^{152} T^{20} - \)\(37\!\cdots\!92\)\( p^{171} T^{21} + \)\(42\!\cdots\!16\)\( p^{190} T^{22} - 7823442559484 p^{209} T^{23} + p^{228} T^{24} \) | |
| 31 | \( 1 + 53403039679182 T + \)\(11\!\cdots\!75\)\( T^{2} + \)\(10\!\cdots\!30\)\( T^{3} + \)\(63\!\cdots\!14\)\( T^{4} - \)\(28\!\cdots\!22\)\( T^{5} + \)\(81\!\cdots\!63\)\( p T^{6} - \)\(20\!\cdots\!46\)\( T^{7} + \)\(81\!\cdots\!72\)\( T^{8} - \)\(82\!\cdots\!02\)\( T^{9} + \)\(21\!\cdots\!15\)\( T^{10} - \)\(24\!\cdots\!10\)\( T^{11} + \)\(50\!\cdots\!16\)\( T^{12} - \)\(24\!\cdots\!10\)\( p^{19} T^{13} + \)\(21\!\cdots\!15\)\( p^{38} T^{14} - \)\(82\!\cdots\!02\)\( p^{57} T^{15} + \)\(81\!\cdots\!72\)\( p^{76} T^{16} - \)\(20\!\cdots\!46\)\( p^{95} T^{17} + \)\(81\!\cdots\!63\)\( p^{115} T^{18} - \)\(28\!\cdots\!22\)\( p^{133} T^{19} + \)\(63\!\cdots\!14\)\( p^{152} T^{20} + \)\(10\!\cdots\!30\)\( p^{171} T^{21} + \)\(11\!\cdots\!75\)\( p^{190} T^{22} + 53403039679182 p^{209} T^{23} + p^{228} T^{24} \) | |
| 37 | \( 1 - 1426583422186452 T + \)\(92\!\cdots\!96\)\( p T^{2} - \)\(39\!\cdots\!04\)\( T^{3} + \)\(62\!\cdots\!22\)\( T^{4} - \)\(62\!\cdots\!72\)\( T^{5} + \)\(79\!\cdots\!84\)\( T^{6} - \)\(72\!\cdots\!12\)\( T^{7} + \)\(79\!\cdots\!15\)\( T^{8} - \)\(17\!\cdots\!12\)\( p T^{9} + \)\(65\!\cdots\!24\)\( T^{10} - \)\(49\!\cdots\!28\)\( T^{11} + \)\(44\!\cdots\!24\)\( T^{12} - \)\(49\!\cdots\!28\)\( p^{19} T^{13} + \)\(65\!\cdots\!24\)\( p^{38} T^{14} - \)\(17\!\cdots\!12\)\( p^{58} T^{15} + \)\(79\!\cdots\!15\)\( p^{76} T^{16} - \)\(72\!\cdots\!12\)\( p^{95} T^{17} + \)\(79\!\cdots\!84\)\( p^{114} T^{18} - \)\(62\!\cdots\!72\)\( p^{133} T^{19} + \)\(62\!\cdots\!22\)\( p^{152} T^{20} - \)\(39\!\cdots\!04\)\( p^{171} T^{21} + \)\(92\!\cdots\!96\)\( p^{191} T^{22} - 1426583422186452 p^{209} T^{23} + p^{228} T^{24} \) | |
| 41 | \( 1 - 4049092818313468 T + \)\(33\!\cdots\!76\)\( T^{2} - \)\(12\!\cdots\!12\)\( T^{3} + \)\(57\!\cdots\!38\)\( T^{4} - \)\(18\!\cdots\!76\)\( T^{5} + \)\(66\!\cdots\!48\)\( T^{6} - \)\(19\!\cdots\!12\)\( T^{7} + \)\(55\!\cdots\!75\)\( T^{8} - \)\(14\!\cdots\!60\)\( T^{9} + \)\(35\!\cdots\!32\)\( T^{10} - \)\(79\!\cdots\!48\)\( T^{11} + \)\(17\!\cdots\!92\)\( T^{12} - \)\(79\!\cdots\!48\)\( p^{19} T^{13} + \)\(35\!\cdots\!32\)\( p^{38} T^{14} - \)\(14\!\cdots\!60\)\( p^{57} T^{15} + \)\(55\!\cdots\!75\)\( p^{76} T^{16} - \)\(19\!\cdots\!12\)\( p^{95} T^{17} + \)\(66\!\cdots\!48\)\( p^{114} T^{18} - \)\(18\!\cdots\!76\)\( p^{133} T^{19} + \)\(57\!\cdots\!38\)\( p^{152} T^{20} - \)\(12\!\cdots\!12\)\( p^{171} T^{21} + \)\(33\!\cdots\!76\)\( p^{190} T^{22} - 4049092818313468 p^{209} T^{23} + p^{228} T^{24} \) | |
| 43 | \( 1 - 7207389682572654 T + \)\(81\!\cdots\!19\)\( T^{2} - \)\(46\!\cdots\!10\)\( T^{3} + \)\(32\!\cdots\!86\)\( T^{4} - \)\(15\!\cdots\!62\)\( T^{5} + \)\(85\!\cdots\!81\)\( T^{6} - \)\(36\!\cdots\!02\)\( T^{7} + \)\(16\!\cdots\!48\)\( T^{8} - \)\(62\!\cdots\!90\)\( T^{9} + \)\(25\!\cdots\!23\)\( T^{10} - \)\(85\!\cdots\!30\)\( T^{11} + \)\(30\!\cdots\!80\)\( T^{12} - \)\(85\!\cdots\!30\)\( p^{19} T^{13} + \)\(25\!\cdots\!23\)\( p^{38} T^{14} - \)\(62\!\cdots\!90\)\( p^{57} T^{15} + \)\(16\!\cdots\!48\)\( p^{76} T^{16} - \)\(36\!\cdots\!02\)\( p^{95} T^{17} + \)\(85\!\cdots\!81\)\( p^{114} T^{18} - \)\(15\!\cdots\!62\)\( p^{133} T^{19} + \)\(32\!\cdots\!86\)\( p^{152} T^{20} - \)\(46\!\cdots\!10\)\( p^{171} T^{21} + \)\(81\!\cdots\!19\)\( p^{190} T^{22} - 7207389682572654 p^{209} T^{23} + p^{228} T^{24} \) | |
| 47 | \( 1 - 10987742991532422 T + \)\(44\!\cdots\!31\)\( T^{2} - \)\(38\!\cdots\!38\)\( T^{3} + \)\(91\!\cdots\!78\)\( T^{4} - \)\(65\!\cdots\!62\)\( T^{5} + \)\(11\!\cdots\!41\)\( T^{6} - \)\(73\!\cdots\!26\)\( T^{7} + \)\(11\!\cdots\!84\)\( T^{8} - \)\(62\!\cdots\!74\)\( T^{9} + \)\(82\!\cdots\!99\)\( T^{10} - \)\(42\!\cdots\!70\)\( T^{11} + \)\(51\!\cdots\!04\)\( T^{12} - \)\(42\!\cdots\!70\)\( p^{19} T^{13} + \)\(82\!\cdots\!99\)\( p^{38} T^{14} - \)\(62\!\cdots\!74\)\( p^{57} T^{15} + \)\(11\!\cdots\!84\)\( p^{76} T^{16} - \)\(73\!\cdots\!26\)\( p^{95} T^{17} + \)\(11\!\cdots\!41\)\( p^{114} T^{18} - \)\(65\!\cdots\!62\)\( p^{133} T^{19} + \)\(91\!\cdots\!78\)\( p^{152} T^{20} - \)\(38\!\cdots\!38\)\( p^{171} T^{21} + \)\(44\!\cdots\!31\)\( p^{190} T^{22} - 10987742991532422 p^{209} T^{23} + p^{228} T^{24} \) | |
| 53 | \( 1 - 51581721294822702 T + \)\(46\!\cdots\!09\)\( T^{2} - \)\(20\!\cdots\!58\)\( T^{3} + \)\(10\!\cdots\!10\)\( T^{4} - \)\(37\!\cdots\!14\)\( T^{5} + \)\(15\!\cdots\!71\)\( T^{6} - \)\(47\!\cdots\!18\)\( T^{7} + \)\(15\!\cdots\!20\)\( T^{8} - \)\(42\!\cdots\!90\)\( T^{9} + \)\(12\!\cdots\!93\)\( T^{10} - \)\(30\!\cdots\!78\)\( T^{11} + \)\(79\!\cdots\!12\)\( T^{12} - \)\(30\!\cdots\!78\)\( p^{19} T^{13} + \)\(12\!\cdots\!93\)\( p^{38} T^{14} - \)\(42\!\cdots\!90\)\( p^{57} T^{15} + \)\(15\!\cdots\!20\)\( p^{76} T^{16} - \)\(47\!\cdots\!18\)\( p^{95} T^{17} + \)\(15\!\cdots\!71\)\( p^{114} T^{18} - \)\(37\!\cdots\!14\)\( p^{133} T^{19} + \)\(10\!\cdots\!10\)\( p^{152} T^{20} - \)\(20\!\cdots\!58\)\( p^{171} T^{21} + \)\(46\!\cdots\!09\)\( p^{190} T^{22} - 51581721294822702 p^{209} T^{23} + p^{228} T^{24} \) | |
| 59 | \( 1 - 92093909668138012 T + \)\(33\!\cdots\!04\)\( T^{2} - \)\(27\!\cdots\!96\)\( T^{3} + \)\(53\!\cdots\!42\)\( T^{4} - \)\(40\!\cdots\!00\)\( T^{5} + \)\(54\!\cdots\!16\)\( T^{6} - \)\(39\!\cdots\!76\)\( T^{7} + \)\(39\!\cdots\!07\)\( T^{8} - \)\(28\!\cdots\!04\)\( T^{9} + \)\(22\!\cdots\!40\)\( T^{10} - \)\(15\!\cdots\!32\)\( T^{11} + \)\(11\!\cdots\!60\)\( T^{12} - \)\(15\!\cdots\!32\)\( p^{19} T^{13} + \)\(22\!\cdots\!40\)\( p^{38} T^{14} - \)\(28\!\cdots\!04\)\( p^{57} T^{15} + \)\(39\!\cdots\!07\)\( p^{76} T^{16} - \)\(39\!\cdots\!76\)\( p^{95} T^{17} + \)\(54\!\cdots\!16\)\( p^{114} T^{18} - \)\(40\!\cdots\!00\)\( p^{133} T^{19} + \)\(53\!\cdots\!42\)\( p^{152} T^{20} - \)\(27\!\cdots\!96\)\( p^{171} T^{21} + \)\(33\!\cdots\!04\)\( p^{190} T^{22} - 92093909668138012 p^{209} T^{23} + p^{228} T^{24} \) | |
| 61 | \( 1 - 392476438629545928 T + \)\(13\!\cdots\!64\)\( T^{2} - \)\(29\!\cdots\!04\)\( T^{3} + \)\(59\!\cdots\!58\)\( T^{4} - \)\(95\!\cdots\!56\)\( T^{5} + \)\(14\!\cdots\!68\)\( T^{6} - \)\(17\!\cdots\!76\)\( T^{7} + \)\(20\!\cdots\!87\)\( T^{8} - \)\(22\!\cdots\!24\)\( T^{9} + \)\(22\!\cdots\!12\)\( T^{10} - \)\(21\!\cdots\!96\)\( T^{11} + \)\(19\!\cdots\!28\)\( T^{12} - \)\(21\!\cdots\!96\)\( p^{19} T^{13} + \)\(22\!\cdots\!12\)\( p^{38} T^{14} - \)\(22\!\cdots\!24\)\( p^{57} T^{15} + \)\(20\!\cdots\!87\)\( p^{76} T^{16} - \)\(17\!\cdots\!76\)\( p^{95} T^{17} + \)\(14\!\cdots\!68\)\( p^{114} T^{18} - \)\(95\!\cdots\!56\)\( p^{133} T^{19} + \)\(59\!\cdots\!58\)\( p^{152} T^{20} - \)\(29\!\cdots\!04\)\( p^{171} T^{21} + \)\(13\!\cdots\!64\)\( p^{190} T^{22} - 392476438629545928 p^{209} T^{23} + p^{228} T^{24} \) | |
| 67 | \( 1 - 59980941838221080 T + \)\(20\!\cdots\!48\)\( T^{2} - \)\(16\!\cdots\!20\)\( T^{3} + \)\(27\!\cdots\!62\)\( T^{4} - \)\(28\!\cdots\!60\)\( T^{5} + \)\(26\!\cdots\!92\)\( T^{6} - \)\(29\!\cdots\!16\)\( T^{7} + \)\(20\!\cdots\!11\)\( T^{8} - \)\(23\!\cdots\!16\)\( T^{9} + \)\(13\!\cdots\!60\)\( T^{10} - \)\(14\!\cdots\!08\)\( T^{11} + \)\(10\!\cdots\!04\)\( p T^{12} - \)\(14\!\cdots\!08\)\( p^{19} T^{13} + \)\(13\!\cdots\!60\)\( p^{38} T^{14} - \)\(23\!\cdots\!16\)\( p^{57} T^{15} + \)\(20\!\cdots\!11\)\( p^{76} T^{16} - \)\(29\!\cdots\!16\)\( p^{95} T^{17} + \)\(26\!\cdots\!92\)\( p^{114} T^{18} - \)\(28\!\cdots\!60\)\( p^{133} T^{19} + \)\(27\!\cdots\!62\)\( p^{152} T^{20} - \)\(16\!\cdots\!20\)\( p^{171} T^{21} + \)\(20\!\cdots\!48\)\( p^{190} T^{22} - 59980941838221080 p^{209} T^{23} + p^{228} T^{24} \) | |
| 71 | \( 1 - 316351220695753092 T + \)\(67\!\cdots\!64\)\( T^{2} - \)\(17\!\cdots\!96\)\( T^{3} + \)\(25\!\cdots\!30\)\( T^{4} - \)\(51\!\cdots\!08\)\( T^{5} + \)\(66\!\cdots\!20\)\( T^{6} - \)\(10\!\cdots\!08\)\( T^{7} + \)\(13\!\cdots\!39\)\( T^{8} - \)\(14\!\cdots\!44\)\( T^{9} + \)\(23\!\cdots\!24\)\( T^{10} - \)\(17\!\cdots\!84\)\( T^{11} + \)\(35\!\cdots\!28\)\( T^{12} - \)\(17\!\cdots\!84\)\( p^{19} T^{13} + \)\(23\!\cdots\!24\)\( p^{38} T^{14} - \)\(14\!\cdots\!44\)\( p^{57} T^{15} + \)\(13\!\cdots\!39\)\( p^{76} T^{16} - \)\(10\!\cdots\!08\)\( p^{95} T^{17} + \)\(66\!\cdots\!20\)\( p^{114} T^{18} - \)\(51\!\cdots\!08\)\( p^{133} T^{19} + \)\(25\!\cdots\!30\)\( p^{152} T^{20} - \)\(17\!\cdots\!96\)\( p^{171} T^{21} + \)\(67\!\cdots\!64\)\( p^{190} T^{22} - 316351220695753092 p^{209} T^{23} + p^{228} T^{24} \) | |
| 73 | \( 1 - 55495866202319660 T + \)\(19\!\cdots\!28\)\( T^{2} - \)\(13\!\cdots\!60\)\( T^{3} + \)\(17\!\cdots\!70\)\( T^{4} - \)\(17\!\cdots\!72\)\( T^{5} + \)\(10\!\cdots\!96\)\( T^{6} - \)\(13\!\cdots\!40\)\( T^{7} + \)\(42\!\cdots\!67\)\( T^{8} - \)\(72\!\cdots\!80\)\( T^{9} + \)\(13\!\cdots\!12\)\( T^{10} - \)\(26\!\cdots\!92\)\( T^{11} + \)\(52\!\cdots\!40\)\( p T^{12} - \)\(26\!\cdots\!92\)\( p^{19} T^{13} + \)\(13\!\cdots\!12\)\( p^{38} T^{14} - \)\(72\!\cdots\!80\)\( p^{57} T^{15} + \)\(42\!\cdots\!67\)\( p^{76} T^{16} - \)\(13\!\cdots\!40\)\( p^{95} T^{17} + \)\(10\!\cdots\!96\)\( p^{114} T^{18} - \)\(17\!\cdots\!72\)\( p^{133} T^{19} + \)\(17\!\cdots\!70\)\( p^{152} T^{20} - \)\(13\!\cdots\!60\)\( p^{171} T^{21} + \)\(19\!\cdots\!28\)\( p^{190} T^{22} - 55495866202319660 p^{209} T^{23} + p^{228} T^{24} \) | |
| 79 | \( 1 - 191684988021168490 T + \)\(61\!\cdots\!47\)\( T^{2} + \)\(31\!\cdots\!18\)\( T^{3} + \)\(16\!\cdots\!66\)\( T^{4} + \)\(74\!\cdots\!02\)\( p T^{5} + \)\(27\!\cdots\!65\)\( T^{6} + \)\(18\!\cdots\!94\)\( T^{7} + \)\(40\!\cdots\!20\)\( T^{8} + \)\(30\!\cdots\!18\)\( T^{9} + \)\(63\!\cdots\!99\)\( T^{10} + \)\(34\!\cdots\!46\)\( T^{11} + \)\(83\!\cdots\!84\)\( T^{12} + \)\(34\!\cdots\!46\)\( p^{19} T^{13} + \)\(63\!\cdots\!99\)\( p^{38} T^{14} + \)\(30\!\cdots\!18\)\( p^{57} T^{15} + \)\(40\!\cdots\!20\)\( p^{76} T^{16} + \)\(18\!\cdots\!94\)\( p^{95} T^{17} + \)\(27\!\cdots\!65\)\( p^{114} T^{18} + \)\(74\!\cdots\!02\)\( p^{134} T^{19} + \)\(16\!\cdots\!66\)\( p^{152} T^{20} + \)\(31\!\cdots\!18\)\( p^{171} T^{21} + \)\(61\!\cdots\!47\)\( p^{190} T^{22} - 191684988021168490 p^{209} T^{23} + p^{228} T^{24} \) | |
| 83 | \( 1 - 4320734092357806060 T + \)\(32\!\cdots\!92\)\( T^{2} - \)\(11\!\cdots\!76\)\( T^{3} + \)\(48\!\cdots\!02\)\( T^{4} - \)\(13\!\cdots\!12\)\( T^{5} + \)\(43\!\cdots\!28\)\( T^{6} - \)\(10\!\cdots\!52\)\( T^{7} + \)\(27\!\cdots\!75\)\( T^{8} - \)\(55\!\cdots\!68\)\( T^{9} + \)\(12\!\cdots\!12\)\( T^{10} - \)\(21\!\cdots\!80\)\( T^{11} + \)\(40\!\cdots\!60\)\( T^{12} - \)\(21\!\cdots\!80\)\( p^{19} T^{13} + \)\(12\!\cdots\!12\)\( p^{38} T^{14} - \)\(55\!\cdots\!68\)\( p^{57} T^{15} + \)\(27\!\cdots\!75\)\( p^{76} T^{16} - \)\(10\!\cdots\!52\)\( p^{95} T^{17} + \)\(43\!\cdots\!28\)\( p^{114} T^{18} - \)\(13\!\cdots\!12\)\( p^{133} T^{19} + \)\(48\!\cdots\!02\)\( p^{152} T^{20} - \)\(11\!\cdots\!76\)\( p^{171} T^{21} + \)\(32\!\cdots\!92\)\( p^{190} T^{22} - 4320734092357806060 p^{209} T^{23} + p^{228} T^{24} \) | |
| 89 | \( 1 - 13713316226476173132 T + \)\(17\!\cdots\!00\)\( T^{2} - \)\(15\!\cdots\!24\)\( T^{3} + \)\(12\!\cdots\!42\)\( T^{4} - \)\(82\!\cdots\!60\)\( T^{5} + \)\(49\!\cdots\!80\)\( T^{6} - \)\(26\!\cdots\!16\)\( T^{7} + \)\(12\!\cdots\!43\)\( T^{8} - \)\(56\!\cdots\!92\)\( T^{9} + \)\(23\!\cdots\!28\)\( T^{10} - \)\(85\!\cdots\!52\)\( T^{11} + \)\(29\!\cdots\!72\)\( T^{12} - \)\(85\!\cdots\!52\)\( p^{19} T^{13} + \)\(23\!\cdots\!28\)\( p^{38} T^{14} - \)\(56\!\cdots\!92\)\( p^{57} T^{15} + \)\(12\!\cdots\!43\)\( p^{76} T^{16} - \)\(26\!\cdots\!16\)\( p^{95} T^{17} + \)\(49\!\cdots\!80\)\( p^{114} T^{18} - \)\(82\!\cdots\!60\)\( p^{133} T^{19} + \)\(12\!\cdots\!42\)\( p^{152} T^{20} - \)\(15\!\cdots\!24\)\( p^{171} T^{21} + \)\(17\!\cdots\!00\)\( p^{190} T^{22} - 13713316226476173132 p^{209} T^{23} + p^{228} T^{24} \) | |
| 97 | \( 1 - 28094407349917720396 T + \)\(82\!\cdots\!24\)\( T^{2} - \)\(14\!\cdots\!28\)\( T^{3} + \)\(26\!\cdots\!26\)\( T^{4} - \)\(36\!\cdots\!60\)\( T^{5} + \)\(49\!\cdots\!84\)\( T^{6} - \)\(55\!\cdots\!04\)\( T^{7} + \)\(60\!\cdots\!71\)\( T^{8} - \)\(58\!\cdots\!44\)\( T^{9} + \)\(53\!\cdots\!44\)\( T^{10} - \)\(45\!\cdots\!08\)\( p T^{11} + \)\(34\!\cdots\!92\)\( T^{12} - \)\(45\!\cdots\!08\)\( p^{20} T^{13} + \)\(53\!\cdots\!44\)\( p^{38} T^{14} - \)\(58\!\cdots\!44\)\( p^{57} T^{15} + \)\(60\!\cdots\!71\)\( p^{76} T^{16} - \)\(55\!\cdots\!04\)\( p^{95} T^{17} + \)\(49\!\cdots\!84\)\( p^{114} T^{18} - \)\(36\!\cdots\!60\)\( p^{133} T^{19} + \)\(26\!\cdots\!26\)\( p^{152} T^{20} - \)\(14\!\cdots\!28\)\( p^{171} T^{21} + \)\(82\!\cdots\!24\)\( p^{190} T^{22} - 28094407349917720396 p^{209} T^{23} + p^{228} T^{24} \) | |
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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
−2.93008802316156956143469298853, −2.42366194278302650949205536554, −2.41201766245914722382911848361, −2.33606627292879800771425380557, −2.30631462088618162049999303306, −2.30137540763821427870286435720, −2.25098822126614411022872888780, −2.17307083983143468381396896682, −2.00257088153994036909552991234, −1.95694007764431123064009728220, −1.93977232461628177690931177536, −1.92959242797912830012429510722, −1.63772562753343272454327600899, −1.48187590442965528404226886115, −1.34633113239839812586262208771, −1.20553494349870945593642810825, −1.02165828677935129832318922894, −0.977833922988226175231442747176, −0.840922233275492410536014279690, −0.76121007726456858460182039483, −0.68740544674146941790628732802, −0.68185115856847051460238549509, −0.45448492670355570121416574782, −0.39534191176548844449645262316, −0.33408420536832338638276445066, 0.33408420536832338638276445066, 0.39534191176548844449645262316, 0.45448492670355570121416574782, 0.68185115856847051460238549509, 0.68740544674146941790628732802, 0.76121007726456858460182039483, 0.840922233275492410536014279690, 0.977833922988226175231442747176, 1.02165828677935129832318922894, 1.20553494349870945593642810825, 1.34633113239839812586262208771, 1.48187590442965528404226886115, 1.63772562753343272454327600899, 1.92959242797912830012429510722, 1.93977232461628177690931177536, 1.95694007764431123064009728220, 2.00257088153994036909552991234, 2.17307083983143468381396896682, 2.25098822126614411022872888780, 2.30137540763821427870286435720, 2.30631462088618162049999303306, 2.33606627292879800771425380557, 2.41201766245914722382911848361, 2.42366194278302650949205536554, 2.93008802316156956143469298853
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.