Dirichlet series
| L(s) = 1 | − 458·2-s − 1.01e5·4-s − 8.07e7·7-s + 2.05e7·8-s + 7.36e9·9-s + 3.69e10·14-s − 1.46e11·16-s + 1.41e10·17-s − 3.37e12·18-s + 2.17e12·23-s + 1.49e14·25-s + 8.16e12·28-s + 4.28e14·31-s + 1.03e14·32-s − 6.46e12·34-s − 7.44e14·36-s + 1.20e15·41-s − 9.97e14·46-s − 1.13e15·47-s − 8.35e16·49-s − 6.84e16·50-s − 1.66e15·56-s − 1.96e17·62-s − 5.94e17·63-s − 3.83e16·64-s − 1.42e15·68-s − 1.59e18·71-s + ⋯ |
| L(s) = 1 | − 0.632·2-s − 0.192·4-s − 0.755·7-s + 0.0542·8-s + 6.33·9-s + 0.478·14-s − 0.531·16-s + 0.0288·17-s − 4.00·18-s + 0.252·23-s + 7.83·25-s + 0.145·28-s + 2.91·31-s + 0.519·32-s − 0.0182·34-s − 1.22·36-s + 0.572·41-s − 0.159·46-s − 0.147·47-s − 7.32·49-s − 4.95·50-s − 0.0409·56-s − 1.84·62-s − 4.78·63-s − 0.266·64-s − 0.00557·68-s − 4.14·71-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{54}\right)^{s/2} \, \Gamma_{\C}(s)^{18} \, L(s)\cr=\mathstrut & \,\Lambda(20-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{54}\right)^{s/2} \, \Gamma_{\C}(s+19/2)^{18} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(36\) |
| Conductor: | \(2^{54}\) |
| Sign: | $1$ |
| Analytic conductor: | \(5.32596\times 10^{22}\) |
| Root analytic conductor: | \(4.27847\) |
| Motivic weight: | \(19\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((36,\ 2^{54} ,\ ( \ : [19/2]^{18} ),\ 1 )\) |
Particular Values
| \(L(10)\) | \(\approx\) | \(49.19997475\) |
| \(L(\frac12)\) | \(\approx\) | \(49.19997475\) |
| \(L(\frac{21}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( 1 + 229 p T + 38867 p^{3} T^{2} + 2627437 p^{6} T^{3} + 239439555 p^{10} T^{4} + 2636101313 p^{15} T^{5} + 46401477395 p^{21} T^{6} - 68780506755 p^{28} T^{7} + 312530586827 p^{36} T^{8} + 778131777713 p^{45} T^{9} + 312530586827 p^{55} T^{10} - 68780506755 p^{66} T^{11} + 46401477395 p^{78} T^{12} + 2636101313 p^{91} T^{13} + 239439555 p^{105} T^{14} + 2627437 p^{120} T^{15} + 38867 p^{136} T^{16} + 229 p^{153} T^{17} + p^{171} T^{18} \) |
| good | 3 | \( 1 - 7360989290 T^{2} + 351145392303782113 p^{4} T^{4} - \)\(94\!\cdots\!08\)\( p^{4} T^{6} + \)\(24\!\cdots\!64\)\( p^{8} T^{8} - \)\(19\!\cdots\!28\)\( p^{15} T^{10} + \)\(11\!\cdots\!60\)\( p^{20} T^{12} - \)\(20\!\cdots\!16\)\( p^{26} T^{14} + \)\(11\!\cdots\!10\)\( p^{33} T^{16} - \)\(53\!\cdots\!36\)\( p^{38} T^{18} + \)\(11\!\cdots\!10\)\( p^{71} T^{20} - \)\(20\!\cdots\!16\)\( p^{102} T^{22} + \)\(11\!\cdots\!60\)\( p^{134} T^{24} - \)\(19\!\cdots\!28\)\( p^{167} T^{26} + \)\(24\!\cdots\!64\)\( p^{198} T^{28} - \)\(94\!\cdots\!08\)\( p^{232} T^{30} + 351145392303782113 p^{270} T^{32} - 7360989290 p^{304} T^{34} + p^{342} T^{36} \) |
| 5 | \( 1 - 149454139847258 T^{2} + \)\(46\!\cdots\!21\)\( p^{2} T^{4} - \)\(19\!\cdots\!08\)\( p^{5} T^{6} + \)\(15\!\cdots\!64\)\( p^{6} T^{8} - \)\(17\!\cdots\!84\)\( p^{11} T^{10} + \)\(24\!\cdots\!76\)\( p^{10} T^{12} - \)\(25\!\cdots\!88\)\( p^{12} T^{14} + \)\(36\!\cdots\!94\)\( p^{18} T^{16} - \)\(47\!\cdots\!96\)\( p^{24} T^{18} + \)\(36\!\cdots\!94\)\( p^{56} T^{20} - \)\(25\!\cdots\!88\)\( p^{88} T^{22} + \)\(24\!\cdots\!76\)\( p^{124} T^{24} - \)\(17\!\cdots\!84\)\( p^{163} T^{26} + \)\(15\!\cdots\!64\)\( p^{196} T^{28} - \)\(19\!\cdots\!08\)\( p^{233} T^{30} + \)\(46\!\cdots\!21\)\( p^{268} T^{32} - 149454139847258 p^{304} T^{34} + p^{342} T^{36} \) | |
| 7 | \( ( 1 + 40353608 T + 44216711939405711 T^{2} + \)\(15\!\cdots\!88\)\( T^{3} + \)\(17\!\cdots\!00\)\( p^{2} T^{4} + \)\(31\!\cdots\!16\)\( p^{2} T^{5} + \)\(31\!\cdots\!60\)\( p^{3} T^{6} - \)\(11\!\cdots\!20\)\( p^{5} T^{7} + \)\(13\!\cdots\!14\)\( p^{7} T^{8} - \)\(13\!\cdots\!32\)\( p^{9} T^{9} + \)\(13\!\cdots\!14\)\( p^{26} T^{10} - \)\(11\!\cdots\!20\)\( p^{43} T^{11} + \)\(31\!\cdots\!60\)\( p^{60} T^{12} + \)\(31\!\cdots\!16\)\( p^{78} T^{13} + \)\(17\!\cdots\!00\)\( p^{97} T^{14} + \)\(15\!\cdots\!88\)\( p^{114} T^{15} + 44216711939405711 p^{133} T^{16} + 40353608 p^{152} T^{17} + p^{171} T^{18} )^{2} \) | |
| 11 | \( 1 - \)\(57\!\cdots\!82\)\( T^{2} + \)\(16\!\cdots\!01\)\( T^{4} - \)\(31\!\cdots\!80\)\( T^{6} + \)\(45\!\cdots\!96\)\( T^{8} - \)\(43\!\cdots\!12\)\( p^{2} T^{10} + \)\(35\!\cdots\!96\)\( p^{4} T^{12} - \)\(24\!\cdots\!20\)\( p^{6} T^{14} + \)\(14\!\cdots\!86\)\( p^{8} T^{16} - \)\(78\!\cdots\!12\)\( p^{10} T^{18} + \)\(14\!\cdots\!86\)\( p^{46} T^{20} - \)\(24\!\cdots\!20\)\( p^{82} T^{22} + \)\(35\!\cdots\!96\)\( p^{118} T^{24} - \)\(43\!\cdots\!12\)\( p^{154} T^{26} + \)\(45\!\cdots\!96\)\( p^{190} T^{28} - \)\(31\!\cdots\!80\)\( p^{228} T^{30} + \)\(16\!\cdots\!01\)\( p^{266} T^{32} - \)\(57\!\cdots\!82\)\( p^{304} T^{34} + p^{342} T^{36} \) | |
| 13 | \( 1 - \)\(13\!\cdots\!18\)\( T^{2} + \)\(75\!\cdots\!05\)\( p T^{4} - \)\(46\!\cdots\!40\)\( T^{6} + \)\(97\!\cdots\!08\)\( p^{2} T^{8} - \)\(16\!\cdots\!88\)\( p^{4} T^{10} + \)\(22\!\cdots\!84\)\( p^{6} T^{12} - \)\(27\!\cdots\!60\)\( p^{8} T^{14} + \)\(28\!\cdots\!22\)\( p^{10} T^{16} - \)\(25\!\cdots\!68\)\( p^{12} T^{18} + \)\(28\!\cdots\!22\)\( p^{48} T^{20} - \)\(27\!\cdots\!60\)\( p^{84} T^{22} + \)\(22\!\cdots\!84\)\( p^{120} T^{24} - \)\(16\!\cdots\!88\)\( p^{156} T^{26} + \)\(97\!\cdots\!08\)\( p^{192} T^{28} - \)\(46\!\cdots\!40\)\( p^{228} T^{30} + \)\(75\!\cdots\!05\)\( p^{267} T^{32} - \)\(13\!\cdots\!18\)\( p^{304} T^{34} + p^{342} T^{36} \) | |
| 17 | \( ( 1 - 7060713346 T + \)\(60\!\cdots\!33\)\( p T^{2} - \)\(71\!\cdots\!12\)\( p^{2} T^{3} + \)\(10\!\cdots\!80\)\( p^{3} T^{4} - \)\(24\!\cdots\!16\)\( p^{4} T^{5} + \)\(11\!\cdots\!52\)\( p^{5} T^{6} - \)\(41\!\cdots\!44\)\( p^{6} T^{7} + \)\(10\!\cdots\!38\)\( p^{7} T^{8} - \)\(42\!\cdots\!56\)\( p^{8} T^{9} + \)\(10\!\cdots\!38\)\( p^{26} T^{10} - \)\(41\!\cdots\!44\)\( p^{44} T^{11} + \)\(11\!\cdots\!52\)\( p^{62} T^{12} - \)\(24\!\cdots\!16\)\( p^{80} T^{13} + \)\(10\!\cdots\!80\)\( p^{98} T^{14} - \)\(71\!\cdots\!12\)\( p^{116} T^{15} + \)\(60\!\cdots\!33\)\( p^{134} T^{16} - 7060713346 p^{152} T^{17} + p^{171} T^{18} )^{2} \) | |
| 19 | \( 1 - \)\(20\!\cdots\!90\)\( T^{2} + \)\(19\!\cdots\!73\)\( T^{4} - \)\(12\!\cdots\!96\)\( T^{6} + \)\(60\!\cdots\!32\)\( T^{8} - \)\(21\!\cdots\!00\)\( T^{10} + \)\(65\!\cdots\!80\)\( T^{12} - \)\(16\!\cdots\!08\)\( T^{14} + \)\(37\!\cdots\!10\)\( T^{16} - \)\(76\!\cdots\!24\)\( T^{18} + \)\(37\!\cdots\!10\)\( p^{38} T^{20} - \)\(16\!\cdots\!08\)\( p^{76} T^{22} + \)\(65\!\cdots\!80\)\( p^{114} T^{24} - \)\(21\!\cdots\!00\)\( p^{152} T^{26} + \)\(60\!\cdots\!32\)\( p^{190} T^{28} - \)\(12\!\cdots\!96\)\( p^{228} T^{30} + \)\(19\!\cdots\!73\)\( p^{266} T^{32} - \)\(20\!\cdots\!90\)\( p^{304} T^{34} + p^{342} T^{36} \) | |
| 23 | \( ( 1 - 1088560791976 T + \)\(30\!\cdots\!15\)\( T^{2} - \)\(83\!\cdots\!76\)\( T^{3} + \)\(57\!\cdots\!52\)\( T^{4} - \)\(16\!\cdots\!56\)\( T^{5} + \)\(73\!\cdots\!88\)\( T^{6} - \)\(22\!\cdots\!32\)\( T^{7} + \)\(71\!\cdots\!38\)\( T^{8} - \)\(18\!\cdots\!56\)\( T^{9} + \)\(71\!\cdots\!38\)\( p^{19} T^{10} - \)\(22\!\cdots\!32\)\( p^{38} T^{11} + \)\(73\!\cdots\!88\)\( p^{57} T^{12} - \)\(16\!\cdots\!56\)\( p^{76} T^{13} + \)\(57\!\cdots\!52\)\( p^{95} T^{14} - \)\(83\!\cdots\!76\)\( p^{114} T^{15} + \)\(30\!\cdots\!15\)\( p^{133} T^{16} - 1088560791976 p^{152} T^{17} + p^{171} T^{18} )^{2} \) | |
| 29 | \( 1 - \)\(49\!\cdots\!38\)\( T^{2} + \)\(12\!\cdots\!17\)\( T^{4} - \)\(21\!\cdots\!28\)\( T^{6} + \)\(32\!\cdots\!00\)\( p^{2} T^{8} - \)\(28\!\cdots\!92\)\( T^{10} + \)\(24\!\cdots\!48\)\( T^{12} - \)\(64\!\cdots\!28\)\( p T^{14} + \)\(12\!\cdots\!54\)\( T^{16} - \)\(81\!\cdots\!20\)\( T^{18} + \)\(12\!\cdots\!54\)\( p^{38} T^{20} - \)\(64\!\cdots\!28\)\( p^{77} T^{22} + \)\(24\!\cdots\!48\)\( p^{114} T^{24} - \)\(28\!\cdots\!92\)\( p^{152} T^{26} + \)\(32\!\cdots\!00\)\( p^{192} T^{28} - \)\(21\!\cdots\!28\)\( p^{228} T^{30} + \)\(12\!\cdots\!17\)\( p^{266} T^{32} - \)\(49\!\cdots\!38\)\( p^{304} T^{34} + p^{342} T^{36} \) | |
| 31 | \( ( 1 - 214252885130208 T + \)\(11\!\cdots\!47\)\( T^{2} - \)\(22\!\cdots\!04\)\( T^{3} + \)\(67\!\cdots\!64\)\( T^{4} - \)\(12\!\cdots\!16\)\( T^{5} + \)\(25\!\cdots\!72\)\( T^{6} - \)\(41\!\cdots\!96\)\( T^{7} + \)\(68\!\cdots\!54\)\( T^{8} - \)\(10\!\cdots\!08\)\( T^{9} + \)\(68\!\cdots\!54\)\( p^{19} T^{10} - \)\(41\!\cdots\!96\)\( p^{38} T^{11} + \)\(25\!\cdots\!72\)\( p^{57} T^{12} - \)\(12\!\cdots\!16\)\( p^{76} T^{13} + \)\(67\!\cdots\!64\)\( p^{95} T^{14} - \)\(22\!\cdots\!04\)\( p^{114} T^{15} + \)\(11\!\cdots\!47\)\( p^{133} T^{16} - 214252885130208 p^{152} T^{17} + p^{171} T^{18} )^{2} \) | |
| 37 | \( 1 - \)\(65\!\cdots\!22\)\( T^{2} + \)\(21\!\cdots\!25\)\( T^{4} - \)\(47\!\cdots\!12\)\( T^{6} + \)\(78\!\cdots\!84\)\( T^{8} - \)\(10\!\cdots\!88\)\( T^{10} + \)\(11\!\cdots\!00\)\( T^{12} - \)\(10\!\cdots\!08\)\( T^{14} + \)\(81\!\cdots\!26\)\( T^{16} - \)\(54\!\cdots\!20\)\( T^{18} + \)\(81\!\cdots\!26\)\( p^{38} T^{20} - \)\(10\!\cdots\!08\)\( p^{76} T^{22} + \)\(11\!\cdots\!00\)\( p^{114} T^{24} - \)\(10\!\cdots\!88\)\( p^{152} T^{26} + \)\(78\!\cdots\!84\)\( p^{190} T^{28} - \)\(47\!\cdots\!12\)\( p^{228} T^{30} + \)\(21\!\cdots\!25\)\( p^{266} T^{32} - \)\(65\!\cdots\!22\)\( p^{304} T^{34} + p^{342} T^{36} \) | |
| 41 | \( ( 1 - 600130010570762 T + \)\(20\!\cdots\!69\)\( T^{2} - \)\(37\!\cdots\!88\)\( T^{3} + \)\(22\!\cdots\!88\)\( T^{4} - \)\(20\!\cdots\!24\)\( T^{5} + \)\(16\!\cdots\!84\)\( T^{6} + \)\(23\!\cdots\!24\)\( T^{7} + \)\(95\!\cdots\!86\)\( T^{8} + \)\(14\!\cdots\!64\)\( T^{9} + \)\(95\!\cdots\!86\)\( p^{19} T^{10} + \)\(23\!\cdots\!24\)\( p^{38} T^{11} + \)\(16\!\cdots\!84\)\( p^{57} T^{12} - \)\(20\!\cdots\!24\)\( p^{76} T^{13} + \)\(22\!\cdots\!88\)\( p^{95} T^{14} - \)\(37\!\cdots\!88\)\( p^{114} T^{15} + \)\(20\!\cdots\!69\)\( p^{133} T^{16} - 600130010570762 p^{152} T^{17} + p^{171} T^{18} )^{2} \) | |
| 43 | \( 1 - \)\(11\!\cdots\!10\)\( T^{2} + \)\(68\!\cdots\!69\)\( T^{4} - \)\(27\!\cdots\!64\)\( T^{6} + \)\(80\!\cdots\!92\)\( T^{8} - \)\(18\!\cdots\!84\)\( T^{10} + \)\(36\!\cdots\!32\)\( T^{12} - \)\(59\!\cdots\!28\)\( T^{14} + \)\(81\!\cdots\!02\)\( T^{16} - \)\(96\!\cdots\!28\)\( T^{18} + \)\(81\!\cdots\!02\)\( p^{38} T^{20} - \)\(59\!\cdots\!28\)\( p^{76} T^{22} + \)\(36\!\cdots\!32\)\( p^{114} T^{24} - \)\(18\!\cdots\!84\)\( p^{152} T^{26} + \)\(80\!\cdots\!92\)\( p^{190} T^{28} - \)\(27\!\cdots\!64\)\( p^{228} T^{30} + \)\(68\!\cdots\!69\)\( p^{266} T^{32} - \)\(11\!\cdots\!10\)\( p^{304} T^{34} + p^{342} T^{36} \) | |
| 47 | \( ( 1 + 567080479996464 T + \)\(16\!\cdots\!87\)\( T^{2} - \)\(11\!\cdots\!44\)\( T^{3} + \)\(12\!\cdots\!28\)\( T^{4} - \)\(14\!\cdots\!56\)\( T^{5} + \)\(68\!\cdots\!36\)\( T^{6} - \)\(54\!\cdots\!48\)\( T^{7} + \)\(34\!\cdots\!90\)\( T^{8} + \)\(33\!\cdots\!44\)\( T^{9} + \)\(34\!\cdots\!90\)\( p^{19} T^{10} - \)\(54\!\cdots\!48\)\( p^{38} T^{11} + \)\(68\!\cdots\!36\)\( p^{57} T^{12} - \)\(14\!\cdots\!56\)\( p^{76} T^{13} + \)\(12\!\cdots\!28\)\( p^{95} T^{14} - \)\(11\!\cdots\!44\)\( p^{114} T^{15} + \)\(16\!\cdots\!87\)\( p^{133} T^{16} + 567080479996464 p^{152} T^{17} + p^{171} T^{18} )^{2} \) | |
| 53 | \( 1 - \)\(65\!\cdots\!74\)\( T^{2} + \)\(21\!\cdots\!45\)\( T^{4} - \)\(48\!\cdots\!04\)\( T^{6} + \)\(79\!\cdots\!52\)\( T^{8} - \)\(10\!\cdots\!88\)\( T^{10} + \)\(10\!\cdots\!48\)\( T^{12} - \)\(94\!\cdots\!68\)\( T^{14} + \)\(69\!\cdots\!22\)\( T^{16} - \)\(43\!\cdots\!92\)\( T^{18} + \)\(69\!\cdots\!22\)\( p^{38} T^{20} - \)\(94\!\cdots\!68\)\( p^{76} T^{22} + \)\(10\!\cdots\!48\)\( p^{114} T^{24} - \)\(10\!\cdots\!88\)\( p^{152} T^{26} + \)\(79\!\cdots\!52\)\( p^{190} T^{28} - \)\(48\!\cdots\!04\)\( p^{228} T^{30} + \)\(21\!\cdots\!45\)\( p^{266} T^{32} - \)\(65\!\cdots\!74\)\( p^{304} T^{34} + p^{342} T^{36} \) | |
| 59 | \( 1 - \)\(45\!\cdots\!86\)\( T^{2} + \)\(10\!\cdots\!73\)\( T^{4} - \)\(16\!\cdots\!08\)\( T^{6} + \)\(19\!\cdots\!88\)\( T^{8} - \)\(18\!\cdots\!24\)\( T^{10} + \)\(14\!\cdots\!08\)\( T^{12} - \)\(92\!\cdots\!00\)\( T^{14} + \)\(51\!\cdots\!06\)\( T^{16} - \)\(24\!\cdots\!76\)\( T^{18} + \)\(51\!\cdots\!06\)\( p^{38} T^{20} - \)\(92\!\cdots\!00\)\( p^{76} T^{22} + \)\(14\!\cdots\!08\)\( p^{114} T^{24} - \)\(18\!\cdots\!24\)\( p^{152} T^{26} + \)\(19\!\cdots\!88\)\( p^{190} T^{28} - \)\(16\!\cdots\!08\)\( p^{228} T^{30} + \)\(10\!\cdots\!73\)\( p^{266} T^{32} - \)\(45\!\cdots\!86\)\( p^{304} T^{34} + p^{342} T^{36} \) | |
| 61 | \( 1 - \)\(87\!\cdots\!86\)\( T^{2} + \)\(38\!\cdots\!13\)\( T^{4} - \)\(11\!\cdots\!68\)\( T^{6} + \)\(24\!\cdots\!08\)\( T^{8} - \)\(41\!\cdots\!84\)\( T^{10} + \)\(58\!\cdots\!08\)\( T^{12} - \)\(70\!\cdots\!00\)\( T^{14} + \)\(72\!\cdots\!46\)\( T^{16} - \)\(65\!\cdots\!16\)\( T^{18} + \)\(72\!\cdots\!46\)\( p^{38} T^{20} - \)\(70\!\cdots\!00\)\( p^{76} T^{22} + \)\(58\!\cdots\!08\)\( p^{114} T^{24} - \)\(41\!\cdots\!84\)\( p^{152} T^{26} + \)\(24\!\cdots\!08\)\( p^{190} T^{28} - \)\(11\!\cdots\!68\)\( p^{228} T^{30} + \)\(38\!\cdots\!13\)\( p^{266} T^{32} - \)\(87\!\cdots\!86\)\( p^{304} T^{34} + p^{342} T^{36} \) | |
| 67 | \( 1 - \)\(87\!\cdots\!38\)\( p T^{2} + \)\(17\!\cdots\!25\)\( T^{4} - \)\(32\!\cdots\!76\)\( T^{6} + \)\(45\!\cdots\!32\)\( T^{8} - \)\(49\!\cdots\!92\)\( T^{10} + \)\(44\!\cdots\!28\)\( T^{12} - \)\(33\!\cdots\!52\)\( T^{14} + \)\(20\!\cdots\!22\)\( T^{16} - \)\(11\!\cdots\!28\)\( T^{18} + \)\(20\!\cdots\!22\)\( p^{38} T^{20} - \)\(33\!\cdots\!52\)\( p^{76} T^{22} + \)\(44\!\cdots\!28\)\( p^{114} T^{24} - \)\(49\!\cdots\!92\)\( p^{152} T^{26} + \)\(45\!\cdots\!32\)\( p^{190} T^{28} - \)\(32\!\cdots\!76\)\( p^{228} T^{30} + \)\(17\!\cdots\!25\)\( p^{266} T^{32} - \)\(87\!\cdots\!38\)\( p^{305} T^{34} + p^{342} T^{36} \) | |
| 71 | \( ( 1 + 799738994599899528 T + \)\(77\!\cdots\!91\)\( T^{2} + \)\(47\!\cdots\!52\)\( T^{3} + \)\(27\!\cdots\!80\)\( T^{4} + \)\(13\!\cdots\!68\)\( T^{5} + \)\(64\!\cdots\!44\)\( T^{6} + \)\(28\!\cdots\!52\)\( T^{7} + \)\(11\!\cdots\!54\)\( T^{8} + \)\(46\!\cdots\!40\)\( T^{9} + \)\(11\!\cdots\!54\)\( p^{19} T^{10} + \)\(28\!\cdots\!52\)\( p^{38} T^{11} + \)\(64\!\cdots\!44\)\( p^{57} T^{12} + \)\(13\!\cdots\!68\)\( p^{76} T^{13} + \)\(27\!\cdots\!80\)\( p^{95} T^{14} + \)\(47\!\cdots\!52\)\( p^{114} T^{15} + \)\(77\!\cdots\!91\)\( p^{133} T^{16} + 799738994599899528 p^{152} T^{17} + p^{171} T^{18} )^{2} \) | |
| 73 | \( ( 1 + 46502149494904534 T + \)\(10\!\cdots\!01\)\( T^{2} + \)\(15\!\cdots\!08\)\( T^{3} + \)\(82\!\cdots\!52\)\( p T^{4} + \)\(12\!\cdots\!44\)\( T^{5} + \)\(24\!\cdots\!36\)\( T^{6} + \)\(49\!\cdots\!92\)\( T^{7} + \)\(77\!\cdots\!02\)\( T^{8} + \)\(13\!\cdots\!04\)\( T^{9} + \)\(77\!\cdots\!02\)\( p^{19} T^{10} + \)\(49\!\cdots\!92\)\( p^{38} T^{11} + \)\(24\!\cdots\!36\)\( p^{57} T^{12} + \)\(12\!\cdots\!44\)\( p^{76} T^{13} + \)\(82\!\cdots\!52\)\( p^{96} T^{14} + \)\(15\!\cdots\!08\)\( p^{114} T^{15} + \)\(10\!\cdots\!01\)\( p^{133} T^{16} + 46502149494904534 p^{152} T^{17} + p^{171} T^{18} )^{2} \) | |
| 79 | \( ( 1 - 906987900512632240 T + \)\(41\!\cdots\!87\)\( T^{2} - \)\(30\!\cdots\!64\)\( T^{3} + \)\(11\!\cdots\!40\)\( T^{4} - \)\(82\!\cdots\!96\)\( T^{5} + \)\(21\!\cdots\!88\)\( T^{6} - \)\(13\!\cdots\!20\)\( T^{7} + \)\(31\!\cdots\!58\)\( T^{8} - \)\(18\!\cdots\!20\)\( T^{9} + \)\(31\!\cdots\!58\)\( p^{19} T^{10} - \)\(13\!\cdots\!20\)\( p^{38} T^{11} + \)\(21\!\cdots\!88\)\( p^{57} T^{12} - \)\(82\!\cdots\!96\)\( p^{76} T^{13} + \)\(11\!\cdots\!40\)\( p^{95} T^{14} - \)\(30\!\cdots\!64\)\( p^{114} T^{15} + \)\(41\!\cdots\!87\)\( p^{133} T^{16} - 906987900512632240 p^{152} T^{17} + p^{171} T^{18} )^{2} \) | |
| 83 | \( 1 - \)\(27\!\cdots\!10\)\( T^{2} + \)\(37\!\cdots\!69\)\( T^{4} - \)\(34\!\cdots\!04\)\( T^{6} + \)\(24\!\cdots\!12\)\( T^{8} - \)\(13\!\cdots\!24\)\( T^{10} + \)\(61\!\cdots\!92\)\( T^{12} - \)\(24\!\cdots\!88\)\( T^{14} + \)\(84\!\cdots\!82\)\( T^{16} - \)\(25\!\cdots\!08\)\( T^{18} + \)\(84\!\cdots\!82\)\( p^{38} T^{20} - \)\(24\!\cdots\!88\)\( p^{76} T^{22} + \)\(61\!\cdots\!92\)\( p^{114} T^{24} - \)\(13\!\cdots\!24\)\( p^{152} T^{26} + \)\(24\!\cdots\!12\)\( p^{190} T^{28} - \)\(34\!\cdots\!04\)\( p^{228} T^{30} + \)\(37\!\cdots\!69\)\( p^{266} T^{32} - \)\(27\!\cdots\!10\)\( p^{304} T^{34} + p^{342} T^{36} \) | |
| 89 | \( ( 1 - 2235254885522871770 T + \)\(41\!\cdots\!77\)\( T^{2} - \)\(60\!\cdots\!24\)\( T^{3} + \)\(86\!\cdots\!00\)\( T^{4} - \)\(68\!\cdots\!56\)\( T^{5} + \)\(12\!\cdots\!08\)\( T^{6} - \)\(38\!\cdots\!00\)\( T^{7} + \)\(15\!\cdots\!98\)\( T^{8} - \)\(15\!\cdots\!80\)\( T^{9} + \)\(15\!\cdots\!98\)\( p^{19} T^{10} - \)\(38\!\cdots\!00\)\( p^{38} T^{11} + \)\(12\!\cdots\!08\)\( p^{57} T^{12} - \)\(68\!\cdots\!56\)\( p^{76} T^{13} + \)\(86\!\cdots\!00\)\( p^{95} T^{14} - \)\(60\!\cdots\!24\)\( p^{114} T^{15} + \)\(41\!\cdots\!77\)\( p^{133} T^{16} - 2235254885522871770 p^{152} T^{17} + p^{171} T^{18} )^{2} \) | |
| 97 | \( ( 1 - 4123214894732058610 T + \)\(14\!\cdots\!05\)\( T^{2} - \)\(82\!\cdots\!32\)\( T^{3} + \)\(90\!\cdots\!72\)\( T^{4} + \)\(10\!\cdots\!00\)\( T^{5} + \)\(42\!\cdots\!92\)\( T^{6} - \)\(76\!\cdots\!44\)\( T^{7} + \)\(29\!\cdots\!38\)\( T^{8} - \)\(14\!\cdots\!80\)\( T^{9} + \)\(29\!\cdots\!38\)\( p^{19} T^{10} - \)\(76\!\cdots\!44\)\( p^{38} T^{11} + \)\(42\!\cdots\!92\)\( p^{57} T^{12} + \)\(10\!\cdots\!00\)\( p^{76} T^{13} + \)\(90\!\cdots\!72\)\( p^{95} T^{14} - \)\(82\!\cdots\!32\)\( p^{114} T^{15} + \)\(14\!\cdots\!05\)\( p^{133} T^{16} - 4123214894732058610 p^{152} T^{17} + p^{171} T^{18} )^{2} \) | |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{36} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−2.90947051724966726143292403601, −2.87620629719959687081800520225, −2.83801492363039183571453687630, −2.81591091515478260787988113159, −2.30380312061133862731177615675, −2.24207716039105267176272163400, −2.12505638181536741177427953186, −1.85174608823908056205057583229, −1.82080734743067700944125352800, −1.79549469338365274062443917029, −1.76751833478733566162641114262, −1.72080640040631325723804110108, −1.28050609307191720766064750124, −1.25113620453891872532878375260, −1.17089537917863706019680589696, −1.16526510462630504903205839916, −1.05237950915081710474232318611, −0.969806628548077081821495839664, −0.833810432074235164452850199942, −0.809453517584328916860452703153, −0.75632830801987315545711665525, −0.39161270295712337687007204622, −0.36444682631327675081448360087, −0.22574297581910259820572998603, −0.15850747222322163598465458566, 0.15850747222322163598465458566, 0.22574297581910259820572998603, 0.36444682631327675081448360087, 0.39161270295712337687007204622, 0.75632830801987315545711665525, 0.809453517584328916860452703153, 0.833810432074235164452850199942, 0.969806628548077081821495839664, 1.05237950915081710474232318611, 1.16526510462630504903205839916, 1.17089537917863706019680589696, 1.25113620453891872532878375260, 1.28050609307191720766064750124, 1.72080640040631325723804110108, 1.76751833478733566162641114262, 1.79549469338365274062443917029, 1.82080734743067700944125352800, 1.85174608823908056205057583229, 2.12505638181536741177427953186, 2.24207716039105267176272163400, 2.30380312061133862731177615675, 2.81591091515478260787988113159, 2.83801492363039183571453687630, 2.87620629719959687081800520225, 2.90947051724966726143292403601
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.