Dirichlet series
| L(s) = 1 | + 4.51e4·9-s − 5.38e4·13-s + 2.55e5·17-s + 6.25e5·25-s − 8.99e5·29-s + 2.54e6·37-s − 1.13e7·41-s + 4.36e7·49-s − 1.24e7·53-s + 1.17e7·61-s − 4.20e7·73-s + 1.00e9·81-s − 1.20e8·89-s − 1.95e8·97-s − 3.65e8·101-s + 5.14e8·109-s + 2.45e8·113-s − 2.42e9·117-s + 1.73e9·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 1.15e10·153-s + 157-s + ⋯ |
| L(s) = 1 | + 6.87·9-s − 1.88·13-s + 3.06·17-s + 8/5·25-s − 1.27·29-s + 1.35·37-s − 4.02·41-s + 7.58·49-s − 1.57·53-s + 0.848·61-s − 1.48·73-s + 23.3·81-s − 1.91·89-s − 2.20·97-s − 3.50·101-s + 3.64·109-s + 1.50·113-s − 12.9·117-s + 8.08·121-s + 21.0·153-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{96} \cdot 5^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(9-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{96} \cdot 5^{16}\right)^{s/2} \, \Gamma_{\C}(s+4)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(32\) |
| Conductor: | \(2^{96} \cdot 5^{16}\) |
| Sign: | $1$ |
| Analytic conductor: | \(6.95618\times 10^{33}\) |
| Root analytic conductor: | \(11.4175\) |
| Motivic weight: | \(8\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((32,\ 2^{96} \cdot 5^{16} ,\ ( \ : [4]^{16} ),\ 1 )\) |
Particular Values
| \(L(\frac{9}{2})\) | \(\approx\) | \(0.04137615387\) |
| \(L(\frac12)\) | \(\approx\) | \(0.04137615387\) |
| \(L(5)\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( 1 \) |
| 5 | \( ( 1 - p^{7} T^{2} )^{8} \) | |
| good | 3 | \( 1 - 15040 p T^{2} + 1032272440 T^{4} - 5358055278656 p T^{6} + 7099812352705940 p^{3} T^{8} - 69250598351440351424 p^{3} T^{10} + \)\(19\!\cdots\!36\)\( p^{4} T^{12} - \)\(17\!\cdots\!80\)\( p^{8} T^{14} + \)\(14\!\cdots\!54\)\( p^{12} T^{16} - \)\(17\!\cdots\!80\)\( p^{24} T^{18} + \)\(19\!\cdots\!36\)\( p^{36} T^{20} - 69250598351440351424 p^{51} T^{22} + 7099812352705940 p^{67} T^{24} - 5358055278656 p^{81} T^{26} + 1032272440 p^{96} T^{28} - 15040 p^{113} T^{30} + p^{128} T^{32} \) |
| 7 | \( 1 - 6242560 p T^{2} + 932883004267960 T^{4} - \)\(26\!\cdots\!92\)\( p^{2} T^{6} + \)\(80\!\cdots\!00\)\( p^{5} T^{8} - \)\(94\!\cdots\!52\)\( p^{6} T^{10} + \)\(13\!\cdots\!56\)\( p^{8} T^{12} - \)\(16\!\cdots\!60\)\( p^{10} T^{14} + \)\(20\!\cdots\!34\)\( p^{12} T^{16} - \)\(16\!\cdots\!60\)\( p^{26} T^{18} + \)\(13\!\cdots\!56\)\( p^{40} T^{20} - \)\(94\!\cdots\!52\)\( p^{54} T^{22} + \)\(80\!\cdots\!00\)\( p^{69} T^{24} - \)\(26\!\cdots\!92\)\( p^{82} T^{26} + 932883004267960 p^{96} T^{28} - 6242560 p^{113} T^{30} + p^{128} T^{32} \) | |
| 11 | \( 1 - 1733041488 T^{2} + 1497224810557544952 T^{4} - \)\(79\!\cdots\!56\)\( p T^{6} + \)\(39\!\cdots\!56\)\( T^{8} - \)\(14\!\cdots\!44\)\( T^{10} + \)\(44\!\cdots\!92\)\( T^{12} - \)\(11\!\cdots\!52\)\( T^{14} + \)\(27\!\cdots\!18\)\( T^{16} - \)\(11\!\cdots\!52\)\( p^{16} T^{18} + \)\(44\!\cdots\!92\)\( p^{32} T^{20} - \)\(14\!\cdots\!44\)\( p^{48} T^{22} + \)\(39\!\cdots\!56\)\( p^{64} T^{24} - \)\(79\!\cdots\!56\)\( p^{81} T^{26} + 1497224810557544952 p^{96} T^{28} - 1733041488 p^{112} T^{30} + p^{128} T^{32} \) | |
| 13 | \( ( 1 + 26912 T + 4462718840 T^{2} + 119734403289184 T^{3} + 772439072580174732 p T^{4} + \)\(24\!\cdots\!92\)\( T^{5} + \)\(14\!\cdots\!28\)\( T^{6} + \)\(30\!\cdots\!96\)\( T^{7} + \)\(14\!\cdots\!58\)\( T^{8} + \)\(30\!\cdots\!96\)\( p^{8} T^{9} + \)\(14\!\cdots\!28\)\( p^{16} T^{10} + \)\(24\!\cdots\!92\)\( p^{24} T^{11} + 772439072580174732 p^{33} T^{12} + 119734403289184 p^{40} T^{13} + 4462718840 p^{48} T^{14} + 26912 p^{56} T^{15} + p^{64} T^{16} )^{2} \) | |
| 17 | \( ( 1 - 127952 T + 26370906232 T^{2} - 2049465150636400 T^{3} + \)\(26\!\cdots\!76\)\( T^{4} - \)\(10\!\cdots\!12\)\( p T^{5} + \)\(25\!\cdots\!88\)\( T^{6} - \)\(18\!\cdots\!36\)\( T^{7} + \)\(22\!\cdots\!06\)\( T^{8} - \)\(18\!\cdots\!36\)\( p^{8} T^{9} + \)\(25\!\cdots\!88\)\( p^{16} T^{10} - \)\(10\!\cdots\!12\)\( p^{25} T^{11} + \)\(26\!\cdots\!76\)\( p^{32} T^{12} - 2049465150636400 p^{40} T^{13} + 26370906232 p^{48} T^{14} - 127952 p^{56} T^{15} + p^{64} T^{16} )^{2} \) | |
| 19 | \( 1 - 86243586704 T^{2} + \)\(32\!\cdots\!00\)\( T^{4} - \)\(72\!\cdots\!60\)\( T^{6} + \)\(11\!\cdots\!00\)\( T^{8} - \)\(10\!\cdots\!92\)\( T^{10} - \)\(26\!\cdots\!92\)\( T^{12} + \)\(32\!\cdots\!00\)\( T^{14} - \)\(71\!\cdots\!70\)\( T^{16} + \)\(32\!\cdots\!00\)\( p^{16} T^{18} - \)\(26\!\cdots\!92\)\( p^{32} T^{20} - \)\(10\!\cdots\!92\)\( p^{48} T^{22} + \)\(11\!\cdots\!00\)\( p^{64} T^{24} - \)\(72\!\cdots\!60\)\( p^{80} T^{26} + \)\(32\!\cdots\!00\)\( p^{96} T^{28} - 86243586704 p^{112} T^{30} + p^{128} T^{32} \) | |
| 23 | \( 1 - 1030870151808 T^{2} + \)\(51\!\cdots\!32\)\( T^{4} - \)\(16\!\cdots\!76\)\( T^{6} + \)\(15\!\cdots\!72\)\( p T^{8} - \)\(62\!\cdots\!84\)\( T^{10} + \)\(84\!\cdots\!92\)\( T^{12} - \)\(90\!\cdots\!32\)\( T^{14} + \)\(78\!\cdots\!58\)\( T^{16} - \)\(90\!\cdots\!32\)\( p^{16} T^{18} + \)\(84\!\cdots\!92\)\( p^{32} T^{20} - \)\(62\!\cdots\!84\)\( p^{48} T^{22} + \)\(15\!\cdots\!72\)\( p^{65} T^{24} - \)\(16\!\cdots\!76\)\( p^{80} T^{26} + \)\(51\!\cdots\!32\)\( p^{96} T^{28} - 1030870151808 p^{112} T^{30} + p^{128} T^{32} \) | |
| 29 | \( ( 1 + 449904 T + 1361406145336 T^{2} + 820715516360870736 T^{3} + \)\(14\!\cdots\!40\)\( T^{4} + \)\(80\!\cdots\!44\)\( T^{5} + \)\(10\!\cdots\!72\)\( T^{6} + \)\(53\!\cdots\!40\)\( T^{7} + \)\(61\!\cdots\!54\)\( T^{8} + \)\(53\!\cdots\!40\)\( p^{8} T^{9} + \)\(10\!\cdots\!72\)\( p^{16} T^{10} + \)\(80\!\cdots\!44\)\( p^{24} T^{11} + \)\(14\!\cdots\!40\)\( p^{32} T^{12} + 820715516360870736 p^{40} T^{13} + 1361406145336 p^{48} T^{14} + 449904 p^{56} T^{15} + p^{64} T^{16} )^{2} \) | |
| 31 | \( 1 - 8556369362896 T^{2} + \)\(36\!\cdots\!88\)\( T^{4} - \)\(10\!\cdots\!16\)\( T^{6} + \)\(21\!\cdots\!24\)\( T^{8} - \)\(34\!\cdots\!36\)\( T^{10} + \)\(46\!\cdots\!76\)\( T^{12} - \)\(16\!\cdots\!12\)\( p T^{14} + \)\(49\!\cdots\!82\)\( p^{2} T^{16} - \)\(16\!\cdots\!12\)\( p^{17} T^{18} + \)\(46\!\cdots\!76\)\( p^{32} T^{20} - \)\(34\!\cdots\!36\)\( p^{48} T^{22} + \)\(21\!\cdots\!24\)\( p^{64} T^{24} - \)\(10\!\cdots\!16\)\( p^{80} T^{26} + \)\(36\!\cdots\!88\)\( p^{96} T^{28} - 8556369362896 p^{112} T^{30} + p^{128} T^{32} \) | |
| 37 | \( ( 1 - 1270496 T + 11006460964728 T^{2} - 15737787526999602592 T^{3} + \)\(75\!\cdots\!88\)\( T^{4} - \)\(10\!\cdots\!72\)\( T^{5} + \)\(10\!\cdots\!16\)\( p T^{6} - \)\(53\!\cdots\!48\)\( T^{7} + \)\(15\!\cdots\!02\)\( T^{8} - \)\(53\!\cdots\!48\)\( p^{8} T^{9} + \)\(10\!\cdots\!16\)\( p^{17} T^{10} - \)\(10\!\cdots\!72\)\( p^{24} T^{11} + \)\(75\!\cdots\!88\)\( p^{32} T^{12} - 15737787526999602592 p^{40} T^{13} + 11006460964728 p^{48} T^{14} - 1270496 p^{56} T^{15} + p^{64} T^{16} )^{2} \) | |
| 41 | \( ( 1 + 5688928 T + 62007681352824 T^{2} + \)\(26\!\cdots\!56\)\( T^{3} + \)\(16\!\cdots\!04\)\( T^{4} + \)\(57\!\cdots\!08\)\( T^{5} + \)\(25\!\cdots\!56\)\( T^{6} + \)\(72\!\cdots\!80\)\( T^{7} + \)\(25\!\cdots\!90\)\( T^{8} + \)\(72\!\cdots\!80\)\( p^{8} T^{9} + \)\(25\!\cdots\!56\)\( p^{16} T^{10} + \)\(57\!\cdots\!08\)\( p^{24} T^{11} + \)\(16\!\cdots\!04\)\( p^{32} T^{12} + \)\(26\!\cdots\!56\)\( p^{40} T^{13} + 62007681352824 p^{48} T^{14} + 5688928 p^{56} T^{15} + p^{64} T^{16} )^{2} \) | |
| 43 | \( 1 - 117833435141056 T^{2} + \)\(64\!\cdots\!88\)\( T^{4} - \)\(21\!\cdots\!76\)\( T^{6} + \)\(24\!\cdots\!36\)\( p^{2} T^{8} - \)\(65\!\cdots\!56\)\( T^{10} + \)\(59\!\cdots\!56\)\( T^{12} - \)\(26\!\cdots\!32\)\( T^{14} + \)\(44\!\cdots\!22\)\( T^{16} - \)\(26\!\cdots\!32\)\( p^{16} T^{18} + \)\(59\!\cdots\!56\)\( p^{32} T^{20} - \)\(65\!\cdots\!56\)\( p^{48} T^{22} + \)\(24\!\cdots\!36\)\( p^{66} T^{24} - \)\(21\!\cdots\!76\)\( p^{80} T^{26} + \)\(64\!\cdots\!88\)\( p^{96} T^{28} - 117833435141056 p^{112} T^{30} + p^{128} T^{32} \) | |
| 47 | \( 1 - 171997151288704 T^{2} + \)\(15\!\cdots\!20\)\( T^{4} - \)\(10\!\cdots\!80\)\( T^{6} + \)\(49\!\cdots\!20\)\( T^{8} - \)\(20\!\cdots\!12\)\( T^{10} + \)\(69\!\cdots\!28\)\( T^{12} - \)\(20\!\cdots\!60\)\( T^{14} + \)\(52\!\cdots\!90\)\( T^{16} - \)\(20\!\cdots\!60\)\( p^{16} T^{18} + \)\(69\!\cdots\!28\)\( p^{32} T^{20} - \)\(20\!\cdots\!12\)\( p^{48} T^{22} + \)\(49\!\cdots\!20\)\( p^{64} T^{24} - \)\(10\!\cdots\!80\)\( p^{80} T^{26} + \)\(15\!\cdots\!20\)\( p^{96} T^{28} - 171997151288704 p^{112} T^{30} + p^{128} T^{32} \) | |
| 53 | \( ( 1 + 6228192 T + 255496320951800 T^{2} + \)\(11\!\cdots\!60\)\( T^{3} + \)\(27\!\cdots\!44\)\( T^{4} + \)\(54\!\cdots\!72\)\( T^{5} + \)\(17\!\cdots\!12\)\( T^{6} - \)\(11\!\cdots\!60\)\( T^{7} + \)\(99\!\cdots\!34\)\( T^{8} - \)\(11\!\cdots\!60\)\( p^{8} T^{9} + \)\(17\!\cdots\!12\)\( p^{16} T^{10} + \)\(54\!\cdots\!72\)\( p^{24} T^{11} + \)\(27\!\cdots\!44\)\( p^{32} T^{12} + \)\(11\!\cdots\!60\)\( p^{40} T^{13} + 255496320951800 p^{48} T^{14} + 6228192 p^{56} T^{15} + p^{64} T^{16} )^{2} \) | |
| 59 | \( 1 - 658614028525456 T^{2} + \)\(23\!\cdots\!76\)\( T^{4} - \)\(59\!\cdots\!12\)\( T^{6} + \)\(12\!\cdots\!32\)\( T^{8} - \)\(23\!\cdots\!56\)\( T^{10} + \)\(41\!\cdots\!72\)\( T^{12} - \)\(69\!\cdots\!72\)\( T^{14} + \)\(30\!\cdots\!46\)\( p^{2} T^{16} - \)\(69\!\cdots\!72\)\( p^{16} T^{18} + \)\(41\!\cdots\!72\)\( p^{32} T^{20} - \)\(23\!\cdots\!56\)\( p^{48} T^{22} + \)\(12\!\cdots\!32\)\( p^{64} T^{24} - \)\(59\!\cdots\!12\)\( p^{80} T^{26} + \)\(23\!\cdots\!76\)\( p^{96} T^{28} - 658614028525456 p^{112} T^{30} + p^{128} T^{32} \) | |
| 61 | \( ( 1 - 5876096 T + 1100803835174328 T^{2} - \)\(61\!\cdots\!00\)\( T^{3} + \)\(56\!\cdots\!28\)\( T^{4} - \)\(29\!\cdots\!24\)\( T^{5} + \)\(17\!\cdots\!04\)\( T^{6} - \)\(88\!\cdots\!84\)\( T^{7} + \)\(39\!\cdots\!50\)\( T^{8} - \)\(88\!\cdots\!84\)\( p^{8} T^{9} + \)\(17\!\cdots\!04\)\( p^{16} T^{10} - \)\(29\!\cdots\!24\)\( p^{24} T^{11} + \)\(56\!\cdots\!28\)\( p^{32} T^{12} - \)\(61\!\cdots\!00\)\( p^{40} T^{13} + 1100803835174328 p^{48} T^{14} - 5876096 p^{56} T^{15} + p^{64} T^{16} )^{2} \) | |
| 67 | \( 1 - 2538946890848704 T^{2} + \)\(37\!\cdots\!76\)\( T^{4} - \)\(38\!\cdots\!60\)\( T^{6} + \)\(31\!\cdots\!60\)\( T^{8} - \)\(20\!\cdots\!36\)\( T^{10} + \)\(11\!\cdots\!00\)\( T^{12} - \)\(59\!\cdots\!60\)\( T^{14} + \)\(25\!\cdots\!06\)\( T^{16} - \)\(59\!\cdots\!60\)\( p^{16} T^{18} + \)\(11\!\cdots\!00\)\( p^{32} T^{20} - \)\(20\!\cdots\!36\)\( p^{48} T^{22} + \)\(31\!\cdots\!60\)\( p^{64} T^{24} - \)\(38\!\cdots\!60\)\( p^{80} T^{26} + \)\(37\!\cdots\!76\)\( p^{96} T^{28} - 2538946890848704 p^{112} T^{30} + p^{128} T^{32} \) | |
| 71 | \( 1 - 3507897904163280 T^{2} + \)\(72\!\cdots\!00\)\( T^{4} - \)\(11\!\cdots\!28\)\( T^{6} + \)\(13\!\cdots\!80\)\( T^{8} - \)\(14\!\cdots\!88\)\( T^{10} + \)\(12\!\cdots\!16\)\( T^{12} - \)\(10\!\cdots\!20\)\( T^{14} + \)\(69\!\cdots\!94\)\( T^{16} - \)\(10\!\cdots\!20\)\( p^{16} T^{18} + \)\(12\!\cdots\!16\)\( p^{32} T^{20} - \)\(14\!\cdots\!88\)\( p^{48} T^{22} + \)\(13\!\cdots\!80\)\( p^{64} T^{24} - \)\(11\!\cdots\!28\)\( p^{80} T^{26} + \)\(72\!\cdots\!00\)\( p^{96} T^{28} - 3507897904163280 p^{112} T^{30} + p^{128} T^{32} \) | |
| 73 | \( ( 1 + 21047536 T + 997518616247288 T^{2} + \)\(17\!\cdots\!40\)\( T^{3} + \)\(96\!\cdots\!68\)\( T^{4} + \)\(12\!\cdots\!24\)\( T^{5} + \)\(89\!\cdots\!64\)\( T^{6} + \)\(10\!\cdots\!84\)\( T^{7} + \)\(41\!\cdots\!70\)\( T^{8} + \)\(10\!\cdots\!84\)\( p^{8} T^{9} + \)\(89\!\cdots\!64\)\( p^{16} T^{10} + \)\(12\!\cdots\!24\)\( p^{24} T^{11} + \)\(96\!\cdots\!68\)\( p^{32} T^{12} + \)\(17\!\cdots\!40\)\( p^{40} T^{13} + 997518616247288 p^{48} T^{14} + 21047536 p^{56} T^{15} + p^{64} T^{16} )^{2} \) | |
| 79 | \( 1 - 8888004577385232 T^{2} + \)\(46\!\cdots\!28\)\( T^{4} - \)\(17\!\cdots\!68\)\( T^{6} + \)\(53\!\cdots\!08\)\( T^{8} - \)\(13\!\cdots\!68\)\( T^{10} + \)\(29\!\cdots\!88\)\( T^{12} - \)\(54\!\cdots\!88\)\( T^{14} + \)\(89\!\cdots\!58\)\( T^{16} - \)\(54\!\cdots\!88\)\( p^{16} T^{18} + \)\(29\!\cdots\!88\)\( p^{32} T^{20} - \)\(13\!\cdots\!68\)\( p^{48} T^{22} + \)\(53\!\cdots\!08\)\( p^{64} T^{24} - \)\(17\!\cdots\!68\)\( p^{80} T^{26} + \)\(46\!\cdots\!28\)\( p^{96} T^{28} - 8888004577385232 p^{112} T^{30} + p^{128} T^{32} \) | |
| 83 | \( 1 - 25133783534610624 T^{2} + \)\(31\!\cdots\!48\)\( T^{4} - \)\(30\!\cdots\!60\)\( p T^{6} + \)\(18\!\cdots\!16\)\( p T^{8} - \)\(70\!\cdots\!36\)\( T^{10} + \)\(26\!\cdots\!84\)\( T^{12} - \)\(78\!\cdots\!36\)\( T^{14} + \)\(19\!\cdots\!10\)\( T^{16} - \)\(78\!\cdots\!36\)\( p^{16} T^{18} + \)\(26\!\cdots\!84\)\( p^{32} T^{20} - \)\(70\!\cdots\!36\)\( p^{48} T^{22} + \)\(18\!\cdots\!16\)\( p^{65} T^{24} - \)\(30\!\cdots\!60\)\( p^{81} T^{26} + \)\(31\!\cdots\!48\)\( p^{96} T^{28} - 25133783534610624 p^{112} T^{30} + p^{128} T^{32} \) | |
| 89 | \( ( 1 + 60068528 T + 12368576167357432 T^{2} + \)\(90\!\cdots\!28\)\( T^{3} + \)\(10\!\cdots\!48\)\( T^{4} + \)\(68\!\cdots\!08\)\( T^{5} + \)\(64\!\cdots\!40\)\( T^{6} + \)\(36\!\cdots\!40\)\( T^{7} + \)\(29\!\cdots\!06\)\( T^{8} + \)\(36\!\cdots\!40\)\( p^{8} T^{9} + \)\(64\!\cdots\!40\)\( p^{16} T^{10} + \)\(68\!\cdots\!08\)\( p^{24} T^{11} + \)\(10\!\cdots\!48\)\( p^{32} T^{12} + \)\(90\!\cdots\!28\)\( p^{40} T^{13} + 12368576167357432 p^{48} T^{14} + 60068528 p^{56} T^{15} + p^{64} T^{16} )^{2} \) | |
| 97 | \( ( 1 + 97624720 T + 42923602103078072 T^{2} + \)\(17\!\cdots\!00\)\( T^{3} + \)\(59\!\cdots\!08\)\( T^{4} - \)\(16\!\cdots\!20\)\( T^{5} + \)\(28\!\cdots\!44\)\( T^{6} - \)\(60\!\cdots\!00\)\( T^{7} + \)\(48\!\cdots\!50\)\( T^{8} - \)\(60\!\cdots\!00\)\( p^{8} T^{9} + \)\(28\!\cdots\!44\)\( p^{16} T^{10} - \)\(16\!\cdots\!20\)\( p^{24} T^{11} + \)\(59\!\cdots\!08\)\( p^{32} T^{12} + \)\(17\!\cdots\!00\)\( p^{40} T^{13} + 42923602103078072 p^{48} T^{14} + 97624720 p^{56} T^{15} + p^{64} T^{16} )^{2} \) | |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{32} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−1.92638764564198073864770423894, −1.77955562953812936476740699182, −1.76510043755206012656502526319, −1.67815648765465346475454831783, −1.61699664903294634439256261523, −1.54983244935172653485293233026, −1.43139981969572877817922203448, −1.40293839567610677259736836917, −1.27900040648478005506697867199, −1.21830887573111463180507176183, −1.15216517430562944685076332878, −1.07348723964496043197220357864, −1.04219053270101161911115187691, −1.03085403104423488493886365717, −0.961864587990048790283083650782, −0.798732967053010703089747564763, −0.73624909324362972114804717463, −0.65517217012758861182727967888, −0.62651432099741481477265065882, −0.54632053072324043390705219019, −0.53354246994954216727344534556, −0.21236200815390454049439198775, −0.14135570421993786409720358584, −0.092991251204926403660187581898, −0.00529815321356503678587332366, 0.00529815321356503678587332366, 0.092991251204926403660187581898, 0.14135570421993786409720358584, 0.21236200815390454049439198775, 0.53354246994954216727344534556, 0.54632053072324043390705219019, 0.62651432099741481477265065882, 0.65517217012758861182727967888, 0.73624909324362972114804717463, 0.798732967053010703089747564763, 0.961864587990048790283083650782, 1.03085403104423488493886365717, 1.04219053270101161911115187691, 1.07348723964496043197220357864, 1.15216517430562944685076332878, 1.21830887573111463180507176183, 1.27900040648478005506697867199, 1.40293839567610677259736836917, 1.43139981969572877817922203448, 1.54983244935172653485293233026, 1.61699664903294634439256261523, 1.67815648765465346475454831783, 1.76510043755206012656502526319, 1.77955562953812936476740699182, 1.92638764564198073864770423894
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.