Dirichlet series
| L(s) = 1 | − 2·5-s + 162·9-s − 28·11-s − 68·19-s − 9·25-s − 340·29-s + 336·31-s + 236·41-s − 324·45-s + 2.64e3·49-s + 56·55-s − 1.75e3·59-s + 780·61-s + 72·71-s − 1.95e3·79-s + 1.25e4·81-s − 292·89-s + 136·95-s − 4.53e3·99-s + 3.72e3·101-s − 1.74e3·109-s − 1.06e4·121-s − 452·125-s + 127-s + 131-s + 137-s + 139-s + ⋯ |
| L(s) = 1 | − 0.178·5-s + 6·9-s − 0.767·11-s − 0.821·19-s − 0.0719·25-s − 2.17·29-s + 1.94·31-s + 0.898·41-s − 1.07·45-s + 54/7·49-s + 0.137·55-s − 3.87·59-s + 1.63·61-s + 0.120·71-s − 2.77·79-s + 17.2·81-s − 0.347·89-s + 0.146·95-s − 4.60·99-s + 3.66·101-s − 1.53·109-s − 8.01·121-s − 0.323·125-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{126} \cdot 5^{18}\right)^{s/2} \, \Gamma_{\C}(s)^{18} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{126} \cdot 5^{18}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{18} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(36\) |
| Conductor: | \(2^{126} \cdot 5^{18}\) |
| Sign: | $1$ |
| Analytic conductor: | \(2.43685\times 10^{28}\) |
| Root analytic conductor: | \(6.14501\) |
| Motivic weight: | \(3\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((36,\ 2^{126} \cdot 5^{18} ,\ ( \ : [3/2]^{18} ),\ 1 )\) |
Particular Values
| \(L(2)\) | \(\approx\) | \(0.06284480792\) |
| \(L(\frac12)\) | \(\approx\) | \(0.06284480792\) |
| \(L(\frac{5}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + 2 T + 13 T^{2} + 496 T^{3} + 10556 T^{4} - 77448 T^{5} + 521596 p T^{6} + 85136 p^{2} T^{7} + 676718 p^{3} T^{8} - 1895412 p^{4} T^{9} + 676718 p^{6} T^{10} + 85136 p^{8} T^{11} + 521596 p^{10} T^{12} - 77448 p^{12} T^{13} + 10556 p^{15} T^{14} + 496 p^{18} T^{15} + 13 p^{21} T^{16} + 2 p^{24} T^{17} + p^{27} T^{18} \) | |
| good | 3 | \( 1 - 2 p^{4} T^{2} + 13697 T^{4} - 89056 p^{2} T^{6} + 11917900 p T^{8} - 1276761368 T^{10} + 12599676220 p T^{12} - 321794417120 p T^{14} + 23009371354142 T^{16} - 581214078652172 T^{18} + 23009371354142 p^{6} T^{20} - 321794417120 p^{13} T^{22} + 12599676220 p^{19} T^{24} - 1276761368 p^{24} T^{26} + 11917900 p^{31} T^{28} - 89056 p^{38} T^{30} + 13697 p^{42} T^{32} - 2 p^{52} T^{34} + p^{54} T^{36} \) |
| 7 | \( 1 - 54 p^{2} T^{2} + 3505273 T^{4} - 3175456896 T^{6} + 2244628262820 T^{8} - 1321783876957640 T^{10} + 671383044235627732 T^{12} - \)\(30\!\cdots\!00\)\( T^{14} + \)\(12\!\cdots\!30\)\( T^{16} - \)\(43\!\cdots\!76\)\( T^{18} + \)\(12\!\cdots\!30\)\( p^{6} T^{20} - \)\(30\!\cdots\!00\)\( p^{12} T^{22} + 671383044235627732 p^{18} T^{24} - 1321783876957640 p^{24} T^{26} + 2244628262820 p^{30} T^{28} - 3175456896 p^{36} T^{30} + 3505273 p^{42} T^{32} - 54 p^{50} T^{34} + p^{54} T^{36} \) | |
| 11 | \( ( 1 + 14 T + 5627 T^{2} + 3120 p T^{3} + 15719060 T^{4} + 15589128 T^{5} + 32297994924 T^{6} + 11904627440 T^{7} + 4972487740586 p T^{8} + 52083142943380 T^{9} + 4972487740586 p^{4} T^{10} + 11904627440 p^{6} T^{11} + 32297994924 p^{9} T^{12} + 15589128 p^{12} T^{13} + 15719060 p^{15} T^{14} + 3120 p^{19} T^{15} + 5627 p^{21} T^{16} + 14 p^{24} T^{17} + p^{27} T^{18} )^{2} \) | |
| 13 | \( 1 - 18410 T^{2} + 168206929 T^{4} - 1021303084400 T^{6} + 4682432877868276 T^{8} - 17526512779132279448 T^{10} + \)\(56\!\cdots\!04\)\( T^{12} - \)\(16\!\cdots\!52\)\( T^{14} + \)\(41\!\cdots\!06\)\( T^{16} - \)\(96\!\cdots\!68\)\( T^{18} + \)\(41\!\cdots\!06\)\( p^{6} T^{20} - \)\(16\!\cdots\!52\)\( p^{12} T^{22} + \)\(56\!\cdots\!04\)\( p^{18} T^{24} - 17526512779132279448 p^{24} T^{26} + 4682432877868276 p^{30} T^{28} - 1021303084400 p^{36} T^{30} + 168206929 p^{42} T^{32} - 18410 p^{48} T^{34} + p^{54} T^{36} \) | |
| 17 | \( 1 - 45282 T^{2} + 1080021241 T^{4} - 17672305163440 T^{6} + 220352751485334260 T^{8} - \)\(22\!\cdots\!96\)\( T^{10} + \)\(18\!\cdots\!40\)\( T^{12} - \)\(13\!\cdots\!44\)\( T^{14} + \)\(79\!\cdots\!74\)\( T^{16} - \)\(42\!\cdots\!36\)\( T^{18} + \)\(79\!\cdots\!74\)\( p^{6} T^{20} - \)\(13\!\cdots\!44\)\( p^{12} T^{22} + \)\(18\!\cdots\!40\)\( p^{18} T^{24} - \)\(22\!\cdots\!96\)\( p^{24} T^{26} + 220352751485334260 p^{30} T^{28} - 17672305163440 p^{36} T^{30} + 1080021241 p^{42} T^{32} - 45282 p^{48} T^{34} + p^{54} T^{36} \) | |
| 19 | \( ( 1 + 34 T + 31907 T^{2} + 1025104 T^{3} + 521944740 T^{4} + 15342161400 T^{5} + 5760096801308 T^{6} + 151273640580272 T^{7} + 48681526084193758 T^{8} + 1148710323738341260 T^{9} + 48681526084193758 p^{3} T^{10} + 151273640580272 p^{6} T^{11} + 5760096801308 p^{9} T^{12} + 15342161400 p^{12} T^{13} + 521944740 p^{15} T^{14} + 1025104 p^{18} T^{15} + 31907 p^{21} T^{16} + 34 p^{24} T^{17} + p^{27} T^{18} )^{2} \) | |
| 23 | \( 1 - 110982 T^{2} + 269302639 p T^{4} - 229175593427776 T^{6} + 6286529227274876740 T^{8} - \)\(13\!\cdots\!12\)\( T^{10} + \)\(24\!\cdots\!72\)\( T^{12} - \)\(37\!\cdots\!36\)\( T^{14} + \)\(51\!\cdots\!06\)\( T^{16} - \)\(64\!\cdots\!68\)\( T^{18} + \)\(51\!\cdots\!06\)\( p^{6} T^{20} - \)\(37\!\cdots\!36\)\( p^{12} T^{22} + \)\(24\!\cdots\!72\)\( p^{18} T^{24} - \)\(13\!\cdots\!12\)\( p^{24} T^{26} + 6286529227274876740 p^{30} T^{28} - 229175593427776 p^{36} T^{30} + 269302639 p^{43} T^{32} - 110982 p^{48} T^{34} + p^{54} T^{36} \) | |
| 29 | \( ( 1 + 170 T + 107517 T^{2} + 13932208 T^{3} + 5725536020 T^{4} + 21104110392 p T^{5} + 214303413402420 T^{6} + 19821854179339344 T^{7} + 6349836904888612174 T^{8} + \)\(52\!\cdots\!60\)\( T^{9} + 6349836904888612174 p^{3} T^{10} + 19821854179339344 p^{6} T^{11} + 214303413402420 p^{9} T^{12} + 21104110392 p^{13} T^{13} + 5725536020 p^{15} T^{14} + 13932208 p^{18} T^{15} + 107517 p^{21} T^{16} + 170 p^{24} T^{17} + p^{27} T^{18} )^{2} \) | |
| 31 | \( ( 1 - 168 T + 132023 T^{2} - 22283328 T^{3} + 10383373956 T^{4} - 1488537215328 T^{5} + 533181488401932 T^{6} - 69306017242405312 T^{7} + 20577323736403541598 T^{8} - \)\(23\!\cdots\!28\)\( T^{9} + 20577323736403541598 p^{3} T^{10} - 69306017242405312 p^{6} T^{11} + 533181488401932 p^{9} T^{12} - 1488537215328 p^{12} T^{13} + 10383373956 p^{15} T^{14} - 22283328 p^{18} T^{15} + 132023 p^{21} T^{16} - 168 p^{24} T^{17} + p^{27} T^{18} )^{2} \) | |
| 37 | \( 1 - 351370 T^{2} + 1777230925 p T^{4} - 8668242846445296 T^{6} + \)\(90\!\cdots\!36\)\( T^{8} - \)\(79\!\cdots\!84\)\( T^{10} + \)\(60\!\cdots\!28\)\( T^{12} - \)\(40\!\cdots\!80\)\( T^{14} + \)\(24\!\cdots\!82\)\( T^{16} - \)\(12\!\cdots\!60\)\( T^{18} + \)\(24\!\cdots\!82\)\( p^{6} T^{20} - \)\(40\!\cdots\!80\)\( p^{12} T^{22} + \)\(60\!\cdots\!28\)\( p^{18} T^{24} - \)\(79\!\cdots\!84\)\( p^{24} T^{26} + \)\(90\!\cdots\!36\)\( p^{30} T^{28} - 8668242846445296 p^{36} T^{30} + 1777230925 p^{43} T^{32} - 351370 p^{48} T^{34} + p^{54} T^{36} \) | |
| 41 | \( ( 1 - 118 T + 306417 T^{2} - 18849056 T^{3} + 42700765332 T^{4} + 723627236344 T^{5} + 3540593544233028 T^{6} + 441956777709999008 T^{7} + \)\(22\!\cdots\!22\)\( T^{8} + \)\(46\!\cdots\!84\)\( T^{9} + \)\(22\!\cdots\!22\)\( p^{3} T^{10} + 441956777709999008 p^{6} T^{11} + 3540593544233028 p^{9} T^{12} + 723627236344 p^{12} T^{13} + 42700765332 p^{15} T^{14} - 18849056 p^{18} T^{15} + 306417 p^{21} T^{16} - 118 p^{24} T^{17} + p^{27} T^{18} )^{2} \) | |
| 43 | \( 1 - 817186 T^{2} + 316383232561 T^{4} - 77191645858921760 T^{6} + \)\(13\!\cdots\!36\)\( T^{8} - \)\(40\!\cdots\!00\)\( p T^{10} + \)\(17\!\cdots\!36\)\( T^{12} - \)\(14\!\cdots\!84\)\( T^{14} + \)\(10\!\cdots\!26\)\( T^{16} - \)\(79\!\cdots\!40\)\( T^{18} + \)\(10\!\cdots\!26\)\( p^{6} T^{20} - \)\(14\!\cdots\!84\)\( p^{12} T^{22} + \)\(17\!\cdots\!36\)\( p^{18} T^{24} - \)\(40\!\cdots\!00\)\( p^{25} T^{26} + \)\(13\!\cdots\!36\)\( p^{30} T^{28} - 77191645858921760 p^{36} T^{30} + 316383232561 p^{42} T^{32} - 817186 p^{48} T^{34} + p^{54} T^{36} \) | |
| 47 | \( 1 - 976438 T^{2} + 463777655113 T^{4} - 142404872157421440 T^{6} + \)\(31\!\cdots\!12\)\( T^{8} - \)\(54\!\cdots\!52\)\( T^{10} + \)\(76\!\cdots\!44\)\( T^{12} - \)\(91\!\cdots\!24\)\( T^{14} + \)\(99\!\cdots\!38\)\( T^{16} - \)\(10\!\cdots\!92\)\( T^{18} + \)\(99\!\cdots\!38\)\( p^{6} T^{20} - \)\(91\!\cdots\!24\)\( p^{12} T^{22} + \)\(76\!\cdots\!44\)\( p^{18} T^{24} - \)\(54\!\cdots\!52\)\( p^{24} T^{26} + \)\(31\!\cdots\!12\)\( p^{30} T^{28} - 142404872157421440 p^{36} T^{30} + 463777655113 p^{42} T^{32} - 976438 p^{48} T^{34} + p^{54} T^{36} \) | |
| 53 | \( 1 - 1520026 T^{2} + 1135748878529 T^{4} - 560763301886123248 T^{6} + \)\(20\!\cdots\!80\)\( T^{8} - \)\(61\!\cdots\!76\)\( T^{10} + \)\(15\!\cdots\!04\)\( T^{12} - \)\(31\!\cdots\!68\)\( T^{14} + \)\(58\!\cdots\!86\)\( T^{16} - \)\(92\!\cdots\!44\)\( T^{18} + \)\(58\!\cdots\!86\)\( p^{6} T^{20} - \)\(31\!\cdots\!68\)\( p^{12} T^{22} + \)\(15\!\cdots\!04\)\( p^{18} T^{24} - \)\(61\!\cdots\!76\)\( p^{24} T^{26} + \)\(20\!\cdots\!80\)\( p^{30} T^{28} - 560763301886123248 p^{36} T^{30} + 1135748878529 p^{42} T^{32} - 1520026 p^{48} T^{34} + p^{54} T^{36} \) | |
| 59 | \( ( 1 + 878 T + 1236955 T^{2} + 620099760 T^{3} + 553755425508 T^{4} + 199575505245768 T^{5} + 172355456546221884 T^{6} + 57782131388837538640 T^{7} + \)\(47\!\cdots\!10\)\( T^{8} + \)\(14\!\cdots\!88\)\( T^{9} + \)\(47\!\cdots\!10\)\( p^{3} T^{10} + 57782131388837538640 p^{6} T^{11} + 172355456546221884 p^{9} T^{12} + 199575505245768 p^{12} T^{13} + 553755425508 p^{15} T^{14} + 620099760 p^{18} T^{15} + 1236955 p^{21} T^{16} + 878 p^{24} T^{17} + p^{27} T^{18} )^{2} \) | |
| 61 | \( ( 1 - 390 T + 946533 T^{2} - 375530288 T^{3} + 444635912140 T^{4} - 173806478525608 T^{5} + 143980830611353404 T^{6} - 52074599746897680592 T^{7} + \)\(37\!\cdots\!62\)\( T^{8} - \)\(12\!\cdots\!24\)\( T^{9} + \)\(37\!\cdots\!62\)\( p^{3} T^{10} - 52074599746897680592 p^{6} T^{11} + 143980830611353404 p^{9} T^{12} - 173806478525608 p^{12} T^{13} + 444635912140 p^{15} T^{14} - 375530288 p^{18} T^{15} + 946533 p^{21} T^{16} - 390 p^{24} T^{17} + p^{27} T^{18} )^{2} \) | |
| 67 | \( 1 - 2700338 T^{2} + 3533629828065 T^{4} - 2996289198536038944 T^{6} + \)\(18\!\cdots\!84\)\( T^{8} - \)\(93\!\cdots\!28\)\( T^{10} + \)\(40\!\cdots\!16\)\( T^{12} - \)\(15\!\cdots\!12\)\( T^{14} + \)\(53\!\cdots\!34\)\( T^{16} - \)\(37\!\cdots\!72\)\( p^{2} T^{18} + \)\(53\!\cdots\!34\)\( p^{6} T^{20} - \)\(15\!\cdots\!12\)\( p^{12} T^{22} + \)\(40\!\cdots\!16\)\( p^{18} T^{24} - \)\(93\!\cdots\!28\)\( p^{24} T^{26} + \)\(18\!\cdots\!84\)\( p^{30} T^{28} - 2996289198536038944 p^{36} T^{30} + 3533629828065 p^{42} T^{32} - 2700338 p^{48} T^{34} + p^{54} T^{36} \) | |
| 71 | \( ( 1 - 36 T + 1841615 T^{2} - 165671520 T^{3} + 1824835269476 T^{4} - 185529115441136 T^{5} + 1209991330264289612 T^{6} - \)\(12\!\cdots\!60\)\( T^{7} + \)\(58\!\cdots\!46\)\( T^{8} - \)\(53\!\cdots\!36\)\( T^{9} + \)\(58\!\cdots\!46\)\( p^{3} T^{10} - \)\(12\!\cdots\!60\)\( p^{6} T^{11} + 1209991330264289612 p^{9} T^{12} - 185529115441136 p^{12} T^{13} + 1824835269476 p^{15} T^{14} - 165671520 p^{18} T^{15} + 1841615 p^{21} T^{16} - 36 p^{24} T^{17} + p^{27} T^{18} )^{2} \) | |
| 73 | \( 1 - 3778834 T^{2} + 6933266462601 T^{4} - 8143954384598370864 T^{6} + \)\(68\!\cdots\!08\)\( T^{8} - \)\(42\!\cdots\!48\)\( T^{10} + \)\(20\!\cdots\!16\)\( T^{12} - \)\(83\!\cdots\!28\)\( T^{14} + \)\(39\!\cdots\!02\)\( p T^{16} - \)\(10\!\cdots\!12\)\( T^{18} + \)\(39\!\cdots\!02\)\( p^{7} T^{20} - \)\(83\!\cdots\!28\)\( p^{12} T^{22} + \)\(20\!\cdots\!16\)\( p^{18} T^{24} - \)\(42\!\cdots\!48\)\( p^{24} T^{26} + \)\(68\!\cdots\!08\)\( p^{30} T^{28} - 8143954384598370864 p^{36} T^{30} + 6933266462601 p^{42} T^{32} - 3778834 p^{48} T^{34} + p^{54} T^{36} \) | |
| 79 | \( ( 1 + 976 T + 1521351 T^{2} + 945533696 T^{3} + 1131624911108 T^{4} + 585675447712192 T^{5} + 718720167172114636 T^{6} + \)\(38\!\cdots\!00\)\( T^{7} + \)\(42\!\cdots\!82\)\( T^{8} + \)\(20\!\cdots\!32\)\( T^{9} + \)\(42\!\cdots\!82\)\( p^{3} T^{10} + \)\(38\!\cdots\!00\)\( p^{6} T^{11} + 718720167172114636 p^{9} T^{12} + 585675447712192 p^{12} T^{13} + 1131624911108 p^{15} T^{14} + 945533696 p^{18} T^{15} + 1521351 p^{21} T^{16} + 976 p^{24} T^{17} + p^{27} T^{18} )^{2} \) | |
| 83 | \( 1 - 6128690 T^{2} + 18682450854913 T^{4} - 37665709554087804320 T^{6} + \)\(56\!\cdots\!84\)\( T^{8} - \)\(66\!\cdots\!56\)\( T^{10} + \)\(65\!\cdots\!32\)\( T^{12} - \)\(53\!\cdots\!76\)\( T^{14} + \)\(38\!\cdots\!34\)\( T^{16} - \)\(23\!\cdots\!36\)\( T^{18} + \)\(38\!\cdots\!34\)\( p^{6} T^{20} - \)\(53\!\cdots\!76\)\( p^{12} T^{22} + \)\(65\!\cdots\!32\)\( p^{18} T^{24} - \)\(66\!\cdots\!56\)\( p^{24} T^{26} + \)\(56\!\cdots\!84\)\( p^{30} T^{28} - 37665709554087804320 p^{36} T^{30} + 18682450854913 p^{42} T^{32} - 6128690 p^{48} T^{34} + p^{54} T^{36} \) | |
| 89 | \( ( 1 + 146 T + 2947297 T^{2} + 762766896 T^{3} + 3967842683252 T^{4} + 1157135497541624 T^{5} + 3380357191051689636 T^{6} + \)\(94\!\cdots\!00\)\( T^{7} + \)\(22\!\cdots\!90\)\( T^{8} + \)\(65\!\cdots\!52\)\( T^{9} + \)\(22\!\cdots\!90\)\( p^{3} T^{10} + \)\(94\!\cdots\!00\)\( p^{6} T^{11} + 3380357191051689636 p^{9} T^{12} + 1157135497541624 p^{12} T^{13} + 3967842683252 p^{15} T^{14} + 762766896 p^{18} T^{15} + 2947297 p^{21} T^{16} + 146 p^{24} T^{17} + p^{27} T^{18} )^{2} \) | |
| 97 | \( 1 - 10032834 T^{2} + 49189296349913 T^{4} - \)\(15\!\cdots\!60\)\( T^{6} + \)\(37\!\cdots\!44\)\( T^{8} - \)\(70\!\cdots\!40\)\( T^{10} + \)\(11\!\cdots\!68\)\( T^{12} - \)\(14\!\cdots\!16\)\( T^{14} + \)\(16\!\cdots\!66\)\( T^{16} - \)\(16\!\cdots\!20\)\( T^{18} + \)\(16\!\cdots\!66\)\( p^{6} T^{20} - \)\(14\!\cdots\!16\)\( p^{12} T^{22} + \)\(11\!\cdots\!68\)\( p^{18} T^{24} - \)\(70\!\cdots\!40\)\( p^{24} T^{26} + \)\(37\!\cdots\!44\)\( p^{30} T^{28} - \)\(15\!\cdots\!60\)\( p^{36} T^{30} + 49189296349913 p^{42} T^{32} - 10032834 p^{48} T^{34} + p^{54} T^{36} \) | |
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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
−2.04287912447139866452109918824, −2.03100533737620034523866477034, −1.99956505739540436389608576615, −1.81827007740901905616334864977, −1.80057522012279434504127529012, −1.78986357142706454380351000185, −1.68337665011858957549486716786, −1.64107525756995137527010571881, −1.53383533941417040400256407281, −1.26017319592086144907705238861, −1.19996960607242203654792337848, −1.12495821068424846665622980358, −1.11742235247817751858929908923, −1.08181103263074801848321188096, −1.03871456211604109219934306147, −0.975184275689600752277835990328, −0.943445877885009947284375611725, −0.925763777207148723006615527972, −0.65147725929841221453964575064, −0.58321847852207006038296654096, −0.45502747907187983524832467085, −0.31813568550163827952201827658, −0.26011434798141376845353399424, −0.04386617744531522675128674035, −0.01662599838095411594775182872, 0.01662599838095411594775182872, 0.04386617744531522675128674035, 0.26011434798141376845353399424, 0.31813568550163827952201827658, 0.45502747907187983524832467085, 0.58321847852207006038296654096, 0.65147725929841221453964575064, 0.925763777207148723006615527972, 0.943445877885009947284375611725, 0.975184275689600752277835990328, 1.03871456211604109219934306147, 1.08181103263074801848321188096, 1.11742235247817751858929908923, 1.12495821068424846665622980358, 1.19996960607242203654792337848, 1.26017319592086144907705238861, 1.53383533941417040400256407281, 1.64107525756995137527010571881, 1.68337665011858957549486716786, 1.78986357142706454380351000185, 1.80057522012279434504127529012, 1.81827007740901905616334864977, 1.99956505739540436389608576615, 2.03100533737620034523866477034, 2.04287912447139866452109918824
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.