Dirichlet series
| L(s) = 1 | + 15·2-s − 1.03e3·4-s + 453·5-s + 343·7-s − 1.68e4·8-s + 6.79e3·10-s + 9.91e4·11-s − 3.24e4·13-s + 5.14e3·14-s + 3.85e5·16-s + 4.15e5·17-s − 8.52e4·19-s − 4.70e5·20-s + 1.48e6·22-s + 1.06e6·23-s − 6.56e6·25-s − 4.86e5·26-s − 3.56e5·28-s − 1.30e6·29-s + 2.35e6·31-s + 7.65e6·32-s + 6.23e6·34-s + 1.55e5·35-s + 8.19e6·37-s − 1.27e6·38-s − 7.63e6·40-s + 5.47e7·41-s + ⋯ |
| L(s) = 1 | + 0.662·2-s − 2.02·4-s + 0.324·5-s + 0.0539·7-s − 1.45·8-s + 0.214·10-s + 2.04·11-s − 0.314·13-s + 0.0357·14-s + 1.47·16-s + 1.20·17-s − 0.150·19-s − 0.657·20-s + 1.35·22-s + 0.793·23-s − 3.36·25-s − 0.208·26-s − 0.109·28-s − 0.343·29-s + 0.458·31-s + 1.29·32-s + 0.799·34-s + 0.0175·35-s + 0.718·37-s − 0.0995·38-s − 0.471·40-s + 3.02·41-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{32}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(10-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{32}\right)^{s/2} \, \Gamma_{\C}(s+9/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(16\) |
| Conductor: | \(3^{32}\) |
| Sign: | $1$ |
| Analytic conductor: | \(9.17444\times 10^{12}\) |
| Root analytic conductor: | \(6.45893\) |
| Motivic weight: | \(9\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((16,\ 3^{32} ,\ ( \ : [9/2]^{8} ),\ 1 )\) |
Particular Values
| \(L(5)\) | \(\approx\) | \(3.743326937\) |
| \(L(\frac12)\) | \(\approx\) | \(3.743326937\) |
| \(L(\frac{11}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 3 | \( 1 \) |
| good | 2 | \( 1 - 15 T + 79 p^{4} T^{2} - 2211 p^{3} T^{3} + 117485 p^{3} T^{4} - 203679 p^{6} T^{5} + 1138817 p^{9} T^{6} - 717735 p^{13} T^{7} + 146093393 p^{11} T^{8} - 717735 p^{22} T^{9} + 1138817 p^{27} T^{10} - 203679 p^{33} T^{11} + 117485 p^{39} T^{12} - 2211 p^{48} T^{13} + 79 p^{58} T^{14} - 15 p^{63} T^{15} + p^{72} T^{16} \) |
| 5 | \( 1 - 453 T + 1353698 p T^{2} - 872788137 T^{3} + 24035149352989 T^{4} - 185526220124556 p T^{5} + 2673762896107831906 p^{2} T^{6} - 42953118968508495918 p^{3} T^{7} + \)\(23\!\cdots\!12\)\( p^{4} T^{8} - 42953118968508495918 p^{12} T^{9} + 2673762896107831906 p^{20} T^{10} - 185526220124556 p^{28} T^{11} + 24035149352989 p^{36} T^{12} - 872788137 p^{45} T^{13} + 1353698 p^{55} T^{14} - 453 p^{63} T^{15} + p^{72} T^{16} \) | |
| 7 | \( 1 - p^{3} T + 169093044 T^{2} - 20053093115 p T^{3} + 6830026874777 p^{4} T^{4} - 50690899225235340 p^{3} T^{5} + \)\(43\!\cdots\!38\)\( p^{4} T^{6} - \)\(69\!\cdots\!90\)\( p^{5} T^{7} + \)\(41\!\cdots\!68\)\( p^{6} T^{8} - \)\(69\!\cdots\!90\)\( p^{14} T^{9} + \)\(43\!\cdots\!38\)\( p^{22} T^{10} - 50690899225235340 p^{30} T^{11} + 6830026874777 p^{40} T^{12} - 20053093115 p^{46} T^{13} + 169093044 p^{54} T^{14} - p^{66} T^{15} + p^{72} T^{16} \) | |
| 11 | \( 1 - 99150 T + 12341403238 T^{2} - 8026435802244 p^{2} T^{3} + 74121043892891542336 T^{4} - \)\(45\!\cdots\!66\)\( T^{5} + \)\(27\!\cdots\!56\)\( T^{6} - \)\(14\!\cdots\!86\)\( T^{7} + \)\(66\!\cdots\!55\)\( p T^{8} - \)\(14\!\cdots\!86\)\( p^{9} T^{9} + \)\(27\!\cdots\!56\)\( p^{18} T^{10} - \)\(45\!\cdots\!66\)\( p^{27} T^{11} + 74121043892891542336 p^{36} T^{12} - 8026435802244 p^{47} T^{13} + 12341403238 p^{54} T^{14} - 99150 p^{63} T^{15} + p^{72} T^{16} \) | |
| 13 | \( 1 + 2495 p T + 50482836288 T^{2} + 2202183313375387 T^{3} + \)\(12\!\cdots\!61\)\( T^{4} + \)\(65\!\cdots\!92\)\( T^{5} + \)\(21\!\cdots\!22\)\( T^{6} + \)\(11\!\cdots\!50\)\( T^{7} + \)\(26\!\cdots\!40\)\( T^{8} + \)\(11\!\cdots\!50\)\( p^{9} T^{9} + \)\(21\!\cdots\!22\)\( p^{18} T^{10} + \)\(65\!\cdots\!92\)\( p^{27} T^{11} + \)\(12\!\cdots\!61\)\( p^{36} T^{12} + 2202183313375387 p^{45} T^{13} + 50482836288 p^{54} T^{14} + 2495 p^{64} T^{15} + p^{72} T^{16} \) | |
| 17 | \( 1 - 415539 T + 581253095413 T^{2} - 156665764490366790 T^{3} + \)\(15\!\cdots\!76\)\( T^{4} - \)\(31\!\cdots\!80\)\( T^{5} + \)\(27\!\cdots\!71\)\( T^{6} - \)\(30\!\cdots\!07\)\( p T^{7} + \)\(13\!\cdots\!22\)\( p^{2} T^{8} - \)\(30\!\cdots\!07\)\( p^{10} T^{9} + \)\(27\!\cdots\!71\)\( p^{18} T^{10} - \)\(31\!\cdots\!80\)\( p^{27} T^{11} + \)\(15\!\cdots\!76\)\( p^{36} T^{12} - 156665764490366790 p^{45} T^{13} + 581253095413 p^{54} T^{14} - 415539 p^{63} T^{15} + p^{72} T^{16} \) | |
| 19 | \( 1 + 85277 T + 1351383718611 T^{2} + 153873747267299692 T^{3} + \)\(91\!\cdots\!74\)\( T^{4} + \)\(11\!\cdots\!14\)\( T^{5} + \)\(43\!\cdots\!25\)\( T^{6} + \)\(56\!\cdots\!39\)\( T^{7} + \)\(15\!\cdots\!78\)\( T^{8} + \)\(56\!\cdots\!39\)\( p^{9} T^{9} + \)\(43\!\cdots\!25\)\( p^{18} T^{10} + \)\(11\!\cdots\!14\)\( p^{27} T^{11} + \)\(91\!\cdots\!74\)\( p^{36} T^{12} + 153873747267299692 p^{45} T^{13} + 1351383718611 p^{54} T^{14} + 85277 p^{63} T^{15} + p^{72} T^{16} \) | |
| 23 | \( 1 - 1064559 T + 8994275273632 T^{2} - 11463521514847317789 T^{3} + \)\(41\!\cdots\!93\)\( T^{4} - \)\(52\!\cdots\!32\)\( T^{5} + \)\(12\!\cdots\!02\)\( T^{6} - \)\(13\!\cdots\!70\)\( T^{7} + \)\(27\!\cdots\!72\)\( T^{8} - \)\(13\!\cdots\!70\)\( p^{9} T^{9} + \)\(12\!\cdots\!02\)\( p^{18} T^{10} - \)\(52\!\cdots\!32\)\( p^{27} T^{11} + \)\(41\!\cdots\!93\)\( p^{36} T^{12} - 11463521514847317789 p^{45} T^{13} + 8994275273632 p^{54} T^{14} - 1064559 p^{63} T^{15} + p^{72} T^{16} \) | |
| 29 | \( 1 + 1309053 T + 56138712865492 T^{2} + \)\(10\!\cdots\!25\)\( T^{3} + \)\(17\!\cdots\!69\)\( T^{4} + \)\(34\!\cdots\!28\)\( T^{5} + \)\(36\!\cdots\!58\)\( T^{6} + \)\(73\!\cdots\!22\)\( T^{7} + \)\(60\!\cdots\!84\)\( T^{8} + \)\(73\!\cdots\!22\)\( p^{9} T^{9} + \)\(36\!\cdots\!58\)\( p^{18} T^{10} + \)\(34\!\cdots\!28\)\( p^{27} T^{11} + \)\(17\!\cdots\!69\)\( p^{36} T^{12} + \)\(10\!\cdots\!25\)\( p^{45} T^{13} + 56138712865492 p^{54} T^{14} + 1309053 p^{63} T^{15} + p^{72} T^{16} \) | |
| 31 | \( 1 - 2359819 T + 111892930980714 T^{2} - 57664071659672156741 T^{3} + \)\(50\!\cdots\!25\)\( T^{4} + \)\(11\!\cdots\!76\)\( T^{5} + \)\(11\!\cdots\!14\)\( T^{6} + \)\(78\!\cdots\!14\)\( T^{7} + \)\(24\!\cdots\!44\)\( T^{8} + \)\(78\!\cdots\!14\)\( p^{9} T^{9} + \)\(11\!\cdots\!14\)\( p^{18} T^{10} + \)\(11\!\cdots\!76\)\( p^{27} T^{11} + \)\(50\!\cdots\!25\)\( p^{36} T^{12} - 57664071659672156741 p^{45} T^{13} + 111892930980714 p^{54} T^{14} - 2359819 p^{63} T^{15} + p^{72} T^{16} \) | |
| 37 | \( 1 - 8195758 T + 504924741456168 T^{2} - 92731899494581349042 p T^{3} + \)\(11\!\cdots\!32\)\( T^{4} - \)\(47\!\cdots\!46\)\( T^{5} + \)\(15\!\cdots\!52\)\( T^{6} - \)\(18\!\cdots\!18\)\( T^{7} + \)\(19\!\cdots\!26\)\( T^{8} - \)\(18\!\cdots\!18\)\( p^{9} T^{9} + \)\(15\!\cdots\!52\)\( p^{18} T^{10} - \)\(47\!\cdots\!46\)\( p^{27} T^{11} + \)\(11\!\cdots\!32\)\( p^{36} T^{12} - 92731899494581349042 p^{46} T^{13} + 504924741456168 p^{54} T^{14} - 8195758 p^{63} T^{15} + p^{72} T^{16} \) | |
| 41 | \( 1 - 54747318 T + 3187182688559482 T^{2} - \)\(11\!\cdots\!28\)\( T^{3} + \)\(40\!\cdots\!52\)\( T^{4} - \)\(11\!\cdots\!78\)\( T^{5} + \)\(27\!\cdots\!80\)\( T^{6} - \)\(59\!\cdots\!94\)\( T^{7} + \)\(11\!\cdots\!73\)\( T^{8} - \)\(59\!\cdots\!94\)\( p^{9} T^{9} + \)\(27\!\cdots\!80\)\( p^{18} T^{10} - \)\(11\!\cdots\!78\)\( p^{27} T^{11} + \)\(40\!\cdots\!52\)\( p^{36} T^{12} - \)\(11\!\cdots\!28\)\( p^{45} T^{13} + 3187182688559482 p^{54} T^{14} - 54747318 p^{63} T^{15} + p^{72} T^{16} \) | |
| 43 | \( 1 + 15249608 T + 2206551401546652 T^{2} + \)\(35\!\cdots\!72\)\( T^{3} + \)\(26\!\cdots\!70\)\( T^{4} + \)\(41\!\cdots\!08\)\( T^{5} + \)\(21\!\cdots\!28\)\( T^{6} + \)\(30\!\cdots\!40\)\( T^{7} + \)\(12\!\cdots\!63\)\( T^{8} + \)\(30\!\cdots\!40\)\( p^{9} T^{9} + \)\(21\!\cdots\!28\)\( p^{18} T^{10} + \)\(41\!\cdots\!08\)\( p^{27} T^{11} + \)\(26\!\cdots\!70\)\( p^{36} T^{12} + \)\(35\!\cdots\!72\)\( p^{45} T^{13} + 2206551401546652 p^{54} T^{14} + 15249608 p^{63} T^{15} + p^{72} T^{16} \) | |
| 47 | \( 1 - 156295545 T + 17249276591906548 T^{2} - \)\(13\!\cdots\!71\)\( T^{3} + \)\(88\!\cdots\!17\)\( T^{4} - \)\(47\!\cdots\!72\)\( T^{5} + \)\(22\!\cdots\!50\)\( T^{6} - \)\(90\!\cdots\!66\)\( T^{7} + \)\(32\!\cdots\!68\)\( T^{8} - \)\(90\!\cdots\!66\)\( p^{9} T^{9} + \)\(22\!\cdots\!50\)\( p^{18} T^{10} - \)\(47\!\cdots\!72\)\( p^{27} T^{11} + \)\(88\!\cdots\!17\)\( p^{36} T^{12} - \)\(13\!\cdots\!71\)\( p^{45} T^{13} + 17249276591906548 p^{54} T^{14} - 156295545 p^{63} T^{15} + p^{72} T^{16} \) | |
| 53 | \( 1 - 262758114 T + 49230989686733464 T^{2} - \)\(65\!\cdots\!90\)\( T^{3} + \)\(72\!\cdots\!16\)\( T^{4} - \)\(66\!\cdots\!42\)\( T^{5} + \)\(10\!\cdots\!68\)\( p T^{6} - \)\(69\!\cdots\!10\)\( p T^{7} + \)\(22\!\cdots\!98\)\( T^{8} - \)\(69\!\cdots\!10\)\( p^{10} T^{9} + \)\(10\!\cdots\!68\)\( p^{19} T^{10} - \)\(66\!\cdots\!42\)\( p^{27} T^{11} + \)\(72\!\cdots\!16\)\( p^{36} T^{12} - \)\(65\!\cdots\!90\)\( p^{45} T^{13} + 49230989686733464 p^{54} T^{14} - 262758114 p^{63} T^{15} + p^{72} T^{16} \) | |
| 59 | \( 1 - 307774074 T + 78120711978580390 T^{2} - \)\(14\!\cdots\!68\)\( T^{3} + \)\(23\!\cdots\!64\)\( T^{4} - \)\(31\!\cdots\!82\)\( T^{5} + \)\(38\!\cdots\!36\)\( T^{6} - \)\(70\!\cdots\!46\)\( p T^{7} + \)\(41\!\cdots\!61\)\( T^{8} - \)\(70\!\cdots\!46\)\( p^{10} T^{9} + \)\(38\!\cdots\!36\)\( p^{18} T^{10} - \)\(31\!\cdots\!82\)\( p^{27} T^{11} + \)\(23\!\cdots\!64\)\( p^{36} T^{12} - \)\(14\!\cdots\!68\)\( p^{45} T^{13} + 78120711978580390 p^{54} T^{14} - 307774074 p^{63} T^{15} + p^{72} T^{16} \) | |
| 61 | \( 1 + 69192125 T + 57236841161614746 T^{2} + \)\(36\!\cdots\!61\)\( T^{3} + \)\(16\!\cdots\!29\)\( T^{4} + \)\(93\!\cdots\!20\)\( T^{5} + \)\(31\!\cdots\!74\)\( T^{6} + \)\(15\!\cdots\!70\)\( T^{7} + \)\(42\!\cdots\!28\)\( T^{8} + \)\(15\!\cdots\!70\)\( p^{9} T^{9} + \)\(31\!\cdots\!74\)\( p^{18} T^{10} + \)\(93\!\cdots\!20\)\( p^{27} T^{11} + \)\(16\!\cdots\!29\)\( p^{36} T^{12} + \)\(36\!\cdots\!61\)\( p^{45} T^{13} + 57236841161614746 p^{54} T^{14} + 69192125 p^{63} T^{15} + p^{72} T^{16} \) | |
| 67 | \( 1 + 14328044 T + 129695852709279504 T^{2} + \)\(61\!\cdots\!24\)\( T^{3} + \)\(81\!\cdots\!30\)\( T^{4} + \)\(57\!\cdots\!04\)\( T^{5} + \)\(34\!\cdots\!64\)\( T^{6} + \)\(26\!\cdots\!48\)\( T^{7} + \)\(10\!\cdots\!11\)\( T^{8} + \)\(26\!\cdots\!48\)\( p^{9} T^{9} + \)\(34\!\cdots\!64\)\( p^{18} T^{10} + \)\(57\!\cdots\!04\)\( p^{27} T^{11} + \)\(81\!\cdots\!30\)\( p^{36} T^{12} + \)\(61\!\cdots\!24\)\( p^{45} T^{13} + 129695852709279504 p^{54} T^{14} + 14328044 p^{63} T^{15} + p^{72} T^{16} \) | |
| 71 | \( 1 - 619800696 T + 342675357334948360 T^{2} - \)\(12\!\cdots\!44\)\( T^{3} + \)\(41\!\cdots\!56\)\( T^{4} - \)\(11\!\cdots\!28\)\( T^{5} + \)\(30\!\cdots\!80\)\( T^{6} - \)\(68\!\cdots\!44\)\( T^{7} + \)\(15\!\cdots\!22\)\( T^{8} - \)\(68\!\cdots\!44\)\( p^{9} T^{9} + \)\(30\!\cdots\!80\)\( p^{18} T^{10} - \)\(11\!\cdots\!28\)\( p^{27} T^{11} + \)\(41\!\cdots\!56\)\( p^{36} T^{12} - \)\(12\!\cdots\!44\)\( p^{45} T^{13} + 342675357334948360 p^{54} T^{14} - 619800696 p^{63} T^{15} + p^{72} T^{16} \) | |
| 73 | \( 1 - 299306599 T + 299926161877414305 T^{2} - \)\(76\!\cdots\!06\)\( T^{3} + \)\(45\!\cdots\!36\)\( T^{4} - \)\(10\!\cdots\!04\)\( T^{5} + \)\(44\!\cdots\!47\)\( T^{6} - \)\(85\!\cdots\!11\)\( T^{7} + \)\(30\!\cdots\!98\)\( T^{8} - \)\(85\!\cdots\!11\)\( p^{9} T^{9} + \)\(44\!\cdots\!47\)\( p^{18} T^{10} - \)\(10\!\cdots\!04\)\( p^{27} T^{11} + \)\(45\!\cdots\!36\)\( p^{36} T^{12} - \)\(76\!\cdots\!06\)\( p^{45} T^{13} + 299926161877414305 p^{54} T^{14} - 299306599 p^{63} T^{15} + p^{72} T^{16} \) | |
| 79 | \( 1 + 30257531 T + 535200167288440026 T^{2} + \)\(22\!\cdots\!37\)\( T^{3} + \)\(15\!\cdots\!13\)\( T^{4} + \)\(73\!\cdots\!24\)\( T^{5} + \)\(30\!\cdots\!30\)\( T^{6} + \)\(13\!\cdots\!06\)\( T^{7} + \)\(43\!\cdots\!52\)\( T^{8} + \)\(13\!\cdots\!06\)\( p^{9} T^{9} + \)\(30\!\cdots\!30\)\( p^{18} T^{10} + \)\(73\!\cdots\!24\)\( p^{27} T^{11} + \)\(15\!\cdots\!13\)\( p^{36} T^{12} + \)\(22\!\cdots\!37\)\( p^{45} T^{13} + 535200167288440026 p^{54} T^{14} + 30257531 p^{63} T^{15} + p^{72} T^{16} \) | |
| 83 | \( 1 - 1176168291 T + 1573181230735958314 T^{2} - \)\(11\!\cdots\!85\)\( T^{3} + \)\(95\!\cdots\!61\)\( T^{4} - \)\(54\!\cdots\!56\)\( T^{5} + \)\(33\!\cdots\!90\)\( T^{6} - \)\(15\!\cdots\!02\)\( T^{7} + \)\(76\!\cdots\!12\)\( T^{8} - \)\(15\!\cdots\!02\)\( p^{9} T^{9} + \)\(33\!\cdots\!90\)\( p^{18} T^{10} - \)\(54\!\cdots\!56\)\( p^{27} T^{11} + \)\(95\!\cdots\!61\)\( p^{36} T^{12} - \)\(11\!\cdots\!85\)\( p^{45} T^{13} + 1573181230735958314 p^{54} T^{14} - 1176168291 p^{63} T^{15} + p^{72} T^{16} \) | |
| 89 | \( 1 - 1658520648 T + 2633786437329298936 T^{2} - \)\(26\!\cdots\!68\)\( T^{3} + \)\(25\!\cdots\!84\)\( T^{4} - \)\(19\!\cdots\!80\)\( T^{5} + \)\(14\!\cdots\!24\)\( T^{6} - \)\(90\!\cdots\!64\)\( T^{7} + \)\(57\!\cdots\!90\)\( T^{8} - \)\(90\!\cdots\!64\)\( p^{9} T^{9} + \)\(14\!\cdots\!24\)\( p^{18} T^{10} - \)\(19\!\cdots\!80\)\( p^{27} T^{11} + \)\(25\!\cdots\!84\)\( p^{36} T^{12} - \)\(26\!\cdots\!68\)\( p^{45} T^{13} + 2633786437329298936 p^{54} T^{14} - 1658520648 p^{63} T^{15} + p^{72} T^{16} \) | |
| 97 | \( 1 - 267311278 T + 4131169348789580526 T^{2} - \)\(21\!\cdots\!64\)\( T^{3} + \)\(77\!\cdots\!04\)\( T^{4} - \)\(57\!\cdots\!86\)\( T^{5} + \)\(90\!\cdots\!16\)\( T^{6} - \)\(77\!\cdots\!10\)\( T^{7} + \)\(77\!\cdots\!17\)\( T^{8} - \)\(77\!\cdots\!10\)\( p^{9} T^{9} + \)\(90\!\cdots\!16\)\( p^{18} T^{10} - \)\(57\!\cdots\!86\)\( p^{27} T^{11} + \)\(77\!\cdots\!04\)\( p^{36} T^{12} - \)\(21\!\cdots\!64\)\( p^{45} T^{13} + 4131169348789580526 p^{54} T^{14} - 267311278 p^{63} T^{15} + p^{72} T^{16} \) | |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−4.97343962532240215808624884004, −4.25205122071406077222340301744, −4.23631583904851376543864310737, −4.12925762901531651788177046240, −4.07440119783667610187292103456, −4.05961178846957704802540579887, −3.91113305022698581968570456926, −3.85699222767582606127405564420, −3.52336730553395921502455801290, −3.15986725694404111905442763365, −2.91735545209692306128489763134, −2.90113392315615546691943121808, −2.57998703204921603664019701632, −2.33396761687780766088022054079, −2.15713746413767328676582118586, −1.90483784958485558506004807337, −1.84944162210392915267930462377, −1.69729299538891996552796459284, −1.10118942545867924182290363623, −0.913182248682956404973070957075, −0.856474554977397540443683623326, −0.851661001140223086632372079579, −0.75991597375676797665497287154, −0.31994093592645233135407272908, −0.12026946828201710960619254326, 0.12026946828201710960619254326, 0.31994093592645233135407272908, 0.75991597375676797665497287154, 0.851661001140223086632372079579, 0.856474554977397540443683623326, 0.913182248682956404973070957075, 1.10118942545867924182290363623, 1.69729299538891996552796459284, 1.84944162210392915267930462377, 1.90483784958485558506004807337, 2.15713746413767328676582118586, 2.33396761687780766088022054079, 2.57998703204921603664019701632, 2.90113392315615546691943121808, 2.91735545209692306128489763134, 3.15986725694404111905442763365, 3.52336730553395921502455801290, 3.85699222767582606127405564420, 3.91113305022698581968570456926, 4.05961178846957704802540579887, 4.07440119783667610187292103456, 4.12925762901531651788177046240, 4.23631583904851376543864310737, 4.25205122071406077222340301744, 4.97343962532240215808624884004
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.