Dirichlet series
| L(s) = 1 | − 1.38e3·5-s + 1.21e3·7-s + 879·11-s + 1.36e4·17-s − 2.92e4·19-s − 3.12e5·23-s − 2.29e4·25-s + 2.89e5·29-s + 2.42e5·31-s − 1.69e6·35-s + 1.91e6·37-s − 8.61e5·43-s + 3.05e5·47-s + 5.65e6·49-s + 1.06e7·53-s − 1.22e6·55-s − 1.84e7·59-s − 1.39e7·61-s − 2.07e7·67-s − 1.13e8·71-s + 4.34e7·73-s + 1.06e6·77-s − 4.21e7·79-s − 1.89e7·85-s − 6.71e7·89-s + 4.06e7·95-s − 1.53e8·101-s + ⋯ |
| L(s) = 1 | − 2.22·5-s + 0.506·7-s + 0.0600·11-s + 0.163·17-s − 0.224·19-s − 1.11·23-s − 0.0586·25-s + 0.409·29-s + 0.262·31-s − 1.12·35-s + 1.02·37-s − 0.252·43-s + 0.0625·47-s + 0.980·49-s + 1.35·53-s − 0.133·55-s − 1.51·59-s − 1.00·61-s − 1.02·67-s − 4.44·71-s + 1.52·73-s + 0.0304·77-s − 1.08·79-s − 0.363·85-s − 1.07·89-s + 0.499·95-s − 1.47·101-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 3^{20} \cdot 7^{10}\right)^{s/2} \, \Gamma_{\C}(s)^{10} \, L(s)\cr=\mathstrut & \,\Lambda(9-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 3^{20} \cdot 7^{10}\right)^{s/2} \, \Gamma_{\C}(s+4)^{10} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(20\) |
| Conductor: | \(2^{20} \cdot 3^{20} \cdot 7^{10}\) |
| Sign: | $1$ |
| Analytic conductor: | \(1.30013\times 10^{20}\) |
| Root analytic conductor: | \(10.1320\) |
| Motivic weight: | \(8\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((20,\ 2^{20} \cdot 3^{20} \cdot 7^{10} ,\ ( \ : [4]^{10} ),\ 1 )\) |
Particular Values
| \(L(\frac{9}{2})\) | \(\approx\) | \(5.395774968\) |
| \(L(\frac12)\) | \(\approx\) | \(5.395774968\) |
| \(L(5)\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) | |
| 7 | \( 1 - 1217 T - 595755 p T^{2} - 2524254 p^{3} T^{3} + 39788085 p^{6} T^{4} + 104772285 p^{10} T^{5} + 39788085 p^{14} T^{6} - 2524254 p^{19} T^{7} - 595755 p^{25} T^{8} - 1217 p^{32} T^{9} + p^{40} T^{10} \) | |
| good | 5 | \( 1 + 1389 T + 1952249 T^{2} + 1818398238 T^{3} + 1512960293514 T^{4} + 47513154784794 p^{2} T^{5} + 35018891259958899 p^{2} T^{6} + 1041117032314478499 p^{4} T^{7} + \)\(74\!\cdots\!59\)\( p^{4} T^{8} + \)\(99\!\cdots\!52\)\( p^{5} T^{9} + \)\(13\!\cdots\!12\)\( p^{6} T^{10} + \)\(99\!\cdots\!52\)\( p^{13} T^{11} + \)\(74\!\cdots\!59\)\( p^{20} T^{12} + 1041117032314478499 p^{28} T^{13} + 35018891259958899 p^{34} T^{14} + 47513154784794 p^{42} T^{15} + 1512960293514 p^{48} T^{16} + 1818398238 p^{56} T^{17} + 1952249 p^{64} T^{18} + 1389 p^{72} T^{19} + p^{80} T^{20} \) |
| 11 | \( 1 - 879 T - 391763711 T^{2} + 12509759470266 T^{3} + 84963250304921850 T^{4} - \)\(46\!\cdots\!02\)\( T^{5} + \)\(41\!\cdots\!75\)\( T^{6} + \)\(97\!\cdots\!21\)\( p T^{7} - \)\(19\!\cdots\!49\)\( T^{8} - \)\(85\!\cdots\!16\)\( T^{9} + \)\(50\!\cdots\!92\)\( T^{10} - \)\(85\!\cdots\!16\)\( p^{8} T^{11} - \)\(19\!\cdots\!49\)\( p^{16} T^{12} + \)\(97\!\cdots\!21\)\( p^{25} T^{13} + \)\(41\!\cdots\!75\)\( p^{32} T^{14} - \)\(46\!\cdots\!02\)\( p^{40} T^{15} + 84963250304921850 p^{48} T^{16} + 12509759470266 p^{56} T^{17} - 391763711 p^{64} T^{18} - 879 p^{72} T^{19} + p^{80} T^{20} \) | |
| 13 | \( 1 - 3412279828 T^{2} + 6756182783182595526 T^{4} - \)\(99\!\cdots\!62\)\( T^{6} + \)\(11\!\cdots\!93\)\( T^{8} - \)\(10\!\cdots\!48\)\( T^{10} + \)\(11\!\cdots\!93\)\( p^{16} T^{12} - \)\(99\!\cdots\!62\)\( p^{32} T^{14} + 6756182783182595526 p^{48} T^{16} - 3412279828 p^{64} T^{18} + p^{80} T^{20} \) | |
| 17 | \( 1 - 13674 T + 29489683421 T^{2} - 402389684116746 T^{3} + \)\(49\!\cdots\!71\)\( T^{4} - \)\(90\!\cdots\!68\)\( T^{5} + \)\(58\!\cdots\!94\)\( T^{6} - \)\(12\!\cdots\!00\)\( T^{7} + \)\(54\!\cdots\!01\)\( T^{8} - \)\(12\!\cdots\!54\)\( T^{9} + \)\(42\!\cdots\!55\)\( T^{10} - \)\(12\!\cdots\!54\)\( p^{8} T^{11} + \)\(54\!\cdots\!01\)\( p^{16} T^{12} - \)\(12\!\cdots\!00\)\( p^{24} T^{13} + \)\(58\!\cdots\!94\)\( p^{32} T^{14} - \)\(90\!\cdots\!68\)\( p^{40} T^{15} + \)\(49\!\cdots\!71\)\( p^{48} T^{16} - 402389684116746 p^{56} T^{17} + 29489683421 p^{64} T^{18} - 13674 p^{72} T^{19} + p^{80} T^{20} \) | |
| 19 | \( 1 + 29268 T + 72222449072 T^{2} + 2105449495460352 T^{3} + \)\(29\!\cdots\!09\)\( T^{4} + \)\(12\!\cdots\!72\)\( T^{5} + \)\(86\!\cdots\!70\)\( T^{6} + \)\(44\!\cdots\!52\)\( T^{7} + \)\(19\!\cdots\!37\)\( T^{8} + \)\(10\!\cdots\!96\)\( T^{9} + \)\(35\!\cdots\!62\)\( T^{10} + \)\(10\!\cdots\!96\)\( p^{8} T^{11} + \)\(19\!\cdots\!37\)\( p^{16} T^{12} + \)\(44\!\cdots\!52\)\( p^{24} T^{13} + \)\(86\!\cdots\!70\)\( p^{32} T^{14} + \)\(12\!\cdots\!72\)\( p^{40} T^{15} + \)\(29\!\cdots\!09\)\( p^{48} T^{16} + 2105449495460352 p^{56} T^{17} + 72222449072 p^{64} T^{18} + 29268 p^{72} T^{19} + p^{80} T^{20} \) | |
| 23 | \( 1 + 312732 T - 137121783101 T^{2} - 81469263279497652 T^{3} - \)\(27\!\cdots\!01\)\( T^{4} + \)\(48\!\cdots\!08\)\( T^{5} + \)\(96\!\cdots\!86\)\( T^{6} + \)\(30\!\cdots\!28\)\( T^{7} + \)\(10\!\cdots\!41\)\( T^{8} - \)\(26\!\cdots\!16\)\( T^{9} - \)\(20\!\cdots\!31\)\( T^{10} - \)\(26\!\cdots\!16\)\( p^{8} T^{11} + \)\(10\!\cdots\!41\)\( p^{16} T^{12} + \)\(30\!\cdots\!28\)\( p^{24} T^{13} + \)\(96\!\cdots\!86\)\( p^{32} T^{14} + \)\(48\!\cdots\!08\)\( p^{40} T^{15} - \)\(27\!\cdots\!01\)\( p^{48} T^{16} - 81469263279497652 p^{56} T^{17} - 137121783101 p^{64} T^{18} + 312732 p^{72} T^{19} + p^{80} T^{20} \) | |
| 29 | \( ( 1 - 144897 T + 1748091076616 T^{2} - 275053027978744863 T^{3} + \)\(14\!\cdots\!87\)\( T^{4} - \)\(20\!\cdots\!44\)\( T^{5} + \)\(14\!\cdots\!87\)\( p^{8} T^{6} - 275053027978744863 p^{16} T^{7} + 1748091076616 p^{24} T^{8} - 144897 p^{32} T^{9} + p^{40} T^{10} )^{2} \) | |
| 31 | \( 1 - 242787 T + 580362887030 T^{2} - 136134161668906809 T^{3} + \)\(22\!\cdots\!74\)\( T^{4} - \)\(80\!\cdots\!21\)\( T^{5} + \)\(12\!\cdots\!68\)\( T^{6} + \)\(40\!\cdots\!11\)\( T^{7} - \)\(56\!\cdots\!71\)\( T^{8} + \)\(40\!\cdots\!70\)\( T^{9} + \)\(33\!\cdots\!16\)\( T^{10} + \)\(40\!\cdots\!70\)\( p^{8} T^{11} - \)\(56\!\cdots\!71\)\( p^{16} T^{12} + \)\(40\!\cdots\!11\)\( p^{24} T^{13} + \)\(12\!\cdots\!68\)\( p^{32} T^{14} - \)\(80\!\cdots\!21\)\( p^{40} T^{15} + \)\(22\!\cdots\!74\)\( p^{48} T^{16} - 136134161668906809 p^{56} T^{17} + 580362887030 p^{64} T^{18} - 242787 p^{72} T^{19} + p^{80} T^{20} \) | |
| 37 | \( 1 - 1913308 T - 5147050155902 T^{2} + 14797555901469154212 T^{3} - \)\(62\!\cdots\!13\)\( T^{4} - \)\(31\!\cdots\!92\)\( T^{5} + \)\(26\!\cdots\!56\)\( T^{6} + \)\(23\!\cdots\!00\)\( T^{7} - \)\(12\!\cdots\!19\)\( T^{8} - \)\(62\!\cdots\!16\)\( T^{9} + \)\(10\!\cdots\!02\)\( T^{10} - \)\(62\!\cdots\!16\)\( p^{8} T^{11} - \)\(12\!\cdots\!19\)\( p^{16} T^{12} + \)\(23\!\cdots\!00\)\( p^{24} T^{13} + \)\(26\!\cdots\!56\)\( p^{32} T^{14} - \)\(31\!\cdots\!92\)\( p^{40} T^{15} - \)\(62\!\cdots\!13\)\( p^{48} T^{16} + 14797555901469154212 p^{56} T^{17} - 5147050155902 p^{64} T^{18} - 1913308 p^{72} T^{19} + p^{80} T^{20} \) | |
| 41 | \( 1 - 35717233702222 T^{2} + \)\(77\!\cdots\!65\)\( T^{4} - \)\(11\!\cdots\!80\)\( T^{6} + \)\(13\!\cdots\!10\)\( T^{8} - \)\(11\!\cdots\!36\)\( T^{10} + \)\(13\!\cdots\!10\)\( p^{16} T^{12} - \)\(11\!\cdots\!80\)\( p^{32} T^{14} + \)\(77\!\cdots\!65\)\( p^{48} T^{16} - 35717233702222 p^{64} T^{18} + p^{80} T^{20} \) | |
| 43 | \( ( 1 + 430924 T + 24581604982782 T^{2} - 12369650421362636934 T^{3} + \)\(46\!\cdots\!81\)\( T^{4} + \)\(71\!\cdots\!84\)\( T^{5} + \)\(46\!\cdots\!81\)\( p^{8} T^{6} - 12369650421362636934 p^{16} T^{7} + 24581604982782 p^{24} T^{8} + 430924 p^{32} T^{9} + p^{40} T^{10} )^{2} \) | |
| 47 | \( 1 - 305448 T + 79866233742617 T^{2} - 24385482086103518952 T^{3} + \)\(31\!\cdots\!83\)\( T^{4} - \)\(44\!\cdots\!44\)\( T^{5} + \)\(10\!\cdots\!54\)\( T^{6} - \)\(45\!\cdots\!84\)\( p T^{7} + \)\(34\!\cdots\!05\)\( T^{8} - \)\(91\!\cdots\!88\)\( p T^{9} + \)\(91\!\cdots\!71\)\( T^{10} - \)\(91\!\cdots\!88\)\( p^{9} T^{11} + \)\(34\!\cdots\!05\)\( p^{16} T^{12} - \)\(45\!\cdots\!84\)\( p^{25} T^{13} + \)\(10\!\cdots\!54\)\( p^{32} T^{14} - \)\(44\!\cdots\!44\)\( p^{40} T^{15} + \)\(31\!\cdots\!83\)\( p^{48} T^{16} - 24385482086103518952 p^{56} T^{17} + 79866233742617 p^{64} T^{18} - 305448 p^{72} T^{19} + p^{80} T^{20} \) | |
| 53 | \( 1 - 10663233 T - 157141440205871 T^{2} + \)\(73\!\cdots\!62\)\( T^{3} + \)\(26\!\cdots\!82\)\( T^{4} - \)\(33\!\cdots\!50\)\( T^{5} - \)\(22\!\cdots\!29\)\( T^{6} - \)\(91\!\cdots\!91\)\( T^{7} + \)\(14\!\cdots\!31\)\( T^{8} + \)\(14\!\cdots\!16\)\( T^{9} - \)\(91\!\cdots\!36\)\( T^{10} + \)\(14\!\cdots\!16\)\( p^{8} T^{11} + \)\(14\!\cdots\!31\)\( p^{16} T^{12} - \)\(91\!\cdots\!91\)\( p^{24} T^{13} - \)\(22\!\cdots\!29\)\( p^{32} T^{14} - \)\(33\!\cdots\!50\)\( p^{40} T^{15} + \)\(26\!\cdots\!82\)\( p^{48} T^{16} + \)\(73\!\cdots\!62\)\( p^{56} T^{17} - 157141440205871 p^{64} T^{18} - 10663233 p^{72} T^{19} + p^{80} T^{20} \) | |
| 59 | \( 1 + 18410871 T + 493241729159489 T^{2} + \)\(70\!\cdots\!82\)\( T^{3} + \)\(12\!\cdots\!54\)\( T^{4} + \)\(12\!\cdots\!46\)\( T^{5} + \)\(88\!\cdots\!83\)\( T^{6} + \)\(14\!\cdots\!69\)\( T^{7} - \)\(15\!\cdots\!45\)\( T^{8} - \)\(27\!\cdots\!88\)\( T^{9} - \)\(51\!\cdots\!96\)\( T^{10} - \)\(27\!\cdots\!88\)\( p^{8} T^{11} - \)\(15\!\cdots\!45\)\( p^{16} T^{12} + \)\(14\!\cdots\!69\)\( p^{24} T^{13} + \)\(88\!\cdots\!83\)\( p^{32} T^{14} + \)\(12\!\cdots\!46\)\( p^{40} T^{15} + \)\(12\!\cdots\!54\)\( p^{48} T^{16} + \)\(70\!\cdots\!82\)\( p^{56} T^{17} + 493241729159489 p^{64} T^{18} + 18410871 p^{72} T^{19} + p^{80} T^{20} \) | |
| 61 | \( 1 + 13937808 T + 538045898663573 T^{2} + \)\(65\!\cdots\!80\)\( T^{3} + \)\(10\!\cdots\!39\)\( T^{4} + \)\(19\!\cdots\!92\)\( T^{5} + \)\(27\!\cdots\!14\)\( T^{6} + \)\(54\!\cdots\!80\)\( T^{7} + \)\(84\!\cdots\!17\)\( T^{8} + \)\(10\!\cdots\!84\)\( T^{9} + \)\(18\!\cdots\!27\)\( T^{10} + \)\(10\!\cdots\!84\)\( p^{8} T^{11} + \)\(84\!\cdots\!17\)\( p^{16} T^{12} + \)\(54\!\cdots\!80\)\( p^{24} T^{13} + \)\(27\!\cdots\!14\)\( p^{32} T^{14} + \)\(19\!\cdots\!92\)\( p^{40} T^{15} + \)\(10\!\cdots\!39\)\( p^{48} T^{16} + \)\(65\!\cdots\!80\)\( p^{56} T^{17} + 538045898663573 p^{64} T^{18} + 13937808 p^{72} T^{19} + p^{80} T^{20} \) | |
| 67 | \( 1 + 20722822 T - 1357114937137634 T^{2} - \)\(22\!\cdots\!20\)\( T^{3} + \)\(11\!\cdots\!59\)\( T^{4} + \)\(13\!\cdots\!72\)\( T^{5} - \)\(81\!\cdots\!08\)\( T^{6} - \)\(45\!\cdots\!46\)\( T^{7} + \)\(46\!\cdots\!21\)\( T^{8} + \)\(74\!\cdots\!28\)\( T^{9} - \)\(21\!\cdots\!26\)\( T^{10} + \)\(74\!\cdots\!28\)\( p^{8} T^{11} + \)\(46\!\cdots\!21\)\( p^{16} T^{12} - \)\(45\!\cdots\!46\)\( p^{24} T^{13} - \)\(81\!\cdots\!08\)\( p^{32} T^{14} + \)\(13\!\cdots\!72\)\( p^{40} T^{15} + \)\(11\!\cdots\!59\)\( p^{48} T^{16} - \)\(22\!\cdots\!20\)\( p^{56} T^{17} - 1357114937137634 p^{64} T^{18} + 20722822 p^{72} T^{19} + p^{80} T^{20} \) | |
| 71 | \( ( 1 + 56516292 T + 2193133095714497 T^{2} + \)\(71\!\cdots\!64\)\( T^{3} + \)\(24\!\cdots\!70\)\( T^{4} + \)\(67\!\cdots\!24\)\( T^{5} + \)\(24\!\cdots\!70\)\( p^{8} T^{6} + \)\(71\!\cdots\!64\)\( p^{16} T^{7} + 2193133095714497 p^{24} T^{8} + 56516292 p^{32} T^{9} + p^{40} T^{10} )^{2} \) | |
| 73 | \( 1 - 43436322 T + 3123294885911732 T^{2} - \)\(10\!\cdots\!88\)\( T^{3} + \)\(51\!\cdots\!13\)\( T^{4} - \)\(18\!\cdots\!16\)\( T^{5} + \)\(71\!\cdots\!94\)\( T^{6} - \)\(24\!\cdots\!62\)\( T^{7} + \)\(76\!\cdots\!65\)\( T^{8} - \)\(24\!\cdots\!04\)\( T^{9} + \)\(67\!\cdots\!86\)\( T^{10} - \)\(24\!\cdots\!04\)\( p^{8} T^{11} + \)\(76\!\cdots\!65\)\( p^{16} T^{12} - \)\(24\!\cdots\!62\)\( p^{24} T^{13} + \)\(71\!\cdots\!94\)\( p^{32} T^{14} - \)\(18\!\cdots\!16\)\( p^{40} T^{15} + \)\(51\!\cdots\!13\)\( p^{48} T^{16} - \)\(10\!\cdots\!88\)\( p^{56} T^{17} + 3123294885911732 p^{64} T^{18} - 43436322 p^{72} T^{19} + p^{80} T^{20} \) | |
| 79 | \( 1 + 42189637 T - 5031952624864598 T^{2} - \)\(14\!\cdots\!69\)\( T^{3} + \)\(17\!\cdots\!02\)\( T^{4} + \)\(27\!\cdots\!71\)\( T^{5} - \)\(47\!\cdots\!72\)\( T^{6} - \)\(40\!\cdots\!53\)\( T^{7} + \)\(99\!\cdots\!21\)\( T^{8} + \)\(30\!\cdots\!22\)\( T^{9} - \)\(16\!\cdots\!48\)\( T^{10} + \)\(30\!\cdots\!22\)\( p^{8} T^{11} + \)\(99\!\cdots\!21\)\( p^{16} T^{12} - \)\(40\!\cdots\!53\)\( p^{24} T^{13} - \)\(47\!\cdots\!72\)\( p^{32} T^{14} + \)\(27\!\cdots\!71\)\( p^{40} T^{15} + \)\(17\!\cdots\!02\)\( p^{48} T^{16} - \)\(14\!\cdots\!69\)\( p^{56} T^{17} - 5031952624864598 p^{64} T^{18} + 42189637 p^{72} T^{19} + p^{80} T^{20} \) | |
| 83 | \( 1 - 13558239844208449 T^{2} + \)\(87\!\cdots\!56\)\( T^{4} - \)\(35\!\cdots\!23\)\( T^{6} + \)\(10\!\cdots\!15\)\( T^{8} - \)\(26\!\cdots\!08\)\( T^{10} + \)\(10\!\cdots\!15\)\( p^{16} T^{12} - \)\(35\!\cdots\!23\)\( p^{32} T^{14} + \)\(87\!\cdots\!56\)\( p^{48} T^{16} - 13558239844208449 p^{64} T^{18} + p^{80} T^{20} \) | |
| 89 | \( 1 + 67171914 T + 11900992272143189 T^{2} + \)\(69\!\cdots\!98\)\( T^{3} + \)\(61\!\cdots\!19\)\( T^{4} + \)\(28\!\cdots\!04\)\( T^{5} + \)\(16\!\cdots\!38\)\( T^{6} + \)\(18\!\cdots\!20\)\( T^{7} + \)\(13\!\cdots\!21\)\( T^{8} - \)\(28\!\cdots\!66\)\( T^{9} - \)\(38\!\cdots\!29\)\( T^{10} - \)\(28\!\cdots\!66\)\( p^{8} T^{11} + \)\(13\!\cdots\!21\)\( p^{16} T^{12} + \)\(18\!\cdots\!20\)\( p^{24} T^{13} + \)\(16\!\cdots\!38\)\( p^{32} T^{14} + \)\(28\!\cdots\!04\)\( p^{40} T^{15} + \)\(61\!\cdots\!19\)\( p^{48} T^{16} + \)\(69\!\cdots\!98\)\( p^{56} T^{17} + 11900992272143189 p^{64} T^{18} + 67171914 p^{72} T^{19} + p^{80} T^{20} \) | |
| 97 | \( 1 - 16661382897278185 T^{2} + \)\(21\!\cdots\!64\)\( T^{4} - \)\(22\!\cdots\!31\)\( T^{6} + \)\(22\!\cdots\!07\)\( T^{8} - \)\(20\!\cdots\!52\)\( T^{10} + \)\(22\!\cdots\!07\)\( p^{16} T^{12} - \)\(22\!\cdots\!31\)\( p^{32} T^{14} + \)\(21\!\cdots\!64\)\( p^{48} T^{16} - 16661382897278185 p^{64} T^{18} + p^{80} T^{20} \) | |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{20} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−3.17692977539222364233257074955, −3.04627645536752097950523886824, −3.03208578737250835964923869779, −2.85154048528282752292011459679, −2.68043971785489176216094337473, −2.61276176915678032728375033338, −2.49345743855186582460025875179, −2.36150495353647679177190594031, −2.30612603347131079024629745364, −2.05012666652134539195221384148, −1.76904780777821428814480518417, −1.72597298447720679869106662332, −1.60984060457892030224392673372, −1.60328560768090220424648091634, −1.42097141891869368431049468948, −1.38015657862067511831156508597, −1.23409709870807695924180265108, −0.867703998631653654227027019435, −0.796454652138627566177251221935, −0.74751191728752220508573861851, −0.48406545776196157820432653833, −0.41884774779709459519760576351, −0.25739576823366117060298201649, −0.25694017352100454447686108361, −0.18739861266678527145880938068, 0.18739861266678527145880938068, 0.25694017352100454447686108361, 0.25739576823366117060298201649, 0.41884774779709459519760576351, 0.48406545776196157820432653833, 0.74751191728752220508573861851, 0.796454652138627566177251221935, 0.867703998631653654227027019435, 1.23409709870807695924180265108, 1.38015657862067511831156508597, 1.42097141891869368431049468948, 1.60328560768090220424648091634, 1.60984060457892030224392673372, 1.72597298447720679869106662332, 1.76904780777821428814480518417, 2.05012666652134539195221384148, 2.30612603347131079024629745364, 2.36150495353647679177190594031, 2.49345743855186582460025875179, 2.61276176915678032728375033338, 2.68043971785489176216094337473, 2.85154048528282752292011459679, 3.03208578737250835964923869779, 3.04627645536752097950523886824, 3.17692977539222364233257074955
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.