Dirichlet series
| L(s) = 1 | − 90·2-s + 2.97e4·4-s − 1.64e6·7-s − 1.80e6·8-s + 7.17e7·9-s + 1.48e8·14-s + 2.20e7·16-s + 7.28e8·17-s − 6.45e9·18-s − 3.55e10·23-s + 1.75e11·25-s − 4.90e10·28-s − 1.05e11·31-s + 1.05e10·32-s − 6.55e10·34-s + 2.13e12·36-s − 5.32e10·41-s + 3.19e12·46-s + 1.25e13·47-s − 2.76e13·49-s − 1.58e13·50-s + 2.97e12·56-s + 9.51e12·62-s − 1.18e14·63-s + 5.58e12·64-s + 2.16e13·68-s − 1.73e14·71-s + ⋯ |
| L(s) = 1 | − 0.497·2-s + 0.908·4-s − 0.755·7-s − 0.304·8-s + 4.99·9-s + 0.375·14-s + 0.0205·16-s + 0.430·17-s − 2.48·18-s − 2.17·23-s + 5.75·25-s − 0.686·28-s − 0.690·31-s + 0.0544·32-s − 0.214·34-s + 4.54·36-s − 0.0426·41-s + 1.08·46-s + 3.60·47-s − 5.82·49-s − 2.86·50-s + 0.230·56-s + 0.343·62-s − 3.77·63-s + 0.158·64-s + 0.391·68-s − 2.26·71-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{42}\right)^{s/2} \, \Gamma_{\C}(s)^{14} \, L(s)\cr=\mathstrut & \,\Lambda(16-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{42}\right)^{s/2} \, \Gamma_{\C}(s+15/2)^{14} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(28\) |
| Conductor: | \(2^{42}\) |
| Sign: | $1$ |
| Analytic conductor: | \(6.38144\times 10^{14}\) |
| Root analytic conductor: | \(3.37868\) |
| Motivic weight: | \(15\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((28,\ 2^{42} ,\ ( \ : [15/2]^{14} ),\ 1 )\) |
Particular Values
| \(L(8)\) | \(\approx\) | \(53.44580903\) |
| \(L(\frac12)\) | \(\approx\) | \(53.44580903\) |
| \(L(\frac{17}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( 1 + 45 p T - 2709 p^{3} T^{2} - 44115 p^{6} T^{3} + 519203 p^{10} T^{4} + 2447097 p^{15} T^{5} - 9426533 p^{21} T^{6} - 28681203 p^{28} T^{7} - 9426533 p^{36} T^{8} + 2447097 p^{45} T^{9} + 519203 p^{55} T^{10} - 44115 p^{66} T^{11} - 2709 p^{78} T^{12} + 45 p^{91} T^{13} + p^{105} T^{14} \) |
| good | 3 | \( 1 - 71744534 T^{2} + 11174092023217 p^{5} T^{4} - \)\(30\!\cdots\!32\)\( p^{5} T^{6} + \)\(22\!\cdots\!17\)\( p^{6} T^{8} - \)\(61\!\cdots\!82\)\( p^{12} T^{10} + \)\(14\!\cdots\!71\)\( p^{18} T^{12} - \)\(30\!\cdots\!76\)\( p^{24} T^{14} + \)\(14\!\cdots\!71\)\( p^{48} T^{16} - \)\(61\!\cdots\!82\)\( p^{72} T^{18} + \)\(22\!\cdots\!17\)\( p^{96} T^{20} - \)\(30\!\cdots\!32\)\( p^{125} T^{22} + 11174092023217 p^{155} T^{24} - 71744534 p^{180} T^{26} + p^{210} T^{28} \) |
| 5 | \( 1 - 175673069478 T^{2} + \)\(66\!\cdots\!83\)\( p^{2} T^{4} - \)\(17\!\cdots\!08\)\( p^{4} T^{6} + \)\(37\!\cdots\!01\)\( p^{6} T^{8} - \)\(66\!\cdots\!98\)\( p^{8} T^{10} + \)\(10\!\cdots\!23\)\( p^{10} T^{12} - \)\(13\!\cdots\!48\)\( p^{12} T^{14} + \)\(10\!\cdots\!23\)\( p^{40} T^{16} - \)\(66\!\cdots\!98\)\( p^{68} T^{18} + \)\(37\!\cdots\!01\)\( p^{96} T^{20} - \)\(17\!\cdots\!08\)\( p^{124} T^{22} + \)\(66\!\cdots\!83\)\( p^{152} T^{24} - 175673069478 p^{180} T^{26} + p^{210} T^{28} \) | |
| 7 | \( ( 1 + 823544 T + 2121247328503 p T^{2} + 57753104805050832 p^{3} T^{3} + \)\(32\!\cdots\!75\)\( p^{3} T^{4} + \)\(63\!\cdots\!12\)\( p^{4} T^{5} + \)\(40\!\cdots\!63\)\( p^{5} T^{6} + \)\(62\!\cdots\!16\)\( p^{6} T^{7} + \)\(40\!\cdots\!63\)\( p^{20} T^{8} + \)\(63\!\cdots\!12\)\( p^{34} T^{9} + \)\(32\!\cdots\!75\)\( p^{48} T^{10} + 57753104805050832 p^{63} T^{11} + 2121247328503 p^{76} T^{12} + 823544 p^{90} T^{13} + p^{105} T^{14} )^{2} \) | |
| 11 | \( 1 - 25585379626376710 T^{2} + \)\(35\!\cdots\!59\)\( T^{4} - \)\(29\!\cdots\!00\)\( p^{2} T^{6} + \)\(18\!\cdots\!77\)\( p^{4} T^{8} - \)\(97\!\cdots\!90\)\( p^{6} T^{10} + \)\(42\!\cdots\!43\)\( p^{8} T^{12} - \)\(16\!\cdots\!00\)\( p^{10} T^{14} + \)\(42\!\cdots\!43\)\( p^{38} T^{16} - \)\(97\!\cdots\!90\)\( p^{66} T^{18} + \)\(18\!\cdots\!77\)\( p^{94} T^{20} - \)\(29\!\cdots\!00\)\( p^{122} T^{22} + \)\(35\!\cdots\!59\)\( p^{150} T^{24} - 25585379626376710 p^{180} T^{26} + p^{210} T^{28} \) | |
| 13 | \( 1 - 364744637033779894 T^{2} + \)\(39\!\cdots\!79\)\( p^{2} T^{4} - \)\(28\!\cdots\!48\)\( p^{4} T^{6} + \)\(15\!\cdots\!73\)\( p^{6} T^{8} - \)\(66\!\cdots\!86\)\( p^{8} T^{10} + \)\(24\!\cdots\!43\)\( p^{10} T^{12} - \)\(79\!\cdots\!24\)\( p^{12} T^{14} + \)\(24\!\cdots\!43\)\( p^{40} T^{16} - \)\(66\!\cdots\!86\)\( p^{68} T^{18} + \)\(15\!\cdots\!73\)\( p^{96} T^{20} - \)\(28\!\cdots\!48\)\( p^{124} T^{22} + \)\(39\!\cdots\!79\)\( p^{152} T^{24} - 364744637033779894 p^{180} T^{26} + p^{210} T^{28} \) | |
| 17 | \( ( 1 - 364277406 T + 11908466948431100331 T^{2} - \)\(11\!\cdots\!28\)\( T^{3} + \)\(67\!\cdots\!81\)\( T^{4} + \)\(10\!\cdots\!18\)\( T^{5} + \)\(24\!\cdots\!95\)\( T^{6} + \)\(62\!\cdots\!56\)\( T^{7} + \)\(24\!\cdots\!95\)\( p^{15} T^{8} + \)\(10\!\cdots\!18\)\( p^{30} T^{9} + \)\(67\!\cdots\!81\)\( p^{45} T^{10} - \)\(11\!\cdots\!28\)\( p^{60} T^{11} + 11908466948431100331 p^{75} T^{12} - 364277406 p^{90} T^{13} + p^{105} T^{14} )^{2} \) | |
| 19 | \( 1 - 86399350183618527862 T^{2} + \)\(42\!\cdots\!63\)\( T^{4} - \)\(15\!\cdots\!36\)\( T^{6} + \)\(42\!\cdots\!85\)\( T^{8} - \)\(98\!\cdots\!10\)\( T^{10} + \)\(19\!\cdots\!59\)\( T^{12} - \)\(31\!\cdots\!00\)\( T^{14} + \)\(19\!\cdots\!59\)\( p^{30} T^{16} - \)\(98\!\cdots\!10\)\( p^{60} T^{18} + \)\(42\!\cdots\!85\)\( p^{90} T^{20} - \)\(15\!\cdots\!36\)\( p^{120} T^{22} + \)\(42\!\cdots\!63\)\( p^{150} T^{24} - 86399350183618527862 p^{180} T^{26} + p^{210} T^{28} \) | |
| 23 | \( ( 1 + 17774408040 T + \)\(11\!\cdots\!57\)\( T^{2} + \)\(11\!\cdots\!28\)\( T^{3} + \)\(52\!\cdots\!45\)\( T^{4} + \)\(21\!\cdots\!64\)\( T^{5} + \)\(13\!\cdots\!65\)\( T^{6} + \)\(10\!\cdots\!44\)\( T^{7} + \)\(13\!\cdots\!65\)\( p^{15} T^{8} + \)\(21\!\cdots\!64\)\( p^{30} T^{9} + \)\(52\!\cdots\!45\)\( p^{45} T^{10} + \)\(11\!\cdots\!28\)\( p^{60} T^{11} + \)\(11\!\cdots\!57\)\( p^{75} T^{12} + 17774408040 p^{90} T^{13} + p^{105} T^{14} )^{2} \) | |
| 29 | \( 1 - \)\(64\!\cdots\!18\)\( T^{2} + \)\(21\!\cdots\!59\)\( T^{4} - \)\(47\!\cdots\!64\)\( T^{6} + \)\(26\!\cdots\!73\)\( p T^{8} - \)\(10\!\cdots\!82\)\( T^{10} + \)\(11\!\cdots\!23\)\( T^{12} - \)\(10\!\cdots\!72\)\( T^{14} + \)\(11\!\cdots\!23\)\( p^{30} T^{16} - \)\(10\!\cdots\!82\)\( p^{60} T^{18} + \)\(26\!\cdots\!73\)\( p^{91} T^{20} - \)\(47\!\cdots\!64\)\( p^{120} T^{22} + \)\(21\!\cdots\!59\)\( p^{150} T^{24} - \)\(64\!\cdots\!18\)\( p^{180} T^{26} + p^{210} T^{28} \) | |
| 31 | \( ( 1 + 52879069408 T + \)\(80\!\cdots\!45\)\( T^{2} - \)\(77\!\cdots\!88\)\( T^{3} + \)\(34\!\cdots\!73\)\( T^{4} - \)\(83\!\cdots\!76\)\( T^{5} + \)\(11\!\cdots\!93\)\( T^{6} - \)\(19\!\cdots\!12\)\( T^{7} + \)\(11\!\cdots\!93\)\( p^{15} T^{8} - \)\(83\!\cdots\!76\)\( p^{30} T^{9} + \)\(34\!\cdots\!73\)\( p^{45} T^{10} - \)\(77\!\cdots\!88\)\( p^{60} T^{11} + \)\(80\!\cdots\!45\)\( p^{75} T^{12} + 52879069408 p^{90} T^{13} + p^{105} T^{14} )^{2} \) | |
| 37 | \( 1 - \)\(26\!\cdots\!54\)\( T^{2} + \)\(36\!\cdots\!75\)\( T^{4} - \)\(33\!\cdots\!68\)\( T^{6} + \)\(23\!\cdots\!01\)\( T^{8} - \)\(12\!\cdots\!42\)\( T^{10} + \)\(55\!\cdots\!15\)\( T^{12} - \)\(20\!\cdots\!72\)\( T^{14} + \)\(55\!\cdots\!15\)\( p^{30} T^{16} - \)\(12\!\cdots\!42\)\( p^{60} T^{18} + \)\(23\!\cdots\!01\)\( p^{90} T^{20} - \)\(33\!\cdots\!68\)\( p^{120} T^{22} + \)\(36\!\cdots\!75\)\( p^{150} T^{24} - \)\(26\!\cdots\!54\)\( p^{180} T^{26} + p^{210} T^{28} \) | |
| 41 | \( ( 1 + 26614592970 T + \)\(90\!\cdots\!27\)\( T^{2} - \)\(98\!\cdots\!32\)\( T^{3} + \)\(36\!\cdots\!69\)\( T^{4} - \)\(65\!\cdots\!58\)\( T^{5} + \)\(86\!\cdots\!79\)\( T^{6} - \)\(15\!\cdots\!12\)\( T^{7} + \)\(86\!\cdots\!79\)\( p^{15} T^{8} - \)\(65\!\cdots\!58\)\( p^{30} T^{9} + \)\(36\!\cdots\!69\)\( p^{45} T^{10} - \)\(98\!\cdots\!32\)\( p^{60} T^{11} + \)\(90\!\cdots\!27\)\( p^{75} T^{12} + 26614592970 p^{90} T^{13} + p^{105} T^{14} )^{2} \) | |
| 43 | \( 1 - \)\(28\!\cdots\!78\)\( T^{2} + \)\(42\!\cdots\!55\)\( T^{4} - \)\(41\!\cdots\!32\)\( T^{6} + \)\(29\!\cdots\!33\)\( T^{8} - \)\(16\!\cdots\!94\)\( T^{10} + \)\(71\!\cdots\!19\)\( T^{12} - \)\(25\!\cdots\!92\)\( T^{14} + \)\(71\!\cdots\!19\)\( p^{30} T^{16} - \)\(16\!\cdots\!94\)\( p^{60} T^{18} + \)\(29\!\cdots\!33\)\( p^{90} T^{20} - \)\(41\!\cdots\!32\)\( p^{120} T^{22} + \)\(42\!\cdots\!55\)\( p^{150} T^{24} - \)\(28\!\cdots\!78\)\( p^{180} T^{26} + p^{210} T^{28} \) | |
| 47 | \( ( 1 - 6263999223216 T + \)\(44\!\cdots\!57\)\( T^{2} - \)\(93\!\cdots\!44\)\( T^{3} + \)\(38\!\cdots\!09\)\( T^{4} + \)\(58\!\cdots\!88\)\( T^{5} + \)\(37\!\cdots\!73\)\( T^{6} - \)\(32\!\cdots\!12\)\( T^{7} + \)\(37\!\cdots\!73\)\( p^{15} T^{8} + \)\(58\!\cdots\!88\)\( p^{30} T^{9} + \)\(38\!\cdots\!09\)\( p^{45} T^{10} - \)\(93\!\cdots\!44\)\( p^{60} T^{11} + \)\(44\!\cdots\!57\)\( p^{75} T^{12} - 6263999223216 p^{90} T^{13} + p^{105} T^{14} )^{2} \) | |
| 53 | \( 1 - \)\(55\!\cdots\!38\)\( T^{2} + \)\(14\!\cdots\!19\)\( T^{4} - \)\(24\!\cdots\!32\)\( T^{6} + \)\(30\!\cdots\!45\)\( T^{8} - \)\(29\!\cdots\!78\)\( T^{10} + \)\(23\!\cdots\!03\)\( T^{12} - \)\(17\!\cdots\!04\)\( T^{14} + \)\(23\!\cdots\!03\)\( p^{30} T^{16} - \)\(29\!\cdots\!78\)\( p^{60} T^{18} + \)\(30\!\cdots\!45\)\( p^{90} T^{20} - \)\(24\!\cdots\!32\)\( p^{120} T^{22} + \)\(14\!\cdots\!19\)\( p^{150} T^{24} - \)\(55\!\cdots\!38\)\( p^{180} T^{26} + p^{210} T^{28} \) | |
| 59 | \( 1 - \)\(20\!\cdots\!14\)\( T^{2} + \)\(25\!\cdots\!23\)\( T^{4} - \)\(21\!\cdots\!72\)\( T^{6} + \)\(14\!\cdots\!29\)\( T^{8} - \)\(83\!\cdots\!02\)\( T^{10} + \)\(39\!\cdots\!79\)\( T^{12} - \)\(44\!\cdots\!48\)\( p^{2} T^{14} + \)\(39\!\cdots\!79\)\( p^{30} T^{16} - \)\(83\!\cdots\!02\)\( p^{60} T^{18} + \)\(14\!\cdots\!29\)\( p^{90} T^{20} - \)\(21\!\cdots\!72\)\( p^{120} T^{22} + \)\(25\!\cdots\!23\)\( p^{150} T^{24} - \)\(20\!\cdots\!14\)\( p^{180} T^{26} + p^{210} T^{28} \) | |
| 61 | \( 1 - \)\(54\!\cdots\!74\)\( T^{2} + \)\(14\!\cdots\!83\)\( T^{4} - \)\(24\!\cdots\!52\)\( T^{6} + \)\(30\!\cdots\!09\)\( T^{8} - \)\(30\!\cdots\!82\)\( T^{10} + \)\(24\!\cdots\!19\)\( T^{12} - \)\(16\!\cdots\!08\)\( T^{14} + \)\(24\!\cdots\!19\)\( p^{30} T^{16} - \)\(30\!\cdots\!82\)\( p^{60} T^{18} + \)\(30\!\cdots\!09\)\( p^{90} T^{20} - \)\(24\!\cdots\!52\)\( p^{120} T^{22} + \)\(14\!\cdots\!83\)\( p^{150} T^{24} - \)\(54\!\cdots\!74\)\( p^{180} T^{26} + p^{210} T^{28} \) | |
| 67 | \( 1 - \)\(15\!\cdots\!62\)\( T^{2} + \)\(12\!\cdots\!19\)\( T^{4} - \)\(70\!\cdots\!08\)\( T^{6} + \)\(29\!\cdots\!65\)\( T^{8} - \)\(10\!\cdots\!02\)\( T^{10} + \)\(30\!\cdots\!43\)\( T^{12} - \)\(78\!\cdots\!56\)\( T^{14} + \)\(30\!\cdots\!43\)\( p^{30} T^{16} - \)\(10\!\cdots\!02\)\( p^{60} T^{18} + \)\(29\!\cdots\!65\)\( p^{90} T^{20} - \)\(70\!\cdots\!08\)\( p^{120} T^{22} + \)\(12\!\cdots\!19\)\( p^{150} T^{24} - \)\(15\!\cdots\!62\)\( p^{180} T^{26} + p^{210} T^{28} \) | |
| 71 | \( ( 1 + 86624963854008 T + \)\(30\!\cdots\!93\)\( T^{2} + \)\(18\!\cdots\!28\)\( T^{3} + \)\(38\!\cdots\!65\)\( T^{4} + \)\(16\!\cdots\!16\)\( T^{5} + \)\(29\!\cdots\!41\)\( T^{6} + \)\(98\!\cdots\!96\)\( T^{7} + \)\(29\!\cdots\!41\)\( p^{15} T^{8} + \)\(16\!\cdots\!16\)\( p^{30} T^{9} + \)\(38\!\cdots\!65\)\( p^{45} T^{10} + \)\(18\!\cdots\!28\)\( p^{60} T^{11} + \)\(30\!\cdots\!93\)\( p^{75} T^{12} + 86624963854008 p^{90} T^{13} + p^{105} T^{14} )^{2} \) | |
| 73 | \( ( 1 + 91028918941098 T + \)\(22\!\cdots\!91\)\( T^{2} - \)\(19\!\cdots\!76\)\( T^{3} + \)\(12\!\cdots\!25\)\( T^{4} - \)\(15\!\cdots\!98\)\( T^{5} + \)\(16\!\cdots\!27\)\( T^{6} - \)\(10\!\cdots\!60\)\( T^{7} + \)\(16\!\cdots\!27\)\( p^{15} T^{8} - \)\(15\!\cdots\!98\)\( p^{30} T^{9} + \)\(12\!\cdots\!25\)\( p^{45} T^{10} - \)\(19\!\cdots\!76\)\( p^{60} T^{11} + \)\(22\!\cdots\!91\)\( p^{75} T^{12} + 91028918941098 p^{90} T^{13} + p^{105} T^{14} )^{2} \) | |
| 79 | \( ( 1 + 147185136635696 T + \)\(16\!\cdots\!85\)\( T^{2} + \)\(24\!\cdots\!92\)\( T^{3} + \)\(13\!\cdots\!25\)\( T^{4} + \)\(16\!\cdots\!32\)\( T^{5} + \)\(60\!\cdots\!33\)\( T^{6} + \)\(64\!\cdots\!60\)\( T^{7} + \)\(60\!\cdots\!33\)\( p^{15} T^{8} + \)\(16\!\cdots\!32\)\( p^{30} T^{9} + \)\(13\!\cdots\!25\)\( p^{45} T^{10} + \)\(24\!\cdots\!92\)\( p^{60} T^{11} + \)\(16\!\cdots\!85\)\( p^{75} T^{12} + 147185136635696 p^{90} T^{13} + p^{105} T^{14} )^{2} \) | |
| 83 | \( 1 - \)\(38\!\cdots\!18\)\( T^{2} + \)\(76\!\cdots\!95\)\( T^{4} - \)\(12\!\cdots\!24\)\( p T^{6} + \)\(11\!\cdots\!53\)\( T^{8} - \)\(10\!\cdots\!54\)\( T^{10} + \)\(78\!\cdots\!39\)\( T^{12} - \)\(51\!\cdots\!72\)\( T^{14} + \)\(78\!\cdots\!39\)\( p^{30} T^{16} - \)\(10\!\cdots\!54\)\( p^{60} T^{18} + \)\(11\!\cdots\!53\)\( p^{90} T^{20} - \)\(12\!\cdots\!24\)\( p^{121} T^{22} + \)\(76\!\cdots\!95\)\( p^{150} T^{24} - \)\(38\!\cdots\!18\)\( p^{180} T^{26} + p^{210} T^{28} \) | |
| 89 | \( ( 1 - 157106975824614 T + \)\(91\!\cdots\!95\)\( T^{2} - \)\(19\!\cdots\!68\)\( T^{3} + \)\(37\!\cdots\!05\)\( T^{4} - \)\(92\!\cdots\!78\)\( T^{5} + \)\(95\!\cdots\!23\)\( T^{6} - \)\(22\!\cdots\!80\)\( T^{7} + \)\(95\!\cdots\!23\)\( p^{15} T^{8} - \)\(92\!\cdots\!78\)\( p^{30} T^{9} + \)\(37\!\cdots\!05\)\( p^{45} T^{10} - \)\(19\!\cdots\!68\)\( p^{60} T^{11} + \)\(91\!\cdots\!95\)\( p^{75} T^{12} - 157106975824614 p^{90} T^{13} + p^{105} T^{14} )^{2} \) | |
| 97 | \( ( 1 + 336287145929618 T + \)\(22\!\cdots\!47\)\( T^{2} + \)\(67\!\cdots\!72\)\( T^{3} + \)\(27\!\cdots\!97\)\( T^{4} + \)\(58\!\cdots\!18\)\( T^{5} + \)\(22\!\cdots\!87\)\( T^{6} + \)\(40\!\cdots\!36\)\( T^{7} + \)\(22\!\cdots\!87\)\( p^{15} T^{8} + \)\(58\!\cdots\!18\)\( p^{30} T^{9} + \)\(27\!\cdots\!97\)\( p^{45} T^{10} + \)\(67\!\cdots\!72\)\( p^{60} T^{11} + \)\(22\!\cdots\!47\)\( p^{75} T^{12} + 336287145929618 p^{90} T^{13} + p^{105} T^{14} )^{2} \) | |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{28} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−4.36342045316022545357964009653, −4.28620802151852017331000577692, −4.13777517808186742392538665525, −3.91394961111727538181404407725, −3.69351757208709180723958128217, −3.36664471257542095012347101100, −3.28388718137026319372159849752, −3.00523991847508938435665195205, −2.94661819094486222538300831854, −2.91556322757769772346147737762, −2.88186768047336702522119847633, −2.26060425701340803843345779057, −2.17396228112553412325439054013, −1.96585061797493061301747041833, −1.73888589386845511023474912886, −1.60125323849680602666766926501, −1.54767798306109025849099011945, −1.53055457201023939509541299948, −1.34431366198644523186534183977, −1.01684451869694760017490180377, −0.76692246943419293797308263204, −0.71952787391488251631604046979, −0.54192841125828026459723361639, −0.47290295042212204902833781802, −0.24502322543592902518104650088, 0.24502322543592902518104650088, 0.47290295042212204902833781802, 0.54192841125828026459723361639, 0.71952787391488251631604046979, 0.76692246943419293797308263204, 1.01684451869694760017490180377, 1.34431366198644523186534183977, 1.53055457201023939509541299948, 1.54767798306109025849099011945, 1.60125323849680602666766926501, 1.73888589386845511023474912886, 1.96585061797493061301747041833, 2.17396228112553412325439054013, 2.26060425701340803843345779057, 2.88186768047336702522119847633, 2.91556322757769772346147737762, 2.94661819094486222538300831854, 3.00523991847508938435665195205, 3.28388718137026319372159849752, 3.36664471257542095012347101100, 3.69351757208709180723958128217, 3.91394961111727538181404407725, 4.13777517808186742392538665525, 4.28620802151852017331000577692, 4.36342045316022545357964009653
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.