Dirichlet series
| L(s) = 1 | − 350·2-s + 1.19e4·3-s + 3.41e4·4-s − 4.18e6·6-s + 5.97e6·8-s − 1.86e8·9-s − 4.91e7·11-s + 4.08e8·12-s + 4.89e8·16-s − 9.98e9·17-s + 6.52e10·18-s + 6.29e10·19-s + 1.71e10·22-s + 7.14e10·24-s + 8.53e11·25-s − 2.75e12·27-s − 6.85e11·32-s − 5.87e11·33-s + 3.49e12·34-s − 6.36e12·36-s − 2.20e13·38-s + 8.71e12·41-s + 1.67e13·43-s − 1.67e12·44-s + 5.85e12·48-s + 1.80e14·49-s − 2.98e14·50-s + ⋯ |
| L(s) = 1 | − 1.36·2-s + 1.82·3-s + 0.520·4-s − 2.49·6-s + 0.356·8-s − 4.33·9-s − 0.229·11-s + 0.949·12-s + 0.114·16-s − 1.43·17-s + 5.92·18-s + 3.70·19-s + 0.313·22-s + 0.649·24-s + 5.59·25-s − 9.75·27-s − 0.623·32-s − 0.418·33-s + 1.95·34-s − 2.25·36-s − 5.06·38-s + 1.09·41-s + 1.43·43-s − 0.119·44-s + 0.207·48-s + 5.42·49-s − 7.64·50-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{42}\right)^{s/2} \, \Gamma_{\C}(s)^{14} \, L(s)\cr=\mathstrut & \,\Lambda(17-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{42}\right)^{s/2} \, \Gamma_{\C}(s+8)^{14} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(28\) |
| Conductor: | \(2^{42}\) |
| Sign: | $1$ |
| Analytic conductor: | \(3.87827\times 10^{15}\) |
| Root analytic conductor: | \(3.60360\) |
| Motivic weight: | \(16\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((28,\ 2^{42} ,\ ( \ : [8]^{14} ),\ 1 )\) |
Particular Values
| \(L(\frac{17}{2})\) | \(\approx\) | \(9.798868192\) |
| \(L(\frac12)\) | \(\approx\) | \(9.798868192\) |
| \(L(9)\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( 1 + 175 p T + 11045 p^{3} T^{2} + 203145 p^{6} T^{3} - 1022889 p^{10} T^{4} - 25153845 p^{15} T^{5} - 83196775 p^{21} T^{6} - 293150725 p^{28} T^{7} - 83196775 p^{37} T^{8} - 25153845 p^{47} T^{9} - 1022889 p^{58} T^{10} + 203145 p^{70} T^{11} + 11045 p^{83} T^{12} + 175 p^{97} T^{13} + p^{112} T^{14} \) |
| good | 3 | \( ( 1 - 1994 p T + 16328483 p^{2} T^{2} - 3420685316 p^{5} T^{3} + 49237882290451 p^{5} T^{4} - 30451660704179398 p^{7} T^{5} + 11750575395184565827 p^{10} T^{6} - \)\(74\!\cdots\!00\)\( p^{14} T^{7} + 11750575395184565827 p^{26} T^{8} - 30451660704179398 p^{39} T^{9} + 49237882290451 p^{53} T^{10} - 3420685316 p^{69} T^{11} + 16328483 p^{82} T^{12} - 1994 p^{97} T^{13} + p^{112} T^{14} )^{2} \) |
| 5 | \( 1 - 34126476734 p^{2} T^{2} + \)\(77\!\cdots\!51\)\( p T^{4} - \)\(97\!\cdots\!56\)\( p^{3} T^{6} + \)\(19\!\cdots\!73\)\( p^{6} T^{8} - \)\(31\!\cdots\!54\)\( p^{9} T^{10} + \)\(44\!\cdots\!87\)\( p^{12} T^{12} - \)\(56\!\cdots\!56\)\( p^{15} T^{14} + \)\(44\!\cdots\!87\)\( p^{44} T^{16} - \)\(31\!\cdots\!54\)\( p^{73} T^{18} + \)\(19\!\cdots\!73\)\( p^{102} T^{20} - \)\(97\!\cdots\!56\)\( p^{131} T^{22} + \)\(77\!\cdots\!51\)\( p^{161} T^{24} - 34126476734 p^{194} T^{26} + p^{224} T^{28} \) | |
| 7 | \( 1 - 180260128286990 T^{2} + \)\(24\!\cdots\!29\)\( p T^{4} - \)\(34\!\cdots\!80\)\( p^{3} T^{6} + \)\(37\!\cdots\!63\)\( p^{5} T^{8} - \)\(50\!\cdots\!42\)\( p^{8} T^{10} + \)\(83\!\cdots\!55\)\( p^{12} T^{12} - \)\(12\!\cdots\!56\)\( p^{16} T^{14} + \)\(83\!\cdots\!55\)\( p^{44} T^{16} - \)\(50\!\cdots\!42\)\( p^{72} T^{18} + \)\(37\!\cdots\!63\)\( p^{101} T^{20} - \)\(34\!\cdots\!80\)\( p^{131} T^{22} + \)\(24\!\cdots\!29\)\( p^{161} T^{24} - 180260128286990 p^{192} T^{26} + p^{224} T^{28} \) | |
| 11 | \( ( 1 + 24570146 T + 160649922269474491 T^{2} + \)\(19\!\cdots\!36\)\( p T^{3} + \)\(10\!\cdots\!61\)\( p^{2} T^{4} - \)\(19\!\cdots\!34\)\( p^{3} T^{5} + \)\(46\!\cdots\!31\)\( p^{4} T^{6} - \)\(47\!\cdots\!44\)\( p^{5} T^{7} + \)\(46\!\cdots\!31\)\( p^{20} T^{8} - \)\(19\!\cdots\!34\)\( p^{35} T^{9} + \)\(10\!\cdots\!61\)\( p^{50} T^{10} + \)\(19\!\cdots\!36\)\( p^{65} T^{11} + 160649922269474491 p^{80} T^{12} + 24570146 p^{96} T^{13} + p^{112} T^{14} )^{2} \) | |
| 13 | \( 1 - 4302368111385722510 T^{2} + \)\(73\!\cdots\!11\)\( p T^{4} - \)\(67\!\cdots\!60\)\( p^{3} T^{6} + \)\(47\!\cdots\!57\)\( p^{5} T^{8} - \)\(27\!\cdots\!26\)\( p^{7} T^{10} + \)\(13\!\cdots\!35\)\( p^{9} T^{12} - \)\(58\!\cdots\!28\)\( p^{11} T^{14} + \)\(13\!\cdots\!35\)\( p^{41} T^{16} - \)\(27\!\cdots\!26\)\( p^{71} T^{18} + \)\(47\!\cdots\!57\)\( p^{101} T^{20} - \)\(67\!\cdots\!60\)\( p^{131} T^{22} + \)\(73\!\cdots\!11\)\( p^{161} T^{24} - 4302368111385722510 p^{192} T^{26} + p^{224} T^{28} \) | |
| 17 | \( ( 1 + 4992568178 T + \)\(17\!\cdots\!87\)\( T^{2} + \)\(72\!\cdots\!16\)\( p T^{3} + \)\(17\!\cdots\!53\)\( T^{4} + \)\(13\!\cdots\!34\)\( T^{5} + \)\(11\!\cdots\!63\)\( T^{6} + \)\(81\!\cdots\!00\)\( T^{7} + \)\(11\!\cdots\!63\)\( p^{16} T^{8} + \)\(13\!\cdots\!34\)\( p^{32} T^{9} + \)\(17\!\cdots\!53\)\( p^{48} T^{10} + \)\(72\!\cdots\!16\)\( p^{65} T^{11} + \)\(17\!\cdots\!87\)\( p^{80} T^{12} + 4992568178 p^{96} T^{13} + p^{112} T^{14} )^{2} \) | |
| 19 | \( ( 1 - 1655845610 p T + \)\(15\!\cdots\!43\)\( T^{2} - \)\(39\!\cdots\!20\)\( T^{3} + \)\(11\!\cdots\!21\)\( T^{4} - \)\(23\!\cdots\!54\)\( T^{5} + \)\(52\!\cdots\!55\)\( T^{6} - \)\(85\!\cdots\!92\)\( T^{7} + \)\(52\!\cdots\!55\)\( p^{16} T^{8} - \)\(23\!\cdots\!54\)\( p^{32} T^{9} + \)\(11\!\cdots\!21\)\( p^{48} T^{10} - \)\(39\!\cdots\!20\)\( p^{64} T^{11} + \)\(15\!\cdots\!43\)\( p^{80} T^{12} - 1655845610 p^{97} T^{13} + p^{112} T^{14} )^{2} \) | |
| 23 | \( 1 - \)\(34\!\cdots\!70\)\( T^{2} + \)\(66\!\cdots\!83\)\( T^{4} - \)\(89\!\cdots\!60\)\( T^{6} + \)\(93\!\cdots\!61\)\( T^{8} - \)\(80\!\cdots\!22\)\( T^{10} + \)\(59\!\cdots\!55\)\( T^{12} - \)\(39\!\cdots\!16\)\( T^{14} + \)\(59\!\cdots\!55\)\( p^{32} T^{16} - \)\(80\!\cdots\!22\)\( p^{64} T^{18} + \)\(93\!\cdots\!61\)\( p^{96} T^{20} - \)\(89\!\cdots\!60\)\( p^{128} T^{22} + \)\(66\!\cdots\!83\)\( p^{160} T^{24} - \)\(34\!\cdots\!70\)\( p^{192} T^{26} + p^{224} T^{28} \) | |
| 29 | \( 1 - \)\(17\!\cdots\!34\)\( T^{2} + \)\(15\!\cdots\!11\)\( T^{4} - \)\(10\!\cdots\!04\)\( T^{6} + \)\(48\!\cdots\!61\)\( T^{8} - \)\(18\!\cdots\!82\)\( T^{10} + \)\(60\!\cdots\!03\)\( T^{12} - \)\(57\!\cdots\!68\)\( p T^{14} + \)\(60\!\cdots\!03\)\( p^{32} T^{16} - \)\(18\!\cdots\!82\)\( p^{64} T^{18} + \)\(48\!\cdots\!61\)\( p^{96} T^{20} - \)\(10\!\cdots\!04\)\( p^{128} T^{22} + \)\(15\!\cdots\!11\)\( p^{160} T^{24} - \)\(17\!\cdots\!34\)\( p^{192} T^{26} + p^{224} T^{28} \) | |
| 31 | \( 1 - \)\(59\!\cdots\!14\)\( T^{2} + \)\(17\!\cdots\!11\)\( T^{4} - \)\(35\!\cdots\!04\)\( T^{6} + \)\(52\!\cdots\!21\)\( T^{8} - \)\(61\!\cdots\!02\)\( T^{10} + \)\(58\!\cdots\!03\)\( T^{12} - \)\(48\!\cdots\!52\)\( p^{2} T^{14} + \)\(58\!\cdots\!03\)\( p^{32} T^{16} - \)\(61\!\cdots\!02\)\( p^{64} T^{18} + \)\(52\!\cdots\!21\)\( p^{96} T^{20} - \)\(35\!\cdots\!04\)\( p^{128} T^{22} + \)\(17\!\cdots\!11\)\( p^{160} T^{24} - \)\(59\!\cdots\!14\)\( p^{192} T^{26} + p^{224} T^{28} \) | |
| 37 | \( 1 - \)\(56\!\cdots\!90\)\( T^{2} + \)\(18\!\cdots\!83\)\( T^{4} - \)\(44\!\cdots\!80\)\( T^{6} + \)\(86\!\cdots\!21\)\( T^{8} - \)\(14\!\cdots\!62\)\( T^{10} + \)\(20\!\cdots\!95\)\( T^{12} - \)\(26\!\cdots\!16\)\( T^{14} + \)\(20\!\cdots\!95\)\( p^{32} T^{16} - \)\(14\!\cdots\!62\)\( p^{64} T^{18} + \)\(86\!\cdots\!21\)\( p^{96} T^{20} - \)\(44\!\cdots\!80\)\( p^{128} T^{22} + \)\(18\!\cdots\!83\)\( p^{160} T^{24} - \)\(56\!\cdots\!90\)\( p^{192} T^{26} + p^{224} T^{28} \) | |
| 41 | \( ( 1 - 4359814536334 T + \)\(27\!\cdots\!11\)\( T^{2} - \)\(85\!\cdots\!84\)\( T^{3} + \)\(34\!\cdots\!81\)\( T^{4} - \)\(62\!\cdots\!34\)\( T^{5} + \)\(27\!\cdots\!51\)\( T^{6} - \)\(32\!\cdots\!04\)\( T^{7} + \)\(27\!\cdots\!51\)\( p^{16} T^{8} - \)\(62\!\cdots\!34\)\( p^{32} T^{9} + \)\(34\!\cdots\!81\)\( p^{48} T^{10} - \)\(85\!\cdots\!84\)\( p^{64} T^{11} + \)\(27\!\cdots\!11\)\( p^{80} T^{12} - 4359814536334 p^{96} T^{13} + p^{112} T^{14} )^{2} \) | |
| 43 | \( ( 1 - 8358154589150 T + \)\(80\!\cdots\!15\)\( T^{2} - \)\(58\!\cdots\!00\)\( T^{3} + \)\(29\!\cdots\!49\)\( T^{4} - \)\(18\!\cdots\!30\)\( T^{5} + \)\(62\!\cdots\!75\)\( T^{6} - \)\(32\!\cdots\!60\)\( T^{7} + \)\(62\!\cdots\!75\)\( p^{16} T^{8} - \)\(18\!\cdots\!30\)\( p^{32} T^{9} + \)\(29\!\cdots\!49\)\( p^{48} T^{10} - \)\(58\!\cdots\!00\)\( p^{64} T^{11} + \)\(80\!\cdots\!15\)\( p^{80} T^{12} - 8358154589150 p^{96} T^{13} + p^{112} T^{14} )^{2} \) | |
| 47 | \( 1 - \)\(51\!\cdots\!10\)\( T^{2} + \)\(13\!\cdots\!63\)\( T^{4} - \)\(22\!\cdots\!40\)\( T^{6} + \)\(27\!\cdots\!61\)\( T^{8} - \)\(27\!\cdots\!82\)\( T^{10} + \)\(21\!\cdots\!75\)\( T^{12} - \)\(13\!\cdots\!76\)\( T^{14} + \)\(21\!\cdots\!75\)\( p^{32} T^{16} - \)\(27\!\cdots\!82\)\( p^{64} T^{18} + \)\(27\!\cdots\!61\)\( p^{96} T^{20} - \)\(22\!\cdots\!40\)\( p^{128} T^{22} + \)\(13\!\cdots\!63\)\( p^{160} T^{24} - \)\(51\!\cdots\!10\)\( p^{192} T^{26} + p^{224} T^{28} \) | |
| 53 | \( 1 - \)\(25\!\cdots\!30\)\( T^{2} + \)\(28\!\cdots\!23\)\( T^{4} - \)\(17\!\cdots\!60\)\( T^{6} + \)\(56\!\cdots\!61\)\( T^{8} + \)\(42\!\cdots\!06\)\( p T^{10} - \)\(12\!\cdots\!85\)\( T^{12} + \)\(67\!\cdots\!04\)\( T^{14} - \)\(12\!\cdots\!85\)\( p^{32} T^{16} + \)\(42\!\cdots\!06\)\( p^{65} T^{18} + \)\(56\!\cdots\!61\)\( p^{96} T^{20} - \)\(17\!\cdots\!60\)\( p^{128} T^{22} + \)\(28\!\cdots\!23\)\( p^{160} T^{24} - \)\(25\!\cdots\!30\)\( p^{192} T^{26} + p^{224} T^{28} \) | |
| 59 | \( ( 1 - 173503665146590 T + \)\(80\!\cdots\!63\)\( T^{2} - \)\(80\!\cdots\!60\)\( T^{3} + \)\(23\!\cdots\!61\)\( T^{4} - \)\(66\!\cdots\!82\)\( T^{5} + \)\(36\!\cdots\!75\)\( T^{6} + \)\(12\!\cdots\!24\)\( T^{7} + \)\(36\!\cdots\!75\)\( p^{16} T^{8} - \)\(66\!\cdots\!82\)\( p^{32} T^{9} + \)\(23\!\cdots\!61\)\( p^{48} T^{10} - \)\(80\!\cdots\!60\)\( p^{64} T^{11} + \)\(80\!\cdots\!63\)\( p^{80} T^{12} - 173503665146590 p^{96} T^{13} + p^{112} T^{14} )^{2} \) | |
| 61 | \( 1 - \)\(16\!\cdots\!34\)\( T^{2} + \)\(14\!\cdots\!51\)\( T^{4} - \)\(90\!\cdots\!04\)\( T^{6} + \)\(50\!\cdots\!61\)\( T^{8} - \)\(24\!\cdots\!22\)\( T^{10} + \)\(10\!\cdots\!83\)\( T^{12} - \)\(10\!\cdots\!12\)\( p^{2} T^{14} + \)\(10\!\cdots\!83\)\( p^{32} T^{16} - \)\(24\!\cdots\!22\)\( p^{64} T^{18} + \)\(50\!\cdots\!61\)\( p^{96} T^{20} - \)\(90\!\cdots\!04\)\( p^{128} T^{22} + \)\(14\!\cdots\!51\)\( p^{160} T^{24} - \)\(16\!\cdots\!34\)\( p^{192} T^{26} + p^{224} T^{28} \) | |
| 67 | \( ( 1 + 307129382984098 T + \)\(32\!\cdots\!07\)\( T^{2} + \)\(83\!\cdots\!32\)\( T^{3} + \)\(54\!\cdots\!13\)\( T^{4} + \)\(18\!\cdots\!74\)\( T^{5} + \)\(16\!\cdots\!43\)\( T^{6} + \)\(41\!\cdots\!00\)\( T^{7} + \)\(16\!\cdots\!43\)\( p^{16} T^{8} + \)\(18\!\cdots\!74\)\( p^{32} T^{9} + \)\(54\!\cdots\!13\)\( p^{48} T^{10} + \)\(83\!\cdots\!32\)\( p^{64} T^{11} + \)\(32\!\cdots\!07\)\( p^{80} T^{12} + 307129382984098 p^{96} T^{13} + p^{112} T^{14} )^{2} \) | |
| 71 | \( 1 - \)\(26\!\cdots\!14\)\( T^{2} + \)\(34\!\cdots\!91\)\( T^{4} - \)\(32\!\cdots\!64\)\( T^{6} + \)\(23\!\cdots\!01\)\( T^{8} - \)\(14\!\cdots\!22\)\( T^{10} + \)\(73\!\cdots\!43\)\( T^{12} - \)\(33\!\cdots\!32\)\( T^{14} + \)\(73\!\cdots\!43\)\( p^{32} T^{16} - \)\(14\!\cdots\!22\)\( p^{64} T^{18} + \)\(23\!\cdots\!01\)\( p^{96} T^{20} - \)\(32\!\cdots\!64\)\( p^{128} T^{22} + \)\(34\!\cdots\!91\)\( p^{160} T^{24} - \)\(26\!\cdots\!14\)\( p^{192} T^{26} + p^{224} T^{28} \) | |
| 73 | \( ( 1 + 271444259185138 T + \)\(22\!\cdots\!27\)\( T^{2} + \)\(81\!\cdots\!72\)\( T^{3} + \)\(30\!\cdots\!53\)\( T^{4} + \)\(10\!\cdots\!74\)\( T^{5} + \)\(27\!\cdots\!43\)\( T^{6} + \)\(85\!\cdots\!00\)\( T^{7} + \)\(27\!\cdots\!43\)\( p^{16} T^{8} + \)\(10\!\cdots\!74\)\( p^{32} T^{9} + \)\(30\!\cdots\!53\)\( p^{48} T^{10} + \)\(81\!\cdots\!72\)\( p^{64} T^{11} + \)\(22\!\cdots\!27\)\( p^{80} T^{12} + 271444259185138 p^{96} T^{13} + p^{112} T^{14} )^{2} \) | |
| 79 | \( 1 - \)\(18\!\cdots\!54\)\( T^{2} + \)\(17\!\cdots\!11\)\( T^{4} - \)\(11\!\cdots\!24\)\( T^{6} + \)\(51\!\cdots\!61\)\( T^{8} - \)\(18\!\cdots\!82\)\( T^{10} + \)\(57\!\cdots\!43\)\( T^{12} - \)\(14\!\cdots\!72\)\( T^{14} + \)\(57\!\cdots\!43\)\( p^{32} T^{16} - \)\(18\!\cdots\!82\)\( p^{64} T^{18} + \)\(51\!\cdots\!61\)\( p^{96} T^{20} - \)\(11\!\cdots\!24\)\( p^{128} T^{22} + \)\(17\!\cdots\!11\)\( p^{160} T^{24} - \)\(18\!\cdots\!54\)\( p^{192} T^{26} + p^{224} T^{28} \) | |
| 83 | \( ( 1 - 3681210965373022 T + \)\(35\!\cdots\!87\)\( T^{2} - \)\(10\!\cdots\!28\)\( T^{3} + \)\(54\!\cdots\!93\)\( T^{4} - \)\(12\!\cdots\!26\)\( T^{5} + \)\(45\!\cdots\!43\)\( T^{6} - \)\(80\!\cdots\!80\)\( T^{7} + \)\(45\!\cdots\!43\)\( p^{16} T^{8} - \)\(12\!\cdots\!26\)\( p^{32} T^{9} + \)\(54\!\cdots\!93\)\( p^{48} T^{10} - \)\(10\!\cdots\!28\)\( p^{64} T^{11} + \)\(35\!\cdots\!87\)\( p^{80} T^{12} - 3681210965373022 p^{96} T^{13} + p^{112} T^{14} )^{2} \) | |
| 89 | \( ( 1 + 204065663682290 T + \)\(83\!\cdots\!63\)\( T^{2} + \)\(84\!\cdots\!40\)\( T^{3} + \)\(32\!\cdots\!01\)\( T^{4} + \)\(20\!\cdots\!86\)\( T^{5} + \)\(77\!\cdots\!95\)\( T^{6} + \)\(35\!\cdots\!68\)\( T^{7} + \)\(77\!\cdots\!95\)\( p^{16} T^{8} + \)\(20\!\cdots\!86\)\( p^{32} T^{9} + \)\(32\!\cdots\!01\)\( p^{48} T^{10} + \)\(84\!\cdots\!40\)\( p^{64} T^{11} + \)\(83\!\cdots\!63\)\( p^{80} T^{12} + 204065663682290 p^{96} T^{13} + p^{112} T^{14} )^{2} \) | |
| 97 | \( ( 1 - 9186068317128590 T + \)\(34\!\cdots\!95\)\( T^{2} - \)\(28\!\cdots\!00\)\( T^{3} + \)\(57\!\cdots\!89\)\( T^{4} - \)\(40\!\cdots\!30\)\( T^{5} + \)\(55\!\cdots\!15\)\( T^{6} - \)\(31\!\cdots\!80\)\( T^{7} + \)\(55\!\cdots\!15\)\( p^{16} T^{8} - \)\(40\!\cdots\!30\)\( p^{32} T^{9} + \)\(57\!\cdots\!89\)\( p^{48} T^{10} - \)\(28\!\cdots\!00\)\( p^{64} T^{11} + \)\(34\!\cdots\!95\)\( p^{80} T^{12} - 9186068317128590 p^{96} T^{13} + p^{112} T^{14} )^{2} \) | |
| show more | ||
| show less | ||
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{28} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−4.12722002721307436939100577985, −3.92448794065837812990122188108, −3.70085141822402114349484022585, −3.38928036440650936417518843262, −3.19574391616774588058806912271, −3.09065411220274613576488858269, −3.07087374864854425702341901013, −3.06897568943048632659149792715, −2.94167002458221154575162569595, −2.57608937500916108963815037448, −2.54630383088831262092897073057, −2.36927336631530891690132004912, −2.35026911402547793720189193262, −2.28847323276074192014519034416, −2.06393102272117719185744713970, −1.49589473164828932950445092735, −1.26839501227598652429847617434, −1.17497124231622392874291644392, −1.11905266998439249552895021679, −0.842620766893523494102672025170, −0.74702477533264377842603447325, −0.65478845134785127410169239897, −0.49387535176397551883411799072, −0.27332495862600666766722802676, −0.23159504115858680143127325697, 0.23159504115858680143127325697, 0.27332495862600666766722802676, 0.49387535176397551883411799072, 0.65478845134785127410169239897, 0.74702477533264377842603447325, 0.842620766893523494102672025170, 1.11905266998439249552895021679, 1.17497124231622392874291644392, 1.26839501227598652429847617434, 1.49589473164828932950445092735, 2.06393102272117719185744713970, 2.28847323276074192014519034416, 2.35026911402547793720189193262, 2.36927336631530891690132004912, 2.54630383088831262092897073057, 2.57608937500916108963815037448, 2.94167002458221154575162569595, 3.06897568943048632659149792715, 3.07087374864854425702341901013, 3.09065411220274613576488858269, 3.19574391616774588058806912271, 3.38928036440650936417518843262, 3.70085141822402114349484022585, 3.92448794065837812990122188108, 4.12722002721307436939100577985
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.