Dirichlet series
| L(s) = 1 | − 9.54e4·13-s − 2.87e5·19-s + 2.24e6·25-s − 3.55e6·31-s − 1.82e5·37-s + 8.47e6·43-s + 6.58e6·49-s + 3.42e7·61-s + 2.26e7·67-s + 2.13e6·73-s − 9.02e7·79-s + 1.34e8·97-s − 1.93e8·103-s + 2.47e8·109-s + 1.25e9·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 1.60e8·169-s + 173-s + 179-s + ⋯ |
| L(s) = 1 | − 3.34·13-s − 2.20·19-s + 5.75·25-s − 3.84·31-s − 0.0976·37-s + 2.47·43-s + 8/7·49-s + 2.47·61-s + 1.12·67-s + 0.0752·73-s − 2.31·79-s + 1.51·97-s − 1.72·103-s + 1.75·109-s + 5.83·121-s − 0.196·169-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{32} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(9-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{32} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s+4)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(32\) |
| Conductor: | \(2^{32} \cdot 3^{32} \cdot 7^{16}\) |
| Sign: | $1$ |
| Analytic conductor: | \(1.52187\times 10^{32}\) |
| Root analytic conductor: | \(10.1320\) |
| Motivic weight: | \(8\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((32,\ 2^{32} \cdot 3^{32} \cdot 7^{16} ,\ ( \ : [4]^{16} ),\ 1 )\) |
Particular Values
| \(L(\frac{9}{2})\) | \(\approx\) | \(12.15896839\) |
| \(L(\frac12)\) | \(\approx\) | \(12.15896839\) |
| \(L(5)\) | 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 | \( ( 1 - p^{7} T^{2} )^{8} \) | |
| good | 5 | \( 1 - 449548 p T^{2} + 2957647251956 T^{4} - 2821584601847157124 T^{6} + \)\(21\!\cdots\!16\)\( T^{8} - \)\(10\!\cdots\!96\)\( p^{3} T^{10} + \)\(12\!\cdots\!24\)\( p^{4} T^{12} - \)\(45\!\cdots\!56\)\( p^{7} T^{14} + \)\(38\!\cdots\!26\)\( p^{8} T^{16} - \)\(45\!\cdots\!56\)\( p^{23} T^{18} + \)\(12\!\cdots\!24\)\( p^{36} T^{20} - \)\(10\!\cdots\!96\)\( p^{51} T^{22} + \)\(21\!\cdots\!16\)\( p^{64} T^{24} - 2821584601847157124 p^{80} T^{26} + 2957647251956 p^{96} T^{28} - 449548 p^{113} T^{30} + p^{128} T^{32} \) |
| 11 | \( 1 - 1251616676 T^{2} + 885499362567905516 T^{4} - \)\(45\!\cdots\!44\)\( T^{6} + \)\(18\!\cdots\!64\)\( T^{8} - \)\(51\!\cdots\!68\)\( p^{2} T^{10} + \)\(16\!\cdots\!68\)\( p T^{12} - \)\(47\!\cdots\!84\)\( T^{14} + \)\(10\!\cdots\!10\)\( T^{16} - \)\(47\!\cdots\!84\)\( p^{16} T^{18} + \)\(16\!\cdots\!68\)\( p^{33} T^{20} - \)\(51\!\cdots\!68\)\( p^{50} T^{22} + \)\(18\!\cdots\!64\)\( p^{64} T^{24} - \)\(45\!\cdots\!44\)\( p^{80} T^{26} + 885499362567905516 p^{96} T^{28} - 1251616676 p^{112} T^{30} + p^{128} T^{32} \) | |
| 13 | \( ( 1 + 47740 T + 3498752228 T^{2} + 10967471516372 p T^{3} + 5587938992569959284 T^{4} + \)\(18\!\cdots\!72\)\( T^{5} + \)\(57\!\cdots\!44\)\( T^{6} + \)\(12\!\cdots\!00\)\( p T^{7} + \)\(48\!\cdots\!98\)\( T^{8} + \)\(12\!\cdots\!00\)\( p^{9} T^{9} + \)\(57\!\cdots\!44\)\( p^{16} T^{10} + \)\(18\!\cdots\!72\)\( p^{24} T^{11} + 5587938992569959284 p^{32} T^{12} + 10967471516372 p^{41} T^{13} + 3498752228 p^{48} T^{14} + 47740 p^{56} T^{15} + p^{64} T^{16} )^{2} \) | |
| 17 | \( 1 - 55387088252 T^{2} + \)\(15\!\cdots\!48\)\( T^{4} - \)\(29\!\cdots\!68\)\( T^{6} + \)\(43\!\cdots\!44\)\( T^{8} - \)\(51\!\cdots\!12\)\( T^{10} + \)\(52\!\cdots\!44\)\( T^{12} - \)\(44\!\cdots\!56\)\( T^{14} + \)\(33\!\cdots\!86\)\( T^{16} - \)\(44\!\cdots\!56\)\( p^{16} T^{18} + \)\(52\!\cdots\!44\)\( p^{32} T^{20} - \)\(51\!\cdots\!12\)\( p^{48} T^{22} + \)\(43\!\cdots\!44\)\( p^{64} T^{24} - \)\(29\!\cdots\!68\)\( p^{80} T^{26} + \)\(15\!\cdots\!48\)\( p^{96} T^{28} - 55387088252 p^{112} T^{30} + p^{128} T^{32} \) | |
| 19 | \( ( 1 + 143780 T + 87605642976 T^{2} + 9775563820503292 T^{3} + \)\(35\!\cdots\!00\)\( T^{4} + \)\(33\!\cdots\!32\)\( T^{5} + \)\(95\!\cdots\!28\)\( T^{6} + \)\(79\!\cdots\!84\)\( T^{7} + \)\(18\!\cdots\!78\)\( T^{8} + \)\(79\!\cdots\!84\)\( p^{8} T^{9} + \)\(95\!\cdots\!28\)\( p^{16} T^{10} + \)\(33\!\cdots\!32\)\( p^{24} T^{11} + \)\(35\!\cdots\!00\)\( p^{32} T^{12} + 9775563820503292 p^{40} T^{13} + 87605642976 p^{48} T^{14} + 143780 p^{56} T^{15} + p^{64} T^{16} )^{2} \) | |
| 23 | \( 1 - 547465040420 T^{2} + \)\(14\!\cdots\!12\)\( T^{4} - \)\(27\!\cdots\!60\)\( T^{6} + \)\(39\!\cdots\!56\)\( T^{8} - \)\(47\!\cdots\!88\)\( T^{10} + \)\(48\!\cdots\!52\)\( T^{12} - \)\(44\!\cdots\!60\)\( T^{14} + \)\(36\!\cdots\!10\)\( T^{16} - \)\(44\!\cdots\!60\)\( p^{16} T^{18} + \)\(48\!\cdots\!52\)\( p^{32} T^{20} - \)\(47\!\cdots\!88\)\( p^{48} T^{22} + \)\(39\!\cdots\!56\)\( p^{64} T^{24} - \)\(27\!\cdots\!60\)\( p^{80} T^{26} + \)\(14\!\cdots\!12\)\( p^{96} T^{28} - 547465040420 p^{112} T^{30} + p^{128} T^{32} \) | |
| 29 | \( 1 - 5391029633072 T^{2} + \)\(14\!\cdots\!68\)\( T^{4} - \)\(25\!\cdots\!84\)\( T^{6} + \)\(32\!\cdots\!84\)\( T^{8} - \)\(33\!\cdots\!72\)\( T^{10} + \)\(26\!\cdots\!44\)\( T^{12} - \)\(17\!\cdots\!56\)\( T^{14} + \)\(11\!\cdots\!30\)\( p^{2} T^{16} - \)\(17\!\cdots\!56\)\( p^{16} T^{18} + \)\(26\!\cdots\!44\)\( p^{32} T^{20} - \)\(33\!\cdots\!72\)\( p^{48} T^{22} + \)\(32\!\cdots\!84\)\( p^{64} T^{24} - \)\(25\!\cdots\!84\)\( p^{80} T^{26} + \)\(14\!\cdots\!68\)\( p^{96} T^{28} - 5391029633072 p^{112} T^{30} + p^{128} T^{32} \) | |
| 31 | \( ( 1 + 1777132 T + 5878234327040 T^{2} + 7849991009326039028 T^{3} + \)\(15\!\cdots\!12\)\( T^{4} + \)\(16\!\cdots\!64\)\( T^{5} + \)\(23\!\cdots\!80\)\( T^{6} + \)\(21\!\cdots\!40\)\( T^{7} + \)\(24\!\cdots\!58\)\( T^{8} + \)\(21\!\cdots\!40\)\( p^{8} T^{9} + \)\(23\!\cdots\!80\)\( p^{16} T^{10} + \)\(16\!\cdots\!64\)\( p^{24} T^{11} + \)\(15\!\cdots\!12\)\( p^{32} T^{12} + 7849991009326039028 p^{40} T^{13} + 5878234327040 p^{48} T^{14} + 1777132 p^{56} T^{15} + p^{64} T^{16} )^{2} \) | |
| 37 | \( ( 1 + 91460 T + 17752512050364 T^{2} + 2363345223380588572 T^{3} + \)\(15\!\cdots\!64\)\( T^{4} + \)\(18\!\cdots\!68\)\( T^{5} + \)\(92\!\cdots\!52\)\( T^{6} + \)\(86\!\cdots\!36\)\( T^{7} + \)\(38\!\cdots\!58\)\( T^{8} + \)\(86\!\cdots\!36\)\( p^{8} T^{9} + \)\(92\!\cdots\!52\)\( p^{16} T^{10} + \)\(18\!\cdots\!68\)\( p^{24} T^{11} + \)\(15\!\cdots\!64\)\( p^{32} T^{12} + 2363345223380588572 p^{40} T^{13} + 17752512050364 p^{48} T^{14} + 91460 p^{56} T^{15} + p^{64} T^{16} )^{2} \) | |
| 41 | \( 1 - 59856667701340 T^{2} + \)\(17\!\cdots\!84\)\( T^{4} - \)\(32\!\cdots\!44\)\( T^{6} + \)\(45\!\cdots\!84\)\( T^{8} - \)\(54\!\cdots\!00\)\( T^{10} + \)\(56\!\cdots\!56\)\( T^{12} - \)\(53\!\cdots\!96\)\( T^{14} + \)\(45\!\cdots\!26\)\( T^{16} - \)\(53\!\cdots\!96\)\( p^{16} T^{18} + \)\(56\!\cdots\!56\)\( p^{32} T^{20} - \)\(54\!\cdots\!00\)\( p^{48} T^{22} + \)\(45\!\cdots\!84\)\( p^{64} T^{24} - \)\(32\!\cdots\!44\)\( p^{80} T^{26} + \)\(17\!\cdots\!84\)\( p^{96} T^{28} - 59856667701340 p^{112} T^{30} + p^{128} T^{32} \) | |
| 43 | \( ( 1 - 4236208 T + 58510508117688 T^{2} - \)\(19\!\cdots\!48\)\( T^{3} + \)\(15\!\cdots\!80\)\( T^{4} - \)\(44\!\cdots\!84\)\( T^{5} + \)\(25\!\cdots\!08\)\( T^{6} - \)\(72\!\cdots\!60\)\( T^{7} + \)\(34\!\cdots\!78\)\( T^{8} - \)\(72\!\cdots\!60\)\( p^{8} T^{9} + \)\(25\!\cdots\!08\)\( p^{16} T^{10} - \)\(44\!\cdots\!84\)\( p^{24} T^{11} + \)\(15\!\cdots\!80\)\( p^{32} T^{12} - \)\(19\!\cdots\!48\)\( p^{40} T^{13} + 58510508117688 p^{48} T^{14} - 4236208 p^{56} T^{15} + p^{64} T^{16} )^{2} \) | |
| 47 | \( 1 - 246961759826368 T^{2} + \)\(30\!\cdots\!84\)\( T^{4} - \)\(24\!\cdots\!84\)\( T^{6} + \)\(14\!\cdots\!28\)\( T^{8} - \)\(15\!\cdots\!68\)\( p T^{10} + \)\(26\!\cdots\!92\)\( T^{12} - \)\(84\!\cdots\!04\)\( T^{14} + \)\(21\!\cdots\!10\)\( T^{16} - \)\(84\!\cdots\!04\)\( p^{16} T^{18} + \)\(26\!\cdots\!92\)\( p^{32} T^{20} - \)\(15\!\cdots\!68\)\( p^{49} T^{22} + \)\(14\!\cdots\!28\)\( p^{64} T^{24} - \)\(24\!\cdots\!84\)\( p^{80} T^{26} + \)\(30\!\cdots\!84\)\( p^{96} T^{28} - 246961759826368 p^{112} T^{30} + p^{128} T^{32} \) | |
| 53 | \( 1 - 611991538560736 T^{2} + \)\(17\!\cdots\!96\)\( T^{4} - \)\(32\!\cdots\!72\)\( T^{6} + \)\(44\!\cdots\!92\)\( T^{8} - \)\(46\!\cdots\!56\)\( T^{10} + \)\(41\!\cdots\!80\)\( T^{12} - \)\(31\!\cdots\!40\)\( T^{14} + \)\(21\!\cdots\!66\)\( T^{16} - \)\(31\!\cdots\!40\)\( p^{16} T^{18} + \)\(41\!\cdots\!80\)\( p^{32} T^{20} - \)\(46\!\cdots\!56\)\( p^{48} T^{22} + \)\(44\!\cdots\!92\)\( p^{64} T^{24} - \)\(32\!\cdots\!72\)\( p^{80} T^{26} + \)\(17\!\cdots\!96\)\( p^{96} T^{28} - 611991538560736 p^{112} T^{30} + p^{128} T^{32} \) | |
| 59 | \( 1 - 1130940700956992 T^{2} + \)\(70\!\cdots\!44\)\( T^{4} - \)\(30\!\cdots\!64\)\( T^{6} + \)\(10\!\cdots\!72\)\( T^{8} - \)\(27\!\cdots\!36\)\( T^{10} + \)\(62\!\cdots\!16\)\( T^{12} - \)\(11\!\cdots\!24\)\( T^{14} + \)\(18\!\cdots\!82\)\( T^{16} - \)\(11\!\cdots\!24\)\( p^{16} T^{18} + \)\(62\!\cdots\!16\)\( p^{32} T^{20} - \)\(27\!\cdots\!36\)\( p^{48} T^{22} + \)\(10\!\cdots\!72\)\( p^{64} T^{24} - \)\(30\!\cdots\!64\)\( p^{80} T^{26} + \)\(70\!\cdots\!44\)\( p^{96} T^{28} - 1130940700956992 p^{112} T^{30} + p^{128} T^{32} \) | |
| 61 | \( ( 1 - 17112284 T + 948627320682344 T^{2} - \)\(19\!\cdots\!28\)\( T^{3} + \)\(48\!\cdots\!56\)\( T^{4} - \)\(92\!\cdots\!40\)\( T^{5} + \)\(17\!\cdots\!84\)\( T^{6} - \)\(25\!\cdots\!80\)\( T^{7} + \)\(40\!\cdots\!98\)\( T^{8} - \)\(25\!\cdots\!80\)\( p^{8} T^{9} + \)\(17\!\cdots\!84\)\( p^{16} T^{10} - \)\(92\!\cdots\!40\)\( p^{24} T^{11} + \)\(48\!\cdots\!56\)\( p^{32} T^{12} - \)\(19\!\cdots\!28\)\( p^{40} T^{13} + 948627320682344 p^{48} T^{14} - 17112284 p^{56} T^{15} + p^{64} T^{16} )^{2} \) | |
| 67 | \( ( 1 - 11341548 T + 1133094675879220 T^{2} - \)\(17\!\cdots\!88\)\( T^{3} + \)\(61\!\cdots\!00\)\( T^{4} - \)\(12\!\cdots\!76\)\( T^{5} + \)\(28\!\cdots\!52\)\( T^{6} - \)\(64\!\cdots\!84\)\( T^{7} + \)\(12\!\cdots\!74\)\( T^{8} - \)\(64\!\cdots\!84\)\( p^{8} T^{9} + \)\(28\!\cdots\!52\)\( p^{16} T^{10} - \)\(12\!\cdots\!76\)\( p^{24} T^{11} + \)\(61\!\cdots\!00\)\( p^{32} T^{12} - \)\(17\!\cdots\!88\)\( p^{40} T^{13} + 1133094675879220 p^{48} T^{14} - 11341548 p^{56} T^{15} + p^{64} T^{16} )^{2} \) | |
| 71 | \( 1 - 6795154850366564 T^{2} + \)\(22\!\cdots\!00\)\( T^{4} - \)\(46\!\cdots\!56\)\( T^{6} + \)\(71\!\cdots\!56\)\( T^{8} - \)\(84\!\cdots\!60\)\( T^{10} + \)\(81\!\cdots\!44\)\( T^{12} - \)\(65\!\cdots\!12\)\( T^{14} + \)\(45\!\cdots\!86\)\( T^{16} - \)\(65\!\cdots\!12\)\( p^{16} T^{18} + \)\(81\!\cdots\!44\)\( p^{32} T^{20} - \)\(84\!\cdots\!60\)\( p^{48} T^{22} + \)\(71\!\cdots\!56\)\( p^{64} T^{24} - \)\(46\!\cdots\!56\)\( p^{80} T^{26} + \)\(22\!\cdots\!00\)\( p^{96} T^{28} - 6795154850366564 p^{112} T^{30} + p^{128} T^{32} \) | |
| 73 | \( ( 1 - 1068648 T + 1077878221991476 T^{2} + \)\(24\!\cdots\!44\)\( T^{3} + \)\(14\!\cdots\!92\)\( T^{4} + \)\(25\!\cdots\!32\)\( T^{5} + \)\(14\!\cdots\!56\)\( T^{6} + \)\(36\!\cdots\!76\)\( T^{7} + \)\(10\!\cdots\!46\)\( T^{8} + \)\(36\!\cdots\!76\)\( p^{8} T^{9} + \)\(14\!\cdots\!56\)\( p^{16} T^{10} + \)\(25\!\cdots\!32\)\( p^{24} T^{11} + \)\(14\!\cdots\!92\)\( p^{32} T^{12} + \)\(24\!\cdots\!44\)\( p^{40} T^{13} + 1077878221991476 p^{48} T^{14} - 1068648 p^{56} T^{15} + p^{64} T^{16} )^{2} \) | |
| 79 | \( ( 1 + 45122812 T + 7276093244745236 T^{2} + \)\(29\!\cdots\!08\)\( T^{3} + \)\(24\!\cdots\!20\)\( T^{4} + \)\(10\!\cdots\!36\)\( T^{5} + \)\(54\!\cdots\!48\)\( T^{6} + \)\(22\!\cdots\!12\)\( T^{7} + \)\(94\!\cdots\!30\)\( T^{8} + \)\(22\!\cdots\!12\)\( p^{8} T^{9} + \)\(54\!\cdots\!48\)\( p^{16} T^{10} + \)\(10\!\cdots\!36\)\( p^{24} T^{11} + \)\(24\!\cdots\!20\)\( p^{32} T^{12} + \)\(29\!\cdots\!08\)\( p^{40} T^{13} + 7276093244745236 p^{48} T^{14} + 45122812 p^{56} T^{15} + p^{64} T^{16} )^{2} \) | |
| 83 | \( 1 - 27150063239267216 T^{2} + \)\(35\!\cdots\!48\)\( T^{4} - \)\(30\!\cdots\!12\)\( T^{6} + \)\(18\!\cdots\!20\)\( T^{8} - \)\(89\!\cdots\!04\)\( T^{10} + \)\(33\!\cdots\!72\)\( T^{12} - \)\(10\!\cdots\!28\)\( T^{14} + \)\(25\!\cdots\!42\)\( T^{16} - \)\(10\!\cdots\!28\)\( p^{16} T^{18} + \)\(33\!\cdots\!72\)\( p^{32} T^{20} - \)\(89\!\cdots\!04\)\( p^{48} T^{22} + \)\(18\!\cdots\!20\)\( p^{64} T^{24} - \)\(30\!\cdots\!12\)\( p^{80} T^{26} + \)\(35\!\cdots\!48\)\( p^{96} T^{28} - 27150063239267216 p^{112} T^{30} + p^{128} T^{32} \) | |
| 89 | \( 1 - 43048251470487868 T^{2} + \)\(90\!\cdots\!08\)\( T^{4} - \)\(14\!\cdots\!40\)\( p T^{6} + \)\(12\!\cdots\!64\)\( T^{8} - \)\(98\!\cdots\!68\)\( T^{10} + \)\(62\!\cdots\!56\)\( T^{12} - \)\(32\!\cdots\!76\)\( T^{14} + \)\(13\!\cdots\!30\)\( T^{16} - \)\(32\!\cdots\!76\)\( p^{16} T^{18} + \)\(62\!\cdots\!56\)\( p^{32} T^{20} - \)\(98\!\cdots\!68\)\( p^{48} T^{22} + \)\(12\!\cdots\!64\)\( p^{64} T^{24} - \)\(14\!\cdots\!40\)\( p^{81} T^{26} + \)\(90\!\cdots\!08\)\( p^{96} T^{28} - 43048251470487868 p^{112} T^{30} + p^{128} T^{32} \) | |
| 97 | \( ( 1 - 67020576 T + 29700021028401412 T^{2} - \)\(27\!\cdots\!68\)\( T^{3} + \)\(52\!\cdots\!32\)\( T^{4} - \)\(46\!\cdots\!52\)\( T^{5} + \)\(66\!\cdots\!20\)\( T^{6} - \)\(50\!\cdots\!48\)\( T^{7} + \)\(61\!\cdots\!54\)\( T^{8} - \)\(50\!\cdots\!48\)\( p^{8} T^{9} + \)\(66\!\cdots\!20\)\( p^{16} T^{10} - \)\(46\!\cdots\!52\)\( p^{24} T^{11} + \)\(52\!\cdots\!32\)\( p^{32} T^{12} - \)\(27\!\cdots\!68\)\( p^{40} T^{13} + 29700021028401412 p^{48} T^{14} - 67020576 p^{56} T^{15} + p^{64} T^{16} )^{2} \) | |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{32} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−2.05237630152015289788238391085, −1.97574438450403793055656299897, −1.96875491658786882858499698085, −1.93925975751902134222226731424, −1.75207335415656024824734129879, −1.63869680012242266053083543562, −1.53725599471069034070629966466, −1.53187903809915818717476687220, −1.28299033993337289642819445271, −1.27483880802214582947497655272, −1.25597391849202029730914831216, −1.17263121574121461282975275848, −0.940079226487895690371920762823, −0.905263087362227383086597661377, −0.863278887719923707765976609656, −0.791819787607958285428916898277, −0.74489617864481587611052330940, −0.67323425946412399544890278251, −0.52029765120857519943884476616, −0.46981552953427539776727856197, −0.32248595840245259041029435080, −0.27753035086624517822821297409, −0.25768752592726922572654975847, −0.17491507130467527216832065476, −0.086539206666428555650084594174, 0.086539206666428555650084594174, 0.17491507130467527216832065476, 0.25768752592726922572654975847, 0.27753035086624517822821297409, 0.32248595840245259041029435080, 0.46981552953427539776727856197, 0.52029765120857519943884476616, 0.67323425946412399544890278251, 0.74489617864481587611052330940, 0.791819787607958285428916898277, 0.863278887719923707765976609656, 0.905263087362227383086597661377, 0.940079226487895690371920762823, 1.17263121574121461282975275848, 1.25597391849202029730914831216, 1.27483880802214582947497655272, 1.28299033993337289642819445271, 1.53187903809915818717476687220, 1.53725599471069034070629966466, 1.63869680012242266053083543562, 1.75207335415656024824734129879, 1.93925975751902134222226731424, 1.96875491658786882858499698085, 1.97574438450403793055656299897, 2.05237630152015289788238391085
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.