Dirichlet series
| L(s) = 1 | + 3·2-s + 3·4-s + 5·5-s − 4·7-s + 3·8-s + 15·10-s + 3·11-s − 7·13-s − 12·14-s + 9·16-s + 5·17-s + 19·19-s + 15·20-s + 9·22-s − 32·23-s + 26·25-s − 21·26-s − 12·28-s − 3·29-s − 7·31-s + 8·32-s + 15·34-s − 20·35-s + 4·37-s + 57·38-s + 15·40-s + 10·41-s + ⋯ |
| L(s) = 1 | + 2.12·2-s + 3/2·4-s + 2.23·5-s − 1.51·7-s + 1.06·8-s + 4.74·10-s + 0.904·11-s − 1.94·13-s − 3.20·14-s + 9/4·16-s + 1.21·17-s + 4.35·19-s + 3.35·20-s + 1.91·22-s − 6.67·23-s + 26/5·25-s − 4.11·26-s − 2.26·28-s − 0.557·29-s − 1.25·31-s + 1.41·32-s + 2.57·34-s − 3.38·35-s + 0.657·37-s + 9.24·38-s + 2.37·40-s + 1.56·41-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{32} \cdot 7^{16} \cdot 11^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{32} \cdot 7^{16} \cdot 11^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(32\) |
| Conductor: | \(3^{32} \cdot 7^{16} \cdot 11^{16}\) |
| Sign: | $1$ |
| Analytic conductor: | \(7.72975\times 10^{11}\) |
| Root analytic conductor: | \(2.35236\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((32,\ 3^{32} \cdot 7^{16} \cdot 11^{16} ,\ ( \ : [1/2]^{16} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(1.947124592\) |
| \(L(\frac12)\) | \(\approx\) | \(1.947124592\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 3 | \( 1 \) |
| 7 | \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{4} \) | |
| 11 | \( 1 - 3 T + 19 T^{2} + 76 T^{3} - 26 p T^{4} + 2245 T^{5} + 1052 T^{6} - 7668 T^{7} + 114073 T^{8} - 7668 p T^{9} + 1052 p^{2} T^{10} + 2245 p^{3} T^{11} - 26 p^{5} T^{12} + 76 p^{5} T^{13} + 19 p^{6} T^{14} - 3 p^{7} T^{15} + p^{8} T^{16} \) | |
| good | 2 | \( 1 - 3 T + 3 p T^{2} - 3 p^{2} T^{3} + 9 p T^{4} - 17 T^{5} + 15 T^{6} + T^{7} - 29 p T^{8} + 119 T^{9} - 185 T^{10} + 73 p^{2} T^{11} - 121 p T^{12} - 33 T^{13} + 39 p^{3} T^{14} - 439 p T^{15} + 1785 T^{16} - 439 p^{2} T^{17} + 39 p^{5} T^{18} - 33 p^{3} T^{19} - 121 p^{5} T^{20} + 73 p^{7} T^{21} - 185 p^{6} T^{22} + 119 p^{7} T^{23} - 29 p^{9} T^{24} + p^{9} T^{25} + 15 p^{10} T^{26} - 17 p^{11} T^{27} + 9 p^{13} T^{28} - 3 p^{15} T^{29} + 3 p^{15} T^{30} - 3 p^{15} T^{31} + p^{16} T^{32} \) |
| 5 | \( 1 - p T - T^{2} + 66 T^{3} - 156 T^{4} - 267 T^{5} + 1789 T^{6} - 1241 T^{7} - 10249 T^{8} + 25094 T^{9} + 25181 T^{10} - 188691 T^{11} + 137316 T^{12} + 851484 T^{13} - 1968914 T^{14} - 1709452 T^{15} + 12417791 T^{16} - 1709452 p T^{17} - 1968914 p^{2} T^{18} + 851484 p^{3} T^{19} + 137316 p^{4} T^{20} - 188691 p^{5} T^{21} + 25181 p^{6} T^{22} + 25094 p^{7} T^{23} - 10249 p^{8} T^{24} - 1241 p^{9} T^{25} + 1789 p^{10} T^{26} - 267 p^{11} T^{27} - 156 p^{12} T^{28} + 66 p^{13} T^{29} - p^{14} T^{30} - p^{16} T^{31} + p^{16} T^{32} \) | |
| 13 | \( 1 + 7 T - p T^{2} - 114 T^{3} + 302 T^{4} + 619 T^{5} - 3203 T^{6} + 11659 T^{7} - 57039 T^{8} - 415732 T^{9} + 983197 T^{10} + 802283 T^{11} - 7060468 T^{12} + 40515806 T^{13} - 160505050 T^{14} - 179149458 T^{15} + 5116354083 T^{16} - 179149458 p T^{17} - 160505050 p^{2} T^{18} + 40515806 p^{3} T^{19} - 7060468 p^{4} T^{20} + 802283 p^{5} T^{21} + 983197 p^{6} T^{22} - 415732 p^{7} T^{23} - 57039 p^{8} T^{24} + 11659 p^{9} T^{25} - 3203 p^{10} T^{26} + 619 p^{11} T^{27} + 302 p^{12} T^{28} - 114 p^{13} T^{29} - p^{15} T^{30} + 7 p^{15} T^{31} + p^{16} T^{32} \) | |
| 17 | \( 1 - 5 T + 13 T^{2} - 163 T^{3} + 1572 T^{4} - 7139 T^{5} + 24321 T^{6} - 178695 T^{7} + 1262499 T^{8} - 5104293 T^{9} + 17336797 T^{10} - 107294017 T^{11} + 637972982 T^{12} - 2309382396 T^{13} + 7576691718 T^{14} - 42485576390 T^{15} + 218631742615 T^{16} - 42485576390 p T^{17} + 7576691718 p^{2} T^{18} - 2309382396 p^{3} T^{19} + 637972982 p^{4} T^{20} - 107294017 p^{5} T^{21} + 17336797 p^{6} T^{22} - 5104293 p^{7} T^{23} + 1262499 p^{8} T^{24} - 178695 p^{9} T^{25} + 24321 p^{10} T^{26} - 7139 p^{11} T^{27} + 1572 p^{12} T^{28} - 163 p^{13} T^{29} + 13 p^{14} T^{30} - 5 p^{15} T^{31} + p^{16} T^{32} \) | |
| 19 | \( 1 - p T + 113 T^{2} + 15 T^{3} - 2596 T^{4} + 2631 T^{5} + 106245 T^{6} - 756527 T^{7} + 1484429 T^{8} + 6206619 T^{9} - 1380795 p T^{10} - 133461535 T^{11} + 1444663142 T^{12} - 5677361136 T^{13} + 11109732198 T^{14} - 13150474478 T^{15} + 38443302841 T^{16} - 13150474478 p T^{17} + 11109732198 p^{2} T^{18} - 5677361136 p^{3} T^{19} + 1444663142 p^{4} T^{20} - 133461535 p^{5} T^{21} - 1380795 p^{7} T^{22} + 6206619 p^{7} T^{23} + 1484429 p^{8} T^{24} - 756527 p^{9} T^{25} + 106245 p^{10} T^{26} + 2631 p^{11} T^{27} - 2596 p^{12} T^{28} + 15 p^{13} T^{29} + 113 p^{14} T^{30} - p^{16} T^{31} + p^{16} T^{32} \) | |
| 23 | \( ( 1 + 16 T + 224 T^{2} + 2234 T^{3} + 19567 T^{4} + 141712 T^{5} + 40290 p T^{6} + 5230592 T^{7} + 26816927 T^{8} + 5230592 p T^{9} + 40290 p^{3} T^{10} + 141712 p^{3} T^{11} + 19567 p^{4} T^{12} + 2234 p^{5} T^{13} + 224 p^{6} T^{14} + 16 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 29 | \( 1 + 3 T + 75 T^{3} + 2441 T^{4} + 473 T^{5} - 37550 T^{6} + 114715 T^{7} + 3177286 T^{8} - 2623287 T^{9} - 24972736 T^{10} + 263064707 T^{11} + 3471423716 T^{12} - 694532172 T^{13} - 11302302466 T^{14} + 87801320252 T^{15} + 2485262854281 T^{16} + 87801320252 p T^{17} - 11302302466 p^{2} T^{18} - 694532172 p^{3} T^{19} + 3471423716 p^{4} T^{20} + 263064707 p^{5} T^{21} - 24972736 p^{6} T^{22} - 2623287 p^{7} T^{23} + 3177286 p^{8} T^{24} + 114715 p^{9} T^{25} - 37550 p^{10} T^{26} + 473 p^{11} T^{27} + 2441 p^{12} T^{28} + 75 p^{13} T^{29} + 3 p^{15} T^{31} + p^{16} T^{32} \) | |
| 31 | \( 1 + 7 T + 13 T^{3} + 77 T^{4} - 8801 T^{5} - 22254 T^{6} + 56119 T^{7} - 947868 T^{8} + 3202093 T^{9} + 57615842 T^{10} - 102279009 T^{11} - 159564504 T^{12} + 5764151836 T^{13} - 36979480986 T^{14} - 181717554252 T^{15} + 841105835367 T^{16} - 181717554252 p T^{17} - 36979480986 p^{2} T^{18} + 5764151836 p^{3} T^{19} - 159564504 p^{4} T^{20} - 102279009 p^{5} T^{21} + 57615842 p^{6} T^{22} + 3202093 p^{7} T^{23} - 947868 p^{8} T^{24} + 56119 p^{9} T^{25} - 22254 p^{10} T^{26} - 8801 p^{11} T^{27} + 77 p^{12} T^{28} + 13 p^{13} T^{29} + 7 p^{15} T^{31} + p^{16} T^{32} \) | |
| 37 | \( 1 - 4 T - 64 T^{2} + 392 T^{3} + 2308 T^{4} - 10820 T^{5} - 47691 T^{6} - 31178 T^{7} + 2748163 T^{8} + 12045860 T^{9} - 124307808 T^{10} - 75116270 T^{11} + 4767089611 T^{12} - 456957494 T^{13} - 122520873497 T^{14} + 217199596514 T^{15} + 4709853143282 T^{16} + 217199596514 p T^{17} - 122520873497 p^{2} T^{18} - 456957494 p^{3} T^{19} + 4767089611 p^{4} T^{20} - 75116270 p^{5} T^{21} - 124307808 p^{6} T^{22} + 12045860 p^{7} T^{23} + 2748163 p^{8} T^{24} - 31178 p^{9} T^{25} - 47691 p^{10} T^{26} - 10820 p^{11} T^{27} + 2308 p^{12} T^{28} + 392 p^{13} T^{29} - 64 p^{14} T^{30} - 4 p^{15} T^{31} + p^{16} T^{32} \) | |
| 41 | \( 1 - 10 T + 32 T^{2} - 192 T^{3} + 5438 T^{4} - 48920 T^{5} + 140400 T^{6} - 769008 T^{7} + 14682039 T^{8} - 96831496 T^{9} + 182717296 T^{10} - 1985070190 T^{11} + 32206741548 T^{12} - 131499578440 T^{13} + 27334729480 T^{14} - 4391234846920 T^{15} + 62202943867805 T^{16} - 4391234846920 p T^{17} + 27334729480 p^{2} T^{18} - 131499578440 p^{3} T^{19} + 32206741548 p^{4} T^{20} - 1985070190 p^{5} T^{21} + 182717296 p^{6} T^{22} - 96831496 p^{7} T^{23} + 14682039 p^{8} T^{24} - 769008 p^{9} T^{25} + 140400 p^{10} T^{26} - 48920 p^{11} T^{27} + 5438 p^{12} T^{28} - 192 p^{13} T^{29} + 32 p^{14} T^{30} - 10 p^{15} T^{31} + p^{16} T^{32} \) | |
| 43 | \( ( 1 + 4 T + 239 T^{2} + 936 T^{3} + 27462 T^{4} + 98508 T^{5} + 2007760 T^{6} + 6271668 T^{7} + 102278657 T^{8} + 6271668 p T^{9} + 2007760 p^{2} T^{10} + 98508 p^{3} T^{11} + 27462 p^{4} T^{12} + 936 p^{5} T^{13} + 239 p^{6} T^{14} + 4 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 47 | \( 1 - 23 T + 186 T^{2} - 322 T^{3} - 4127 T^{4} + 30138 T^{5} - 70920 T^{6} - 1094344 T^{7} + 18527582 T^{8} - 119631636 T^{9} + 262096090 T^{10} + 2200503287 T^{11} - 30693701772 T^{12} + 88025997822 T^{13} + 1410058928022 T^{14} - 15969608141898 T^{15} + 104939392199145 T^{16} - 15969608141898 p T^{17} + 1410058928022 p^{2} T^{18} + 88025997822 p^{3} T^{19} - 30693701772 p^{4} T^{20} + 2200503287 p^{5} T^{21} + 262096090 p^{6} T^{22} - 119631636 p^{7} T^{23} + 18527582 p^{8} T^{24} - 1094344 p^{9} T^{25} - 70920 p^{10} T^{26} + 30138 p^{11} T^{27} - 4127 p^{12} T^{28} - 322 p^{13} T^{29} + 186 p^{14} T^{30} - 23 p^{15} T^{31} + p^{16} T^{32} \) | |
| 53 | \( 1 + 4 T - 22 T^{2} + 440 T^{3} + 660 T^{4} - 49868 T^{5} + 140113 T^{6} + 2285596 T^{7} - 15873165 T^{8} + 76638860 T^{9} + 1323931162 T^{10} - 10376469922 T^{11} - 35381234389 T^{12} + 446229246850 T^{13} - 3132652820265 T^{14} - 9035271835264 T^{15} + 292112805840914 T^{16} - 9035271835264 p T^{17} - 3132652820265 p^{2} T^{18} + 446229246850 p^{3} T^{19} - 35381234389 p^{4} T^{20} - 10376469922 p^{5} T^{21} + 1323931162 p^{6} T^{22} + 76638860 p^{7} T^{23} - 15873165 p^{8} T^{24} + 2285596 p^{9} T^{25} + 140113 p^{10} T^{26} - 49868 p^{11} T^{27} + 660 p^{12} T^{28} + 440 p^{13} T^{29} - 22 p^{14} T^{30} + 4 p^{15} T^{31} + p^{16} T^{32} \) | |
| 59 | \( 1 + 17 T - 50 T^{2} - 1985 T^{3} + 451 T^{4} + 76307 T^{5} - 426220 T^{6} + 1429235 T^{7} + 60743716 T^{8} - 309308083 T^{9} - 1984484046 T^{10} + 33011892753 T^{11} - 101353409674 T^{12} - 2740413120218 T^{13} + 8464635562844 T^{14} + 86256705224328 T^{15} - 317724345351369 T^{16} + 86256705224328 p T^{17} + 8464635562844 p^{2} T^{18} - 2740413120218 p^{3} T^{19} - 101353409674 p^{4} T^{20} + 33011892753 p^{5} T^{21} - 1984484046 p^{6} T^{22} - 309308083 p^{7} T^{23} + 60743716 p^{8} T^{24} + 1429235 p^{9} T^{25} - 426220 p^{10} T^{26} + 76307 p^{11} T^{27} + 451 p^{12} T^{28} - 1985 p^{13} T^{29} - 50 p^{14} T^{30} + 17 p^{15} T^{31} + p^{16} T^{32} \) | |
| 61 | \( 1 + 7 T - 20 T^{2} + 403 T^{3} + 10067 T^{4} + 799 p T^{5} - 116314 T^{6} + 1172749 T^{7} + 26082242 T^{8} + 24039263 T^{9} - 379016678 T^{10} - 5077729829 T^{11} - 113601259714 T^{12} - 664603591174 T^{13} + 819501759744 T^{14} - 46004665392052 T^{15} - 871290620425703 T^{16} - 46004665392052 p T^{17} + 819501759744 p^{2} T^{18} - 664603591174 p^{3} T^{19} - 113601259714 p^{4} T^{20} - 5077729829 p^{5} T^{21} - 379016678 p^{6} T^{22} + 24039263 p^{7} T^{23} + 26082242 p^{8} T^{24} + 1172749 p^{9} T^{25} - 116314 p^{10} T^{26} + 799 p^{12} T^{27} + 10067 p^{12} T^{28} + 403 p^{13} T^{29} - 20 p^{14} T^{30} + 7 p^{15} T^{31} + p^{16} T^{32} \) | |
| 67 | \( ( 1 + 19 T + 520 T^{2} + 7751 T^{3} + 119005 T^{4} + 1415171 T^{5} + 15702408 T^{6} + 150444400 T^{7} + 1308159665 T^{8} + 150444400 p T^{9} + 15702408 p^{2} T^{10} + 1415171 p^{3} T^{11} + 119005 p^{4} T^{12} + 7751 p^{5} T^{13} + 520 p^{6} T^{14} + 19 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 71 | \( 1 - 14 T + 140 T^{2} - 763 T^{3} + 147 p T^{4} - 25566 T^{5} + 32310 T^{6} + 3862236 T^{7} + 10586088 T^{8} + 330657408 T^{9} - 2081383949 T^{10} + 39603959100 T^{11} + 15217861524 T^{12} + 1721196695007 T^{13} + 1593568125167 T^{14} + 22773634065006 T^{15} + 1780879231377332 T^{16} + 22773634065006 p T^{17} + 1593568125167 p^{2} T^{18} + 1721196695007 p^{3} T^{19} + 15217861524 p^{4} T^{20} + 39603959100 p^{5} T^{21} - 2081383949 p^{6} T^{22} + 330657408 p^{7} T^{23} + 10586088 p^{8} T^{24} + 3862236 p^{9} T^{25} + 32310 p^{10} T^{26} - 25566 p^{11} T^{27} + 147 p^{13} T^{28} - 763 p^{13} T^{29} + 140 p^{14} T^{30} - 14 p^{15} T^{31} + p^{16} T^{32} \) | |
| 73 | \( 1 + 35 T + 622 T^{2} + 8881 T^{3} + 123217 T^{4} + 1506367 T^{5} + 15880754 T^{6} + 159131205 T^{7} + 1535260024 T^{8} + 13342640169 T^{9} + 108277262748 T^{10} + 880406179959 T^{11} + 6960680213462 T^{12} + 52467783616198 T^{13} + 405906809006112 T^{14} + 3284917386947140 T^{15} + 27500270697009885 T^{16} + 3284917386947140 p T^{17} + 405906809006112 p^{2} T^{18} + 52467783616198 p^{3} T^{19} + 6960680213462 p^{4} T^{20} + 880406179959 p^{5} T^{21} + 108277262748 p^{6} T^{22} + 13342640169 p^{7} T^{23} + 1535260024 p^{8} T^{24} + 159131205 p^{9} T^{25} + 15880754 p^{10} T^{26} + 1506367 p^{11} T^{27} + 123217 p^{12} T^{28} + 8881 p^{13} T^{29} + 622 p^{14} T^{30} + 35 p^{15} T^{31} + p^{16} T^{32} \) | |
| 79 | \( 1 - 15 T + 94 T^{2} - 2495 T^{3} + 38855 T^{4} - 414785 T^{5} + 5676295 T^{6} - 62322940 T^{7} + 690670255 T^{8} - 8423294280 T^{9} + 83337602737 T^{10} - 856819337855 T^{11} + 8989269459608 T^{12} - 82967566755485 T^{13} + 808287092858990 T^{14} - 7591629044633355 T^{15} + 65763672844350810 T^{16} - 7591629044633355 p T^{17} + 808287092858990 p^{2} T^{18} - 82967566755485 p^{3} T^{19} + 8989269459608 p^{4} T^{20} - 856819337855 p^{5} T^{21} + 83337602737 p^{6} T^{22} - 8423294280 p^{7} T^{23} + 690670255 p^{8} T^{24} - 62322940 p^{9} T^{25} + 5676295 p^{10} T^{26} - 414785 p^{11} T^{27} + 38855 p^{12} T^{28} - 2495 p^{13} T^{29} + 94 p^{14} T^{30} - 15 p^{15} T^{31} + p^{16} T^{32} \) | |
| 83 | \( 1 + 5 T - 318 T^{2} - 3741 T^{3} + 48517 T^{4} + 994133 T^{5} - 3255786 T^{6} - 170309105 T^{7} - 263285806 T^{8} + 21750909411 T^{9} + 1436748826 p T^{10} - 2097202277489 T^{11} - 20999477370338 T^{12} + 141328121959992 T^{13} + 2545394345691772 T^{14} - 4513257769614610 T^{15} - 237031046695965905 T^{16} - 4513257769614610 p T^{17} + 2545394345691772 p^{2} T^{18} + 141328121959992 p^{3} T^{19} - 20999477370338 p^{4} T^{20} - 2097202277489 p^{5} T^{21} + 1436748826 p^{7} T^{22} + 21750909411 p^{7} T^{23} - 263285806 p^{8} T^{24} - 170309105 p^{9} T^{25} - 3255786 p^{10} T^{26} + 994133 p^{11} T^{27} + 48517 p^{12} T^{28} - 3741 p^{13} T^{29} - 318 p^{14} T^{30} + 5 p^{15} T^{31} + p^{16} T^{32} \) | |
| 89 | \( ( 1 + 37 T + 1032 T^{2} + 20901 T^{3} + 357823 T^{4} + 5152639 T^{5} + 737152 p T^{6} + 731972999 T^{7} + 7339853952 T^{8} + 731972999 p T^{9} + 737152 p^{3} T^{10} + 5152639 p^{3} T^{11} + 357823 p^{4} T^{12} + 20901 p^{5} T^{13} + 1032 p^{6} T^{14} + 37 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 97 | \( 1 - 20 T + 123 T^{2} + 28 T^{3} + 4997 T^{4} - 239296 T^{5} + 2683386 T^{6} - 13918670 T^{7} + 12108314 T^{8} + 81926108 T^{9} + 9118270907 T^{10} - 123570735168 T^{11} + 816960697702 T^{12} + 3552108922056 T^{13} - 86512314985382 T^{14} - 43450963654390 T^{15} + 5424229096525225 T^{16} - 43450963654390 p T^{17} - 86512314985382 p^{2} T^{18} + 3552108922056 p^{3} T^{19} + 816960697702 p^{4} T^{20} - 123570735168 p^{5} T^{21} + 9118270907 p^{6} T^{22} + 81926108 p^{7} T^{23} + 12108314 p^{8} T^{24} - 13918670 p^{9} T^{25} + 2683386 p^{10} T^{26} - 239296 p^{11} T^{27} + 4997 p^{12} T^{28} + 28 p^{13} T^{29} + 123 p^{14} T^{30} - 20 p^{15} T^{31} + p^{16} T^{32} \) | |
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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
−2.84046279342465533992178435930, −2.75860037067622386924619303465, −2.75854737808194058897202836827, −2.72576157625927281509584688199, −2.71559041963088538153763844214, −2.52846524867477749231537838947, −2.26063302256924063709212464886, −2.11122368624654462820966833509, −2.04155489971544831492316267957, −2.00736597968515148287647716497, −1.92141272011380821852846426029, −1.91159391296781962544641443635, −1.80662876067279089010283857602, −1.68462006511209760999439951442, −1.49334885507516025453478534445, −1.43579519062344877744091210025, −1.32111391590392936561389170428, −1.31769905208989281753633268975, −1.27167923188188725928020150029, −1.17008457610744164994574902648, −0.907668805281116667998653514345, −0.78667343189598338377842745382, −0.38839322476315926135571284020, −0.35271257959197038152938814155, −0.06211053370300269099319149531, 0.06211053370300269099319149531, 0.35271257959197038152938814155, 0.38839322476315926135571284020, 0.78667343189598338377842745382, 0.907668805281116667998653514345, 1.17008457610744164994574902648, 1.27167923188188725928020150029, 1.31769905208989281753633268975, 1.32111391590392936561389170428, 1.43579519062344877744091210025, 1.49334885507516025453478534445, 1.68462006511209760999439951442, 1.80662876067279089010283857602, 1.91159391296781962544641443635, 1.92141272011380821852846426029, 2.00736597968515148287647716497, 2.04155489971544831492316267957, 2.11122368624654462820966833509, 2.26063302256924063709212464886, 2.52846524867477749231537838947, 2.71559041963088538153763844214, 2.72576157625927281509584688199, 2.75854737808194058897202836827, 2.75860037067622386924619303465, 2.84046279342465533992178435930
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.