Dirichlet series
| L(s) = 1 | − 3·2-s − 2·3-s + 3·4-s − 5·5-s + 6·6-s − 4·7-s − 3·8-s + 2·9-s + 15·10-s − 3·11-s − 6·12-s − 7·13-s + 12·14-s + 10·15-s + 9·16-s − 5·17-s − 6·18-s + 19·19-s − 15·20-s + 8·21-s + 9·22-s + 32·23-s + 6·24-s + 26·25-s + 21·26-s + 4·27-s − 12·28-s + ⋯ |
| L(s) = 1 | − 2.12·2-s − 1.15·3-s + 3/2·4-s − 2.23·5-s + 2.44·6-s − 1.51·7-s − 1.06·8-s + 2/3·9-s + 4.74·10-s − 0.904·11-s − 1.73·12-s − 1.94·13-s + 3.20·14-s + 2.58·15-s + 9/4·16-s − 1.21·17-s − 1.41·18-s + 4.35·19-s − 3.35·20-s + 1.74·21-s + 1.91·22-s + 6.67·23-s + 1.22·24-s + 26/5·25-s + 4.11·26-s + 0.769·27-s − 2.26·28-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{16} \cdot 11^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{16} \cdot 11^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(32\) |
| Conductor: | \(7^{16} \cdot 11^{16}\) |
| Sign: | $1$ |
| Analytic conductor: | \(0.000417143\) |
| Root analytic conductor: | \(0.784122\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((32,\ 7^{16} \cdot 11^{16} ,\ ( \ : [1/2]^{16} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(0.02931180404\) |
| \(L(\frac12)\) | \(\approx\) | \(0.02931180404\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 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 | \( 1 + 3 T + 3 p T^{2} + 3 p^{2} T^{3} + 9 p T^{4} + 17 T^{5} + 15 T^{6} - T^{7} - 29 p T^{8} - 119 T^{9} - 185 T^{10} - 73 p^{2} T^{11} - 121 p T^{12} + 33 T^{13} + 39 p^{3} T^{14} + 439 p T^{15} + 1785 T^{16} + 439 p^{2} T^{17} + 39 p^{5} T^{18} + 33 p^{3} T^{19} - 121 p^{5} T^{20} - 73 p^{7} T^{21} - 185 p^{6} T^{22} - 119 p^{7} T^{23} - 29 p^{9} T^{24} - p^{9} T^{25} + 15 p^{10} T^{26} + 17 p^{11} T^{27} + 9 p^{13} T^{28} + 3 p^{15} T^{29} + 3 p^{15} T^{30} + 3 p^{15} T^{31} + p^{16} T^{32} \) |
| 3 | \( 1 + 2 T + 2 T^{2} - 4 T^{3} - 8 T^{4} + 4 T^{5} + 14 p T^{6} + 104 T^{7} + 161 T^{8} + 8 T^{9} - 278 T^{10} - 122 T^{11} + 544 p T^{12} + 148 p^{3} T^{13} + 4420 T^{14} - 908 T^{15} - 3527 T^{16} - 908 p T^{17} + 4420 p^{2} T^{18} + 148 p^{6} T^{19} + 544 p^{5} T^{20} - 122 p^{5} T^{21} - 278 p^{6} T^{22} + 8 p^{7} T^{23} + 161 p^{8} T^{24} + 104 p^{9} T^{25} + 14 p^{11} T^{26} + 4 p^{11} T^{27} - 8 p^{12} T^{28} - 4 p^{13} T^{29} + 2 p^{14} T^{30} + 2 p^{15} T^{31} + p^{16} T^{32} \) | |
| 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
−4.76547158668519222644871402436, −4.68509719340527348716189789709, −4.65188983933093131117510533483, −4.39491536689549981562605544598, −4.21801943229774669434406899361, −4.11910068654873997734361548081, −4.09718573179734231953423801607, −3.61528827076840361165939371500, −3.52475014859992153315838817505, −3.49036956941392011556591088240, −3.45841311397960636141667034005, −3.37342464068301906088963377283, −3.18143608442819947579568218588, −3.13333730230601008321607943850, −2.96459395047780766182849500593, −2.86717031095119522655279819678, −2.81582577645182690231715345674, −2.78399977587664358326351861803, −2.59430838486175112491784265723, −2.39789737135663244476015465875, −1.94291250384609553130849150420, −1.46380038616775393221737670558, −1.38230635777839221939653233589, −1.01474788977872755794881375848, −0.75956679774041450742393153840, 0.75956679774041450742393153840, 1.01474788977872755794881375848, 1.38230635777839221939653233589, 1.46380038616775393221737670558, 1.94291250384609553130849150420, 2.39789737135663244476015465875, 2.59430838486175112491784265723, 2.78399977587664358326351861803, 2.81582577645182690231715345674, 2.86717031095119522655279819678, 2.96459395047780766182849500593, 3.13333730230601008321607943850, 3.18143608442819947579568218588, 3.37342464068301906088963377283, 3.45841311397960636141667034005, 3.49036956941392011556591088240, 3.52475014859992153315838817505, 3.61528827076840361165939371500, 4.09718573179734231953423801607, 4.11910068654873997734361548081, 4.21801943229774669434406899361, 4.39491536689549981562605544598, 4.65188983933093131117510533483, 4.68509719340527348716189789709, 4.76547158668519222644871402436
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.