| L(s) = 1 | + (0.958 + 0.958i)3-s + (−1.04 − 24.9i)5-s + (−26.0 + 26.0i)7-s − 79.1i·9-s − 75.9·11-s + (−99.1 − 99.1i)13-s + (22.9 − 24.9i)15-s + (77.9 − 77.9i)17-s − 640. i·19-s − 49.9·21-s + (−117. − 117. i)23-s + (−622. + 52.0i)25-s + (153. − 153. i)27-s + 752. i·29-s − 708.·31-s + ⋯ |
| L(s) = 1 | + (0.106 + 0.106i)3-s + (−0.0416 − 0.999i)5-s + (−0.531 + 0.531i)7-s − 0.977i·9-s − 0.627·11-s + (−0.586 − 0.586i)13-s + (0.101 − 0.110i)15-s + (0.269 − 0.269i)17-s − 1.77i·19-s − 0.113·21-s + (−0.221 − 0.221i)23-s + (−0.996 + 0.0832i)25-s + (0.210 − 0.210i)27-s + 0.894i·29-s − 0.736·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.489 + 0.871i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.489 + 0.871i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(0.527708 - 0.901789i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.527708 - 0.901789i\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + (1.04 + 24.9i)T \) |
| good | 3 | \( 1 + (-0.958 - 0.958i)T + 81iT^{2} \) |
| 7 | \( 1 + (26.0 - 26.0i)T - 2.40e3iT^{2} \) |
| 11 | \( 1 + 75.9T + 1.46e4T^{2} \) |
| 13 | \( 1 + (99.1 + 99.1i)T + 2.85e4iT^{2} \) |
| 17 | \( 1 + (-77.9 + 77.9i)T - 8.35e4iT^{2} \) |
| 19 | \( 1 + 640. iT - 1.30e5T^{2} \) |
| 23 | \( 1 + (117. + 117. i)T + 2.79e5iT^{2} \) |
| 29 | \( 1 - 752. iT - 7.07e5T^{2} \) |
| 31 | \( 1 + 708.T + 9.23e5T^{2} \) |
| 37 | \( 1 + (-1.17e3 + 1.17e3i)T - 1.87e6iT^{2} \) |
| 41 | \( 1 - 1.82e3T + 2.82e6T^{2} \) |
| 43 | \( 1 + (-2.37e3 - 2.37e3i)T + 3.41e6iT^{2} \) |
| 47 | \( 1 + (2.45e3 - 2.45e3i)T - 4.87e6iT^{2} \) |
| 53 | \( 1 + (-2.11e3 - 2.11e3i)T + 7.89e6iT^{2} \) |
| 59 | \( 1 + 5.25e3iT - 1.21e7T^{2} \) |
| 61 | \( 1 + 1.31e3T + 1.38e7T^{2} \) |
| 67 | \( 1 + (-5.56e3 + 5.56e3i)T - 2.01e7iT^{2} \) |
| 71 | \( 1 + 7.55e3T + 2.54e7T^{2} \) |
| 73 | \( 1 + (-1.20e3 - 1.20e3i)T + 2.83e7iT^{2} \) |
| 79 | \( 1 + 3.39e3iT - 3.89e7T^{2} \) |
| 83 | \( 1 + (928. + 928. i)T + 4.74e7iT^{2} \) |
| 89 | \( 1 + 1.19e4iT - 6.27e7T^{2} \) |
| 97 | \( 1 + (-4.68e3 + 4.68e3i)T - 8.85e7iT^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−12.84887932471180872111723350584, −12.59481689992552011854242439442, −11.18722451514396397444311028041, −9.574828850261999111891192215249, −9.006660288490275853507381135416, −7.56490303886254670391438860643, −5.95810166984472806623665259098, −4.69039238448878153584298311987, −2.87798910150210049080076702484, −0.49920449577049735180187990288,
2.28290608373619159146096367311, 3.88396801592360327910131518632, 5.75572365685735759473980997616, 7.16989957992045042142017976932, 8.008755714615810080742866102064, 9.913741254974132819580436701332, 10.50999703343274851843753675985, 11.76664618023962405459371646664, 13.10308801944172424054691363105, 14.02220187409144363120295706550
