L(s) = 1 | + (2.82 − 2.82i)2-s − 16.0i·4-s + (−50.1 − 50.1i)7-s + (−45.2 − 45.2i)8-s + 659. i·11-s + (517. − 517. i)13-s − 283.·14-s − 256.·16-s + (−62.7 + 62.7i)17-s + 38.1i·19-s + (1.86e3 + 1.86e3i)22-s + (455. + 455. i)23-s − 2.92e3i·26-s + (−802. + 802. i)28-s + 4.59e3·29-s + ⋯ |
L(s) = 1 | + (0.499 − 0.499i)2-s − 0.500i·4-s + (−0.386 − 0.386i)7-s + (−0.250 − 0.250i)8-s + 1.64i·11-s + (0.849 − 0.849i)13-s − 0.386·14-s − 0.250·16-s + (−0.0526 + 0.0526i)17-s + 0.0242i·19-s + (0.821 + 0.821i)22-s + (0.179 + 0.179i)23-s − 0.849i·26-s + (−0.193 + 0.193i)28-s + 1.01·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.279 + 0.960i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.279 + 0.960i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(3)\) |
\(\approx\) |
\(2.674659445\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.674659445\) |
\(L(\frac{7}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-2.82 + 2.82i)T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
good | 7 | \( 1 + (50.1 + 50.1i)T + 1.68e4iT^{2} \) |
| 11 | \( 1 - 659. iT - 1.61e5T^{2} \) |
| 13 | \( 1 + (-517. + 517. i)T - 3.71e5iT^{2} \) |
| 17 | \( 1 + (62.7 - 62.7i)T - 1.41e6iT^{2} \) |
| 19 | \( 1 - 38.1iT - 2.47e6T^{2} \) |
| 23 | \( 1 + (-455. - 455. i)T + 6.43e6iT^{2} \) |
| 29 | \( 1 - 4.59e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 2.69e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + (-1.48e3 - 1.48e3i)T + 6.93e7iT^{2} \) |
| 41 | \( 1 - 194. iT - 1.15e8T^{2} \) |
| 43 | \( 1 + (-9.86e3 + 9.86e3i)T - 1.47e8iT^{2} \) |
| 47 | \( 1 + (-5.30e3 + 5.30e3i)T - 2.29e8iT^{2} \) |
| 53 | \( 1 + (2.05e4 + 2.05e4i)T + 4.18e8iT^{2} \) |
| 59 | \( 1 - 1.86e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 4.48e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + (-2.57e4 - 2.57e4i)T + 1.35e9iT^{2} \) |
| 71 | \( 1 + 6.14e4iT - 1.80e9T^{2} \) |
| 73 | \( 1 + (-2.72e4 + 2.72e4i)T - 2.07e9iT^{2} \) |
| 79 | \( 1 - 7.33e4iT - 3.07e9T^{2} \) |
| 83 | \( 1 + (-5.89e4 - 5.89e4i)T + 3.93e9iT^{2} \) |
| 89 | \( 1 - 1.12e5T + 5.58e9T^{2} \) |
| 97 | \( 1 + (8.79e4 + 8.79e4i)T + 8.58e9iT^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.20572845440993160524322430516, −9.517026707739922954690390139361, −8.296955944261758218787500655356, −7.19326643360938552544461002082, −6.31367309481035262983326928362, −5.13947778298245463393968239905, −4.20249942803887813348647675264, −3.17051354536169264977282404248, −1.94078259036216266452430301106, −0.67591130455786755124519104060,
0.940569567668271333858021144027, 2.69787054705816065397443405621, 3.63560203237202469638604461451, 4.76317313830161770187728699079, 6.10809982577963363487157692273, 6.31000693344072444734266300976, 7.72550466762344821554655131868, 8.659863609261679317707084070161, 9.251027470351590332372470944788, 10.68432682218208991230445618780