| L(s) = 1 | + 5.21i·2-s + 100.·4-s − 125·5-s + (−839. − 344. i)7-s + 1.19e3i·8-s − 651. i·10-s + 709. i·11-s + 6.69e3i·13-s + (1.79e3 − 4.37e3i)14-s + 6.69e3·16-s + 2.98e4·17-s − 1.33e4i·19-s − 1.26e4·20-s − 3.69e3·22-s − 1.00e5i·23-s + ⋯ |
| L(s) = 1 | + 0.460i·2-s + 0.787·4-s − 0.447·5-s + (−0.924 − 0.380i)7-s + 0.823i·8-s − 0.205i·10-s + 0.160i·11-s + 0.845i·13-s + (0.175 − 0.426i)14-s + 0.408·16-s + 1.47·17-s − 0.447i·19-s − 0.352·20-s − 0.0740·22-s − 1.72i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.223 - 0.974i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.223 - 0.974i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(1.910587004\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.910587004\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 + 125T \) |
| 7 | \( 1 + (839. + 344. i)T \) |
| good | 2 | \( 1 - 5.21iT - 128T^{2} \) |
| 11 | \( 1 - 709. iT - 1.94e7T^{2} \) |
| 13 | \( 1 - 6.69e3iT - 6.27e7T^{2} \) |
| 17 | \( 1 - 2.98e4T + 4.10e8T^{2} \) |
| 19 | \( 1 + 1.33e4iT - 8.93e8T^{2} \) |
| 23 | \( 1 + 1.00e5iT - 3.40e9T^{2} \) |
| 29 | \( 1 + 8.13e4iT - 1.72e10T^{2} \) |
| 31 | \( 1 - 1.54e5iT - 2.75e10T^{2} \) |
| 37 | \( 1 - 3.58e5T + 9.49e10T^{2} \) |
| 41 | \( 1 + 6.13e5T + 1.94e11T^{2} \) |
| 43 | \( 1 + 2.21e5T + 2.71e11T^{2} \) |
| 47 | \( 1 + 8.79e5T + 5.06e11T^{2} \) |
| 53 | \( 1 - 8.68e5iT - 1.17e12T^{2} \) |
| 59 | \( 1 - 1.46e6T + 2.48e12T^{2} \) |
| 61 | \( 1 - 2.39e6iT - 3.14e12T^{2} \) |
| 67 | \( 1 + 3.20e6T + 6.06e12T^{2} \) |
| 71 | \( 1 - 7.36e5iT - 9.09e12T^{2} \) |
| 73 | \( 1 - 2.17e6iT - 1.10e13T^{2} \) |
| 79 | \( 1 - 5.94e6T + 1.92e13T^{2} \) |
| 83 | \( 1 - 2.91e6T + 2.71e13T^{2} \) |
| 89 | \( 1 - 5.40e6T + 4.42e13T^{2} \) |
| 97 | \( 1 - 1.23e7iT - 8.07e13T^{2} \) |
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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
−10.63942977797510749723489275114, −9.902205039097930891964954655389, −8.661191605578569569255177921423, −7.66360581698842205094471667328, −6.81770688674856731184559322337, −6.17647233593343169899152756098, −4.81823248886773054178771571046, −3.52462572270330187174795544522, −2.49837322970957238749908196689, −0.995783576471691844829670405936,
0.46059605737595590996431895524, 1.68676069099846775375525173799, 3.18493582859593111685891718353, 3.47517066849021020578936373289, 5.39625347785361428227178430994, 6.23868157109708570098221559011, 7.37313409558386033852103777145, 8.125547200171575415948665319032, 9.634861103968051882602488326768, 10.10338488000573422295412270731
