| L(s) = 1 | + (6.79 − 6.79i)3-s + (158. − 158. i)5-s + 213.·7-s + 636. i·9-s + (−1.30e3 − 1.30e3i)11-s + (11.6 + 11.6i)13-s − 2.15e3i·15-s + 2.83e3·17-s + (7.19e3 − 7.19e3i)19-s + (1.45e3 − 1.45e3i)21-s − 7.83e3·23-s − 3.45e4i·25-s + (9.28e3 + 9.28e3i)27-s + (−112. − 112. i)29-s − 7.87e3i·31-s + ⋯ |
| L(s) = 1 | + (0.251 − 0.251i)3-s + (1.26 − 1.26i)5-s + 0.622·7-s + 0.873i·9-s + (−0.983 − 0.983i)11-s + (0.00529 + 0.00529i)13-s − 0.638i·15-s + 0.576·17-s + (1.04 − 1.04i)19-s + (0.156 − 0.156i)21-s − 0.644·23-s − 2.21i·25-s + (0.471 + 0.471i)27-s + (−0.00459 − 0.00459i)29-s − 0.264i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 64 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.325 + 0.945i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 64 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.325 + 0.945i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(2.00184 - 1.42760i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.00184 - 1.42760i\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| good | 3 | \( 1 + (-6.79 + 6.79i)T - 729iT^{2} \) |
| 5 | \( 1 + (-158. + 158. i)T - 1.56e4iT^{2} \) |
| 7 | \( 1 - 213.T + 1.17e5T^{2} \) |
| 11 | \( 1 + (1.30e3 + 1.30e3i)T + 1.77e6iT^{2} \) |
| 13 | \( 1 + (-11.6 - 11.6i)T + 4.82e6iT^{2} \) |
| 17 | \( 1 - 2.83e3T + 2.41e7T^{2} \) |
| 19 | \( 1 + (-7.19e3 + 7.19e3i)T - 4.70e7iT^{2} \) |
| 23 | \( 1 + 7.83e3T + 1.48e8T^{2} \) |
| 29 | \( 1 + (112. + 112. i)T + 5.94e8iT^{2} \) |
| 31 | \( 1 + 7.87e3iT - 8.87e8T^{2} \) |
| 37 | \( 1 + (-2.27e4 + 2.27e4i)T - 2.56e9iT^{2} \) |
| 41 | \( 1 + 2.02e4iT - 4.75e9T^{2} \) |
| 43 | \( 1 + (2.96e4 + 2.96e4i)T + 6.32e9iT^{2} \) |
| 47 | \( 1 - 7.96e4iT - 1.07e10T^{2} \) |
| 53 | \( 1 + (1.89e5 - 1.89e5i)T - 2.21e10iT^{2} \) |
| 59 | \( 1 + (-2.51e5 - 2.51e5i)T + 4.21e10iT^{2} \) |
| 61 | \( 1 + (-1.97e5 - 1.97e5i)T + 5.15e10iT^{2} \) |
| 67 | \( 1 + (-2.14e5 + 2.14e5i)T - 9.04e10iT^{2} \) |
| 71 | \( 1 - 2.75e5T + 1.28e11T^{2} \) |
| 73 | \( 1 + 1.06e5iT - 1.51e11T^{2} \) |
| 79 | \( 1 - 6.37e5iT - 2.43e11T^{2} \) |
| 83 | \( 1 + (4.67e5 - 4.67e5i)T - 3.26e11iT^{2} \) |
| 89 | \( 1 - 2.40e5iT - 4.96e11T^{2} \) |
| 97 | \( 1 - 8.69e5T + 8.32e11T^{2} \) |
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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
−13.57643876744580837681510019944, −12.68681863548227844220923797711, −11.16756543612732159858260897889, −9.898791333262845358949059330880, −8.698272390593372133862707535977, −7.78060060104057624400221938800, −5.68937418090705728823981897724, −4.95267833375881569326027502506, −2.43658541623012784684605320336, −1.04417290886186892095826442916,
1.87732562191052933570215880138, 3.27417496911859018970768545596, 5.31303890049982615996333524452, 6.59091785980906752972077851768, 7.900328455755979824906893031735, 9.806328315401509214158611771845, 10.07547771252543739518103595992, 11.54734638059059156370891375777, 12.93999581132022406532030562257, 14.33386571539281965578520557777
