| L(s) = 1 | + 3-s − 3.23·5-s − 1.61·7-s + 9-s − 2·11-s − 1.23·13-s − 3.23·15-s + 4.47·17-s + 6.47·19-s − 1.61·21-s + 8.94·23-s + 5.47·25-s + 27-s − 2.61·29-s − 10.6·31-s − 2·33-s + 5.23·35-s − 8·37-s − 1.23·39-s − 1.09·41-s + 8.94·43-s − 3.23·45-s − 8.61·47-s − 4.38·49-s + 4.47·51-s − 3.61·53-s + 6.47·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 1.44·5-s − 0.611·7-s + 0.333·9-s − 0.603·11-s − 0.342·13-s − 0.835·15-s + 1.08·17-s + 1.48·19-s − 0.353·21-s + 1.86·23-s + 1.09·25-s + 0.192·27-s − 0.486·29-s − 1.90·31-s − 0.348·33-s + 0.885·35-s − 1.31·37-s − 0.197·39-s − 0.170·41-s + 1.36·43-s − 0.482·45-s − 1.25·47-s − 0.625·49-s + 0.626·51-s − 0.496·53-s + 0.872·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5052 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5052 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 421 | \( 1 + T \) |
| good | 5 | \( 1 + 3.23T + 5T^{2} \) |
| 7 | \( 1 + 1.61T + 7T^{2} \) |
| 11 | \( 1 + 2T + 11T^{2} \) |
| 13 | \( 1 + 1.23T + 13T^{2} \) |
| 17 | \( 1 - 4.47T + 17T^{2} \) |
| 19 | \( 1 - 6.47T + 19T^{2} \) |
| 23 | \( 1 - 8.94T + 23T^{2} \) |
| 29 | \( 1 + 2.61T + 29T^{2} \) |
| 31 | \( 1 + 10.6T + 31T^{2} \) |
| 37 | \( 1 + 8T + 37T^{2} \) |
| 41 | \( 1 + 1.09T + 41T^{2} \) |
| 43 | \( 1 - 8.94T + 43T^{2} \) |
| 47 | \( 1 + 8.61T + 47T^{2} \) |
| 53 | \( 1 + 3.61T + 53T^{2} \) |
| 59 | \( 1 - 4.61T + 59T^{2} \) |
| 61 | \( 1 - 7.23T + 61T^{2} \) |
| 67 | \( 1 + 3.90T + 67T^{2} \) |
| 71 | \( 1 - 7.61T + 71T^{2} \) |
| 73 | \( 1 - 9.23T + 73T^{2} \) |
| 79 | \( 1 + 5.09T + 79T^{2} \) |
| 83 | \( 1 - 1.14T + 83T^{2} \) |
| 89 | \( 1 - 2T + 89T^{2} \) |
| 97 | \( 1 + 19.0T + 97T^{2} \) |
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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
−7.74352449392757349682853717269, −7.34602084665853768460558291203, −6.81049868604162186826559318459, −5.39629899229291060663021182825, −5.04883518287873073938759349116, −3.75632177918213904129192165190, −3.42842569397933790886637532429, −2.72338192403924548091642364767, −1.24302435981079440958072077896, 0,
1.24302435981079440958072077896, 2.72338192403924548091642364767, 3.42842569397933790886637532429, 3.75632177918213904129192165190, 5.04883518287873073938759349116, 5.39629899229291060663021182825, 6.81049868604162186826559318459, 7.34602084665853768460558291203, 7.74352449392757349682853717269
