| L(s) = 1 | + 2-s + 4-s − 3.77·5-s + 0.814i·7-s + 8-s − 3.77·10-s + (2.67 − 1.95i)11-s + 2.43·13-s + 0.814i·14-s + 16-s − 3.57i·17-s + (−1.78 + 3.97i)19-s − 3.77·20-s + (2.67 − 1.95i)22-s − 6.28·23-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.5·4-s − 1.68·5-s + 0.308i·7-s + 0.353·8-s − 1.19·10-s + (0.806 − 0.590i)11-s + 0.676·13-s + 0.217i·14-s + 0.250·16-s − 0.867i·17-s + (−0.408 + 0.912i)19-s − 0.843·20-s + (0.570 − 0.417i)22-s − 1.31·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3762 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.209 + 0.977i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3762 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.209 + 0.977i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.652480663\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.652480663\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 \) |
| 11 | \( 1 + (-2.67 + 1.95i)T \) |
| 19 | \( 1 + (1.78 - 3.97i)T \) |
| good | 5 | \( 1 + 3.77T + 5T^{2} \) |
| 7 | \( 1 - 0.814iT - 7T^{2} \) |
| 13 | \( 1 - 2.43T + 13T^{2} \) |
| 17 | \( 1 + 3.57iT - 17T^{2} \) |
| 23 | \( 1 + 6.28T + 23T^{2} \) |
| 29 | \( 1 + 6.64T + 29T^{2} \) |
| 31 | \( 1 - 2.09iT - 31T^{2} \) |
| 37 | \( 1 - 1.67iT - 37T^{2} \) |
| 41 | \( 1 - 6.59T + 41T^{2} \) |
| 43 | \( 1 + 4.68iT - 43T^{2} \) |
| 47 | \( 1 - 4.49T + 47T^{2} \) |
| 53 | \( 1 + 11.4iT - 53T^{2} \) |
| 59 | \( 1 + 9.88iT - 59T^{2} \) |
| 61 | \( 1 + 15.1iT - 61T^{2} \) |
| 67 | \( 1 + 2.32iT - 67T^{2} \) |
| 71 | \( 1 - 13.9iT - 71T^{2} \) |
| 73 | \( 1 + 1.29iT - 73T^{2} \) |
| 79 | \( 1 + 5.80T + 79T^{2} \) |
| 83 | \( 1 + 7.69iT - 83T^{2} \) |
| 89 | \( 1 + 5.58iT - 89T^{2} \) |
| 97 | \( 1 + 2.59iT - 97T^{2} \) |
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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
−8.240135120882820376334678708218, −7.61526132723776002076712638393, −6.85633853630423419097400182700, −6.07713809149534451773394822677, −5.32065389640354918578823904383, −4.21837413635892332014295015941, −3.81899006838387817317476979799, −3.19271196860033330199874447371, −1.87112309543056908968465796222, −0.43194598031274989968834818488,
1.09116383056809330029538527979, 2.40453957073597764262169129884, 3.59496129429621326143719504844, 4.17898331541468232412131889811, 4.37117956512073971668868287161, 5.75328254799127259825248396548, 6.40141667502436352285847830487, 7.45389267995886204388835683024, 7.53971705906820343099446389663, 8.590465845592430158701588009578
