| L(s) = 1 | − 2.51·2-s + 4.32·4-s − 5-s − 0.514·7-s − 5.83·8-s + 2.51·10-s + 3.32·11-s − 1.32·13-s + 1.29·14-s + 6.02·16-s + 3.32·17-s − 1.32·19-s − 4.32·20-s − 8.34·22-s − 4.12·23-s + 25-s + 3.32·26-s − 2.22·28-s + 1.38·29-s + 8.73·31-s − 3.48·32-s − 8.34·34-s + 0.514·35-s + 0.292·37-s + 3.32·38-s + 5.83·40-s + 11.3·41-s + ⋯ |
| L(s) = 1 | − 1.77·2-s + 2.16·4-s − 0.447·5-s − 0.194·7-s − 2.06·8-s + 0.795·10-s + 1.00·11-s − 0.366·13-s + 0.345·14-s + 1.50·16-s + 0.805·17-s − 0.303·19-s − 0.966·20-s − 1.78·22-s − 0.860·23-s + 0.200·25-s + 0.651·26-s − 0.419·28-s + 0.257·29-s + 1.56·31-s − 0.616·32-s − 1.43·34-s + 0.0869·35-s + 0.0481·37-s + 0.538·38-s + 0.922·40-s + 1.77·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5616801339\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5616801339\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| good | 2 | \( 1 + 2.51T + 2T^{2} \) |
| 7 | \( 1 + 0.514T + 7T^{2} \) |
| 11 | \( 1 - 3.32T + 11T^{2} \) |
| 13 | \( 1 + 1.32T + 13T^{2} \) |
| 17 | \( 1 - 3.32T + 17T^{2} \) |
| 19 | \( 1 + 1.32T + 19T^{2} \) |
| 23 | \( 1 + 4.12T + 23T^{2} \) |
| 29 | \( 1 - 1.38T + 29T^{2} \) |
| 31 | \( 1 - 8.73T + 31T^{2} \) |
| 37 | \( 1 - 0.292T + 37T^{2} \) |
| 41 | \( 1 - 11.3T + 41T^{2} \) |
| 43 | \( 1 - 10.3T + 43T^{2} \) |
| 47 | \( 1 - 4.86T + 47T^{2} \) |
| 53 | \( 1 - 5.02T + 53T^{2} \) |
| 59 | \( 1 - 5.02T + 59T^{2} \) |
| 61 | \( 1 - 7.34T + 61T^{2} \) |
| 67 | \( 1 - 9.44T + 67T^{2} \) |
| 71 | \( 1 + 8.99T + 71T^{2} \) |
| 73 | \( 1 - 6.05T + 73T^{2} \) |
| 79 | \( 1 + 8.05T + 79T^{2} \) |
| 83 | \( 1 + 1.54T + 83T^{2} \) |
| 89 | \( 1 - 3T + 89T^{2} \) |
| 97 | \( 1 + 12.2T + 97T^{2} \) |
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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
−11.04285306493371508566371080667, −10.07146441959455442372232616676, −9.476007814914580912086630166011, −8.533713877497509955171401547219, −7.79205900265118743413620976189, −6.91898632223280747240435368876, −5.98567514029642307861749193439, −4.11886575434530500089847515341, −2.55449225267992608826531435509, −0.954570366014384757362295281472,
0.954570366014384757362295281472, 2.55449225267992608826531435509, 4.11886575434530500089847515341, 5.98567514029642307861749193439, 6.91898632223280747240435368876, 7.79205900265118743413620976189, 8.533713877497509955171401547219, 9.476007814914580912086630166011, 10.07146441959455442372232616676, 11.04285306493371508566371080667
