| L(s) = 1 | − 1.82·2-s − 3-s + 1.31·4-s − 5-s + 1.82·6-s − 0.729·7-s + 1.24·8-s + 9-s + 1.82·10-s + 0.729·11-s − 1.31·12-s + 3.38·13-s + 1.32·14-s + 15-s − 4.90·16-s − 5.74·17-s − 1.82·18-s + 6.11·19-s − 1.31·20-s + 0.729·21-s − 1.32·22-s − 9.48·23-s − 1.24·24-s + 25-s − 6.15·26-s − 27-s − 0.957·28-s + ⋯ |
| L(s) = 1 | − 1.28·2-s − 0.577·3-s + 0.656·4-s − 0.447·5-s + 0.743·6-s − 0.275·7-s + 0.441·8-s + 0.333·9-s + 0.575·10-s + 0.219·11-s − 0.379·12-s + 0.938·13-s + 0.354·14-s + 0.258·15-s − 1.22·16-s − 1.39·17-s − 0.429·18-s + 1.40·19-s − 0.293·20-s + 0.159·21-s − 0.282·22-s − 1.97·23-s − 0.255·24-s + 0.200·25-s − 1.20·26-s − 0.192·27-s − 0.180·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 435 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 435 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 29 | \( 1 + T \) |
| good | 2 | \( 1 + 1.82T + 2T^{2} \) |
| 7 | \( 1 + 0.729T + 7T^{2} \) |
| 11 | \( 1 - 0.729T + 11T^{2} \) |
| 13 | \( 1 - 3.38T + 13T^{2} \) |
| 17 | \( 1 + 5.74T + 17T^{2} \) |
| 19 | \( 1 - 6.11T + 19T^{2} \) |
| 23 | \( 1 + 9.48T + 23T^{2} \) |
| 31 | \( 1 - 5.48T + 31T^{2} \) |
| 37 | \( 1 + 10.2T + 37T^{2} \) |
| 41 | \( 1 + 11.3T + 41T^{2} \) |
| 43 | \( 1 + 10.1T + 43T^{2} \) |
| 47 | \( 1 + 1.89T + 47T^{2} \) |
| 53 | \( 1 - 8.14T + 53T^{2} \) |
| 59 | \( 1 - 8.68T + 59T^{2} \) |
| 61 | \( 1 + 15.5T + 61T^{2} \) |
| 67 | \( 1 + 2.55T + 67T^{2} \) |
| 71 | \( 1 + 4.83T + 71T^{2} \) |
| 73 | \( 1 - 6.29T + 73T^{2} \) |
| 79 | \( 1 + 5.39T + 79T^{2} \) |
| 83 | \( 1 - 0.0848T + 83T^{2} \) |
| 89 | \( 1 - 4.63T + 89T^{2} \) |
| 97 | \( 1 - 1.30T + 97T^{2} \) |
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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
−10.45183838134413542320854461322, −9.890716923780104667221332176754, −8.808432889635575266950827412996, −8.172824529198231833814855520098, −7.08830874552896084502705225365, −6.29960085958085693732772872556, −4.87743515483435024311432960468, −3.66294992069654980628268562429, −1.64831522406643035234155751022, 0,
1.64831522406643035234155751022, 3.66294992069654980628268562429, 4.87743515483435024311432960468, 6.29960085958085693732772872556, 7.08830874552896084502705225365, 8.172824529198231833814855520098, 8.808432889635575266950827412996, 9.890716923780104667221332176754, 10.45183838134413542320854461322
