| L(s) = 1 | − 1.78·2-s + 1.17·3-s + 1.19·4-s − 1.95·5-s − 2.10·6-s − 1.57·7-s + 1.44·8-s − 1.61·9-s + 3.48·10-s + 3.10·11-s + 1.40·12-s + 2.22·13-s + 2.81·14-s − 2.29·15-s − 4.96·16-s + 17-s + 2.88·18-s + 3.29·19-s − 2.32·20-s − 1.85·21-s − 5.54·22-s − 3.85·23-s + 1.69·24-s − 1.19·25-s − 3.98·26-s − 5.43·27-s − 1.88·28-s + ⋯ |
| L(s) = 1 | − 1.26·2-s + 0.680·3-s + 0.597·4-s − 0.872·5-s − 0.859·6-s − 0.595·7-s + 0.509·8-s − 0.537·9-s + 1.10·10-s + 0.935·11-s + 0.406·12-s + 0.617·13-s + 0.753·14-s − 0.593·15-s − 1.24·16-s + 0.242·17-s + 0.679·18-s + 0.755·19-s − 0.520·20-s − 0.405·21-s − 1.18·22-s − 0.803·23-s + 0.346·24-s − 0.239·25-s − 0.780·26-s − 1.04·27-s − 0.355·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 17 | \( 1 - T \) |
| 43 | \( 1 - T \) |
| good | 2 | \( 1 + 1.78T + 2T^{2} \) |
| 3 | \( 1 - 1.17T + 3T^{2} \) |
| 5 | \( 1 + 1.95T + 5T^{2} \) |
| 7 | \( 1 + 1.57T + 7T^{2} \) |
| 11 | \( 1 - 3.10T + 11T^{2} \) |
| 13 | \( 1 - 2.22T + 13T^{2} \) |
| 19 | \( 1 - 3.29T + 19T^{2} \) |
| 23 | \( 1 + 3.85T + 23T^{2} \) |
| 29 | \( 1 + 0.0511T + 29T^{2} \) |
| 31 | \( 1 + 9.84T + 31T^{2} \) |
| 37 | \( 1 + 11.4T + 37T^{2} \) |
| 41 | \( 1 + 2.81T + 41T^{2} \) |
| 47 | \( 1 - 0.384T + 47T^{2} \) |
| 53 | \( 1 - 6.87T + 53T^{2} \) |
| 59 | \( 1 - 0.424T + 59T^{2} \) |
| 61 | \( 1 - 0.494T + 61T^{2} \) |
| 67 | \( 1 + 6.29T + 67T^{2} \) |
| 71 | \( 1 + 12.7T + 71T^{2} \) |
| 73 | \( 1 + 0.682T + 73T^{2} \) |
| 79 | \( 1 + 0.156T + 79T^{2} \) |
| 83 | \( 1 - 0.486T + 83T^{2} \) |
| 89 | \( 1 + 7.53T + 89T^{2} \) |
| 97 | \( 1 + 1.92T + 97T^{2} \) |
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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
−9.659867138723846907361277012557, −9.004941003597572768898957505270, −8.439537787307021626455806322210, −7.63698999423401812221725858425, −6.90454608314614365185867196549, −5.64101816712825746121218380500, −4.01117253454398004000958957933, −3.32479346588805464178294087668, −1.68712996882066248150478259170, 0,
1.68712996882066248150478259170, 3.32479346588805464178294087668, 4.01117253454398004000958957933, 5.64101816712825746121218380500, 6.90454608314614365185867196549, 7.63698999423401812221725858425, 8.439537787307021626455806322210, 9.004941003597572768898957505270, 9.659867138723846907361277012557
