| L(s) = 1 | + 20.2·2-s + 729·3-s − 7.78e3·4-s + 3.72e4·5-s + 1.47e4·6-s − 2.60e5·7-s − 3.23e5·8-s + 5.31e5·9-s + 7.54e5·10-s − 3.65e6·11-s − 5.67e6·12-s − 1.12e7·13-s − 5.27e6·14-s + 2.71e7·15-s + 5.71e7·16-s + 1.19e6·17-s + 1.07e7·18-s + 2.27e8·19-s − 2.89e8·20-s − 1.89e8·21-s − 7.40e7·22-s + 8.27e8·23-s − 2.36e8·24-s + 1.65e8·25-s − 2.28e8·26-s + 3.87e8·27-s + 2.02e9·28-s + ⋯ |
| L(s) = 1 | + 0.223·2-s + 0.577·3-s − 0.949·4-s + 1.06·5-s + 0.129·6-s − 0.835·7-s − 0.436·8-s + 0.333·9-s + 0.238·10-s − 0.621·11-s − 0.548·12-s − 0.647·13-s − 0.187·14-s + 0.615·15-s + 0.852·16-s + 0.0120·17-s + 0.0746·18-s + 1.10·19-s − 1.01·20-s − 0.482·21-s − 0.139·22-s + 1.16·23-s − 0.252·24-s + 0.135·25-s − 0.145·26-s + 0.192·27-s + 0.794·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(14-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+13/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(7)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{15}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - 729T \) |
| 59 | \( 1 - 4.21e10T \) |
| good | 2 | \( 1 - 20.2T + 8.19e3T^{2} \) |
| 5 | \( 1 - 3.72e4T + 1.22e9T^{2} \) |
| 7 | \( 1 + 2.60e5T + 9.68e10T^{2} \) |
| 11 | \( 1 + 3.65e6T + 3.45e13T^{2} \) |
| 13 | \( 1 + 1.12e7T + 3.02e14T^{2} \) |
| 17 | \( 1 - 1.19e6T + 9.90e15T^{2} \) |
| 19 | \( 1 - 2.27e8T + 4.20e16T^{2} \) |
| 23 | \( 1 - 8.27e8T + 5.04e17T^{2} \) |
| 29 | \( 1 - 1.28e9T + 1.02e19T^{2} \) |
| 31 | \( 1 - 3.95e9T + 2.44e19T^{2} \) |
| 37 | \( 1 + 9.79e9T + 2.43e20T^{2} \) |
| 41 | \( 1 + 3.95e10T + 9.25e20T^{2} \) |
| 43 | \( 1 - 6.25e10T + 1.71e21T^{2} \) |
| 47 | \( 1 + 8.64e10T + 5.46e21T^{2} \) |
| 53 | \( 1 + 6.34e10T + 2.60e22T^{2} \) |
| 61 | \( 1 - 2.34e11T + 1.61e23T^{2} \) |
| 67 | \( 1 + 1.03e12T + 5.48e23T^{2} \) |
| 71 | \( 1 + 1.82e12T + 1.16e24T^{2} \) |
| 73 | \( 1 - 2.32e12T + 1.67e24T^{2} \) |
| 79 | \( 1 + 8.66e11T + 4.66e24T^{2} \) |
| 83 | \( 1 - 4.79e12T + 8.87e24T^{2} \) |
| 89 | \( 1 + 1.11e12T + 2.19e25T^{2} \) |
| 97 | \( 1 + 5.79e12T + 6.73e25T^{2} \) |
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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
−9.677094075065509742817286726443, −9.139889323112457045394193152241, −7.973851510380976543349832948913, −6.73731536042881440199782261188, −5.55282529442105532471757292435, −4.77837547927584047848389327568, −3.35450222698237068632568795787, −2.63572909884115572016938983702, −1.22972387435969187277819529747, 0,
1.22972387435969187277819529747, 2.63572909884115572016938983702, 3.35450222698237068632568795787, 4.77837547927584047848389327568, 5.55282529442105532471757292435, 6.73731536042881440199782261188, 7.973851510380976543349832948913, 9.139889323112457045394193152241, 9.677094075065509742817286726443
