| L(s) = 1 | − 169.·2-s − 729·3-s + 2.06e4·4-s + 4.14e4·5-s + 1.23e5·6-s + 2.22e5·7-s − 2.12e6·8-s + 5.31e5·9-s − 7.05e6·10-s − 1.07e7·11-s − 1.50e7·12-s + 2.00e7·13-s − 3.77e7·14-s − 3.02e7·15-s + 1.91e8·16-s + 1.01e8·17-s − 9.02e7·18-s − 3.33e8·19-s + 8.58e8·20-s − 1.62e8·21-s + 1.83e9·22-s + 1.06e9·23-s + 1.54e9·24-s + 5.01e8·25-s − 3.40e9·26-s − 3.87e8·27-s + 4.59e9·28-s + ⋯ |
| L(s) = 1 | − 1.87·2-s − 0.577·3-s + 2.52·4-s + 1.18·5-s + 1.08·6-s + 0.714·7-s − 2.86·8-s + 0.333·9-s − 2.22·10-s − 1.83·11-s − 1.45·12-s + 1.15·13-s − 1.34·14-s − 0.685·15-s + 2.84·16-s + 1.02·17-s − 0.625·18-s − 1.62·19-s + 2.99·20-s − 0.412·21-s + 3.44·22-s + 1.50·23-s + 1.65·24-s + 0.410·25-s − 2.16·26-s − 0.192·27-s + 1.80·28-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 | \( 1 + 169.T + 8.19e3T^{2} \) |
| 5 | \( 1 - 4.14e4T + 1.22e9T^{2} \) |
| 7 | \( 1 - 2.22e5T + 9.68e10T^{2} \) |
| 11 | \( 1 + 1.07e7T + 3.45e13T^{2} \) |
| 13 | \( 1 - 2.00e7T + 3.02e14T^{2} \) |
| 17 | \( 1 - 1.01e8T + 9.90e15T^{2} \) |
| 19 | \( 1 + 3.33e8T + 4.20e16T^{2} \) |
| 23 | \( 1 - 1.06e9T + 5.04e17T^{2} \) |
| 29 | \( 1 - 5.94e9T + 1.02e19T^{2} \) |
| 31 | \( 1 + 6.31e9T + 2.44e19T^{2} \) |
| 37 | \( 1 + 1.05e10T + 2.43e20T^{2} \) |
| 41 | \( 1 + 3.83e10T + 9.25e20T^{2} \) |
| 43 | \( 1 - 1.84e10T + 1.71e21T^{2} \) |
| 47 | \( 1 - 2.22e10T + 5.46e21T^{2} \) |
| 53 | \( 1 - 1.89e10T + 2.60e22T^{2} \) |
| 61 | \( 1 - 3.61e11T + 1.61e23T^{2} \) |
| 67 | \( 1 + 1.10e12T + 5.48e23T^{2} \) |
| 71 | \( 1 + 5.32e11T + 1.16e24T^{2} \) |
| 73 | \( 1 - 8.30e11T + 1.67e24T^{2} \) |
| 79 | \( 1 + 2.73e12T + 4.66e24T^{2} \) |
| 83 | \( 1 - 3.89e12T + 8.87e24T^{2} \) |
| 89 | \( 1 + 6.16e12T + 2.19e25T^{2} \) |
| 97 | \( 1 + 1.11e13T + 6.73e25T^{2} \) |
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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
−10.04121549508203014904021193416, −8.774100329642621225433142002594, −8.158249074451148021592205581468, −7.01880538221701582156459303553, −6.01818004004802432151470133566, −5.16322477056362259240053695512, −2.84842803410853174377621576584, −1.84882266289069097346414280332, −1.11420570534007834789373047447, 0,
1.11420570534007834789373047447, 1.84882266289069097346414280332, 2.84842803410853174377621576584, 5.16322477056362259240053695512, 6.01818004004802432151470133566, 7.01880538221701582156459303553, 8.158249074451148021592205581468, 8.774100329642621225433142002594, 10.04121549508203014904021193416
