| L(s) = 1 | + 112.·2-s − 729·3-s + 4.53e3·4-s + 2.84e4·5-s − 8.22e4·6-s − 2.24e5·7-s − 4.12e5·8-s + 5.31e5·9-s + 3.20e6·10-s + 5.94e6·11-s − 3.30e6·12-s − 2.08e7·13-s − 2.53e7·14-s − 2.07e7·15-s − 8.36e7·16-s + 1.51e8·17-s + 5.99e7·18-s − 1.15e7·19-s + 1.28e8·20-s + 1.63e8·21-s + 6.70e8·22-s + 1.14e9·23-s + 3.00e8·24-s − 4.11e8·25-s − 2.35e9·26-s − 3.87e8·27-s − 1.01e9·28-s + ⋯ |
| L(s) = 1 | + 1.24·2-s − 0.577·3-s + 0.553·4-s + 0.814·5-s − 0.719·6-s − 0.720·7-s − 0.556·8-s + 0.333·9-s + 1.01·10-s + 1.01·11-s − 0.319·12-s − 1.19·13-s − 0.898·14-s − 0.470·15-s − 1.24·16-s + 1.51·17-s + 0.415·18-s − 0.0565·19-s + 0.450·20-s + 0.416·21-s + 1.26·22-s + 1.60·23-s + 0.321·24-s − 0.336·25-s − 1.49·26-s − 0.192·27-s − 0.399·28-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 | \( 1 - 112.T + 8.19e3T^{2} \) |
| 5 | \( 1 - 2.84e4T + 1.22e9T^{2} \) |
| 7 | \( 1 + 2.24e5T + 9.68e10T^{2} \) |
| 11 | \( 1 - 5.94e6T + 3.45e13T^{2} \) |
| 13 | \( 1 + 2.08e7T + 3.02e14T^{2} \) |
| 17 | \( 1 - 1.51e8T + 9.90e15T^{2} \) |
| 19 | \( 1 + 1.15e7T + 4.20e16T^{2} \) |
| 23 | \( 1 - 1.14e9T + 5.04e17T^{2} \) |
| 29 | \( 1 + 1.30e8T + 1.02e19T^{2} \) |
| 31 | \( 1 + 8.79e9T + 2.44e19T^{2} \) |
| 37 | \( 1 - 2.75e10T + 2.43e20T^{2} \) |
| 41 | \( 1 + 5.59e8T + 9.25e20T^{2} \) |
| 43 | \( 1 - 1.45e10T + 1.71e21T^{2} \) |
| 47 | \( 1 + 5.89e10T + 5.46e21T^{2} \) |
| 53 | \( 1 + 2.21e10T + 2.60e22T^{2} \) |
| 61 | \( 1 + 5.30e11T + 1.61e23T^{2} \) |
| 67 | \( 1 + 4.06e11T + 5.48e23T^{2} \) |
| 71 | \( 1 + 1.24e12T + 1.16e24T^{2} \) |
| 73 | \( 1 - 2.17e12T + 1.67e24T^{2} \) |
| 79 | \( 1 + 3.22e12T + 4.66e24T^{2} \) |
| 83 | \( 1 - 5.27e12T + 8.87e24T^{2} \) |
| 89 | \( 1 + 3.76e11T + 2.19e25T^{2} \) |
| 97 | \( 1 + 8.14e12T + 6.73e25T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.689465499239806549308928351291, −9.330443248949308223132789600074, −7.35895584698779103503367428105, −6.36588292236417122314185889960, −5.64404162517254491542095228123, −4.83215205464810337974178414101, −3.65874127865053183012501455867, −2.71505729608588780292699002392, −1.33418984134318495104325129895, 0,
1.33418984134318495104325129895, 2.71505729608588780292699002392, 3.65874127865053183012501455867, 4.83215205464810337974178414101, 5.64404162517254491542095228123, 6.36588292236417122314185889960, 7.35895584698779103503367428105, 9.330443248949308223132789600074, 9.689465499239806549308928351291
