| L(s) = 1 | − 27.7·2-s − 729·3-s − 7.42e3·4-s − 6.27e4·5-s + 2.02e4·6-s − 9.85e4·7-s + 4.33e5·8-s + 5.31e5·9-s + 1.74e6·10-s + 1.01e7·11-s + 5.40e6·12-s + 5.60e6·13-s + 2.73e6·14-s + 4.57e7·15-s + 4.87e7·16-s + 1.47e8·17-s − 1.47e7·18-s − 3.38e8·19-s + 4.65e8·20-s + 7.18e7·21-s − 2.82e8·22-s + 1.06e9·23-s − 3.16e8·24-s + 2.72e9·25-s − 1.55e8·26-s − 3.87e8·27-s + 7.30e8·28-s + ⋯ |
| L(s) = 1 | − 0.306·2-s − 0.577·3-s − 0.905·4-s − 1.79·5-s + 0.177·6-s − 0.316·7-s + 0.584·8-s + 0.333·9-s + 0.551·10-s + 1.73·11-s + 0.522·12-s + 0.322·13-s + 0.0971·14-s + 1.03·15-s + 0.726·16-s + 1.48·17-s − 0.102·18-s − 1.65·19-s + 1.62·20-s + 0.182·21-s − 0.531·22-s + 1.49·23-s − 0.337·24-s + 2.22·25-s − 0.0988·26-s − 0.192·27-s + 0.286·28-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
| \(L(7)\) |
\(\approx\) |
\(0.8799745895\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8799745895\) |
| \(L(\frac{15}{2})\) |
|
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 | \( 1 + 27.7T + 8.19e3T^{2} \) |
| 5 | \( 1 + 6.27e4T + 1.22e9T^{2} \) |
| 7 | \( 1 + 9.85e4T + 9.68e10T^{2} \) |
| 11 | \( 1 - 1.01e7T + 3.45e13T^{2} \) |
| 13 | \( 1 - 5.60e6T + 3.02e14T^{2} \) |
| 17 | \( 1 - 1.47e8T + 9.90e15T^{2} \) |
| 19 | \( 1 + 3.38e8T + 4.20e16T^{2} \) |
| 23 | \( 1 - 1.06e9T + 5.04e17T^{2} \) |
| 29 | \( 1 - 3.26e9T + 1.02e19T^{2} \) |
| 31 | \( 1 + 3.41e9T + 2.44e19T^{2} \) |
| 37 | \( 1 + 1.70e10T + 2.43e20T^{2} \) |
| 41 | \( 1 - 2.90e10T + 9.25e20T^{2} \) |
| 43 | \( 1 + 1.26e10T + 1.71e21T^{2} \) |
| 47 | \( 1 - 1.29e11T + 5.46e21T^{2} \) |
| 53 | \( 1 + 1.26e11T + 2.60e22T^{2} \) |
| 61 | \( 1 - 4.72e11T + 1.61e23T^{2} \) |
| 67 | \( 1 - 6.43e11T + 5.48e23T^{2} \) |
| 71 | \( 1 + 2.05e11T + 1.16e24T^{2} \) |
| 73 | \( 1 - 3.09e11T + 1.67e24T^{2} \) |
| 79 | \( 1 + 6.23e11T + 4.66e24T^{2} \) |
| 83 | \( 1 - 5.06e12T + 8.87e24T^{2} \) |
| 89 | \( 1 - 5.09e11T + 2.19e25T^{2} \) |
| 97 | \( 1 - 6.06e12T + 6.73e25T^{2} \) |
| 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
−10.44702872097134720377803194742, −9.137348628750563112101698473527, −8.465181416202328246463210922390, −7.40457918566975057598590131803, −6.45441311710189983871884024931, −4.95324936441018112215463973424, −4.01060909770137163415604489181, −3.49523150856881445229931207256, −1.14958690340196951947998423701, −0.54626157636839431337625059713,
0.54626157636839431337625059713, 1.14958690340196951947998423701, 3.49523150856881445229931207256, 4.01060909770137163415604489181, 4.95324936441018112215463973424, 6.45441311710189983871884024931, 7.40457918566975057598590131803, 8.465181416202328246463210922390, 9.137348628750563112101698473527, 10.44702872097134720377803194742
