| L(s) = 1 | + 163.·2-s − 729·3-s + 1.85e4·4-s + 2.04e4·5-s − 1.19e5·6-s − 1.41e5·7-s + 1.68e6·8-s + 5.31e5·9-s + 3.34e6·10-s + 9.53e6·11-s − 1.34e7·12-s − 2.74e7·13-s − 2.31e7·14-s − 1.49e7·15-s + 1.24e8·16-s − 1.51e8·17-s + 8.68e7·18-s − 3.23e8·19-s + 3.79e8·20-s + 1.03e8·21-s + 1.55e9·22-s + 1.57e8·23-s − 1.22e9·24-s − 8.01e8·25-s − 4.48e9·26-s − 3.87e8·27-s − 2.62e9·28-s + ⋯ |
| L(s) = 1 | + 1.80·2-s − 0.577·3-s + 2.26·4-s + 0.586·5-s − 1.04·6-s − 0.455·7-s + 2.27·8-s + 0.333·9-s + 1.05·10-s + 1.62·11-s − 1.30·12-s − 1.57·13-s − 0.821·14-s − 0.338·15-s + 1.84·16-s − 1.51·17-s + 0.601·18-s − 1.57·19-s + 1.32·20-s + 0.262·21-s + 2.92·22-s + 0.222·23-s − 1.31·24-s − 0.656·25-s − 2.84·26-s − 0.192·27-s − 1.02·28-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 | \( 1 - 163.T + 8.19e3T^{2} \) |
| 5 | \( 1 - 2.04e4T + 1.22e9T^{2} \) |
| 7 | \( 1 + 1.41e5T + 9.68e10T^{2} \) |
| 11 | \( 1 - 9.53e6T + 3.45e13T^{2} \) |
| 13 | \( 1 + 2.74e7T + 3.02e14T^{2} \) |
| 17 | \( 1 + 1.51e8T + 9.90e15T^{2} \) |
| 19 | \( 1 + 3.23e8T + 4.20e16T^{2} \) |
| 23 | \( 1 - 1.57e8T + 5.04e17T^{2} \) |
| 29 | \( 1 - 2.89e9T + 1.02e19T^{2} \) |
| 31 | \( 1 + 2.87e9T + 2.44e19T^{2} \) |
| 37 | \( 1 + 2.22e10T + 2.43e20T^{2} \) |
| 41 | \( 1 + 2.17e10T + 9.25e20T^{2} \) |
| 43 | \( 1 - 4.89e10T + 1.71e21T^{2} \) |
| 47 | \( 1 - 6.68e10T + 5.46e21T^{2} \) |
| 53 | \( 1 - 7.25e10T + 2.60e22T^{2} \) |
| 61 | \( 1 - 5.91e11T + 1.61e23T^{2} \) |
| 67 | \( 1 - 2.90e10T + 5.48e23T^{2} \) |
| 71 | \( 1 + 2.23e11T + 1.16e24T^{2} \) |
| 73 | \( 1 + 1.85e12T + 1.67e24T^{2} \) |
| 79 | \( 1 - 2.22e12T + 4.66e24T^{2} \) |
| 83 | \( 1 + 3.73e12T + 8.87e24T^{2} \) |
| 89 | \( 1 + 4.14e12T + 2.19e25T^{2} \) |
| 97 | \( 1 + 6.55e12T + 6.73e25T^{2} \) |
| 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
−10.19331962001843908880808034730, −8.996858432132817849461654033637, −6.91756335890837720940526113745, −6.62552321493277046324672577101, −5.65209276116516029153374116355, −4.57642151122946265303221458371, −3.95449853010045314670760380289, −2.51350490754077170278006039872, −1.75864557355029219164378079104, 0,
1.75864557355029219164378079104, 2.51350490754077170278006039872, 3.95449853010045314670760380289, 4.57642151122946265303221458371, 5.65209276116516029153374116355, 6.62552321493277046324672577101, 6.91756335890837720940526113745, 8.996858432132817849461654033637, 10.19331962001843908880808034730
