| L(s) = 1 | + 168.·2-s − 729·3-s + 2.02e4·4-s − 5.06e4·5-s − 1.22e5·6-s + 2.03e5·7-s + 2.02e6·8-s + 5.31e5·9-s − 8.54e6·10-s − 3.23e6·11-s − 1.47e7·12-s + 1.43e7·13-s + 3.42e7·14-s + 3.69e7·15-s + 1.75e8·16-s − 9.48e7·17-s + 8.95e7·18-s − 7.38e7·19-s − 1.02e9·20-s − 1.48e8·21-s − 5.44e8·22-s + 2.17e8·23-s − 1.47e9·24-s + 1.34e9·25-s + 2.41e9·26-s − 3.87e8·27-s + 4.10e9·28-s + ⋯ |
| L(s) = 1 | + 1.86·2-s − 0.577·3-s + 2.46·4-s − 1.45·5-s − 1.07·6-s + 0.652·7-s + 2.72·8-s + 0.333·9-s − 2.70·10-s − 0.550·11-s − 1.42·12-s + 0.824·13-s + 1.21·14-s + 0.837·15-s + 2.61·16-s − 0.953·17-s + 0.620·18-s − 0.359·19-s − 3.57·20-s − 0.376·21-s − 1.02·22-s + 0.306·23-s − 1.57·24-s + 1.10·25-s + 1.53·26-s − 0.192·27-s + 1.60·28-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 | \( 1 - 168.T + 8.19e3T^{2} \) |
| 5 | \( 1 + 5.06e4T + 1.22e9T^{2} \) |
| 7 | \( 1 - 2.03e5T + 9.68e10T^{2} \) |
| 11 | \( 1 + 3.23e6T + 3.45e13T^{2} \) |
| 13 | \( 1 - 1.43e7T + 3.02e14T^{2} \) |
| 17 | \( 1 + 9.48e7T + 9.90e15T^{2} \) |
| 19 | \( 1 + 7.38e7T + 4.20e16T^{2} \) |
| 23 | \( 1 - 2.17e8T + 5.04e17T^{2} \) |
| 29 | \( 1 - 1.43e9T + 1.02e19T^{2} \) |
| 31 | \( 1 + 2.08e9T + 2.44e19T^{2} \) |
| 37 | \( 1 + 8.62e9T + 2.43e20T^{2} \) |
| 41 | \( 1 - 1.89e10T + 9.25e20T^{2} \) |
| 43 | \( 1 + 2.90e10T + 1.71e21T^{2} \) |
| 47 | \( 1 - 3.92e10T + 5.46e21T^{2} \) |
| 53 | \( 1 - 1.89e10T + 2.60e22T^{2} \) |
| 61 | \( 1 + 3.32e11T + 1.61e23T^{2} \) |
| 67 | \( 1 - 4.65e11T + 5.48e23T^{2} \) |
| 71 | \( 1 + 1.25e12T + 1.16e24T^{2} \) |
| 73 | \( 1 + 4.92e11T + 1.67e24T^{2} \) |
| 79 | \( 1 + 1.86e12T + 4.66e24T^{2} \) |
| 83 | \( 1 + 5.31e11T + 8.87e24T^{2} \) |
| 89 | \( 1 + 9.16e12T + 2.19e25T^{2} \) |
| 97 | \( 1 + 5.98e12T + 6.73e25T^{2} \) |
| 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
−10.71382023058523055875900714901, −8.431597086709317073244650408207, −7.42726932439706376079995003320, −6.55880534281965863574518902468, −5.43515003681295096196461686999, −4.51868582657865484934049213326, −3.95415967349468284081938747393, −2.84390108913191062328585931169, −1.49740868921118380096545498133, 0,
1.49740868921118380096545498133, 2.84390108913191062328585931169, 3.95415967349468284081938747393, 4.51868582657865484934049213326, 5.43515003681295096196461686999, 6.55880534281965863574518902468, 7.42726932439706376079995003320, 8.431597086709317073244650408207, 10.71382023058523055875900714901
