| L(s) = 1 | + 151.·2-s − 729·3-s + 1.46e4·4-s + 6.83e4·5-s − 1.10e5·6-s − 3.03e5·7-s + 9.73e5·8-s + 5.31e5·9-s + 1.03e7·10-s − 7.32e6·11-s − 1.06e7·12-s + 3.60e6·13-s − 4.59e7·14-s − 4.98e7·15-s + 2.71e7·16-s − 1.04e7·17-s + 8.02e7·18-s − 3.54e7·19-s + 9.99e8·20-s + 2.21e8·21-s − 1.10e9·22-s − 7.59e8·23-s − 7.09e8·24-s + 3.44e9·25-s + 5.44e8·26-s − 3.87e8·27-s − 4.44e9·28-s + ⋯ |
| L(s) = 1 | + 1.66·2-s − 0.577·3-s + 1.78·4-s + 1.95·5-s − 0.963·6-s − 0.976·7-s + 1.31·8-s + 0.333·9-s + 3.26·10-s − 1.24·11-s − 1.03·12-s + 0.207·13-s − 1.63·14-s − 1.12·15-s + 0.404·16-s − 0.104·17-s + 0.556·18-s − 0.172·19-s + 3.49·20-s + 0.563·21-s − 2.08·22-s − 1.06·23-s − 0.757·24-s + 2.82·25-s + 0.345·26-s − 0.192·27-s − 1.74·28-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 | \( 1 - 151.T + 8.19e3T^{2} \) |
| 5 | \( 1 - 6.83e4T + 1.22e9T^{2} \) |
| 7 | \( 1 + 3.03e5T + 9.68e10T^{2} \) |
| 11 | \( 1 + 7.32e6T + 3.45e13T^{2} \) |
| 13 | \( 1 - 3.60e6T + 3.02e14T^{2} \) |
| 17 | \( 1 + 1.04e7T + 9.90e15T^{2} \) |
| 19 | \( 1 + 3.54e7T + 4.20e16T^{2} \) |
| 23 | \( 1 + 7.59e8T + 5.04e17T^{2} \) |
| 29 | \( 1 + 3.30e9T + 1.02e19T^{2} \) |
| 31 | \( 1 + 8.30e9T + 2.44e19T^{2} \) |
| 37 | \( 1 + 3.43e9T + 2.43e20T^{2} \) |
| 41 | \( 1 + 3.61e10T + 9.25e20T^{2} \) |
| 43 | \( 1 + 1.31e10T + 1.71e21T^{2} \) |
| 47 | \( 1 - 1.29e11T + 5.46e21T^{2} \) |
| 53 | \( 1 - 6.61e10T + 2.60e22T^{2} \) |
| 61 | \( 1 - 7.54e10T + 1.61e23T^{2} \) |
| 67 | \( 1 + 9.97e11T + 5.48e23T^{2} \) |
| 71 | \( 1 - 1.02e11T + 1.16e24T^{2} \) |
| 73 | \( 1 - 4.69e11T + 1.67e24T^{2} \) |
| 79 | \( 1 + 7.23e11T + 4.66e24T^{2} \) |
| 83 | \( 1 - 4.73e12T + 8.87e24T^{2} \) |
| 89 | \( 1 + 2.71e12T + 2.19e25T^{2} \) |
| 97 | \( 1 - 6.53e12T + 6.73e25T^{2} \) |
| 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
−10.18961419700850941662881513624, −9.163860601191037442601435260776, −7.15255854636060560353826557600, −6.18556374539716649105783044549, −5.68790222003144965049751039118, −5.03672238098448213380448804279, −3.60720756193356327368805861583, −2.50677895902203529736554584536, −1.79012829936041050583216076981, 0,
1.79012829936041050583216076981, 2.50677895902203529736554584536, 3.60720756193356327368805861583, 5.03672238098448213380448804279, 5.68790222003144965049751039118, 6.18556374539716649105783044549, 7.15255854636060560353826557600, 9.163860601191037442601435260776, 10.18961419700850941662881513624
