| L(s) = 1 | + 43.5·2-s + 729·3-s − 6.29e3·4-s − 5.38e4·5-s + 3.17e4·6-s − 6.00e5·7-s − 6.31e5·8-s + 5.31e5·9-s − 2.34e6·10-s + 4.53e6·11-s − 4.58e6·12-s + 1.89e7·13-s − 2.61e7·14-s − 3.92e7·15-s + 2.40e7·16-s + 5.77e7·17-s + 2.31e7·18-s − 1.29e7·19-s + 3.38e8·20-s − 4.37e8·21-s + 1.97e8·22-s + 5.60e8·23-s − 4.60e8·24-s + 1.67e9·25-s + 8.27e8·26-s + 3.87e8·27-s + 3.77e9·28-s + ⋯ |
| L(s) = 1 | + 0.481·2-s + 0.577·3-s − 0.768·4-s − 1.54·5-s + 0.277·6-s − 1.92·7-s − 0.851·8-s + 0.333·9-s − 0.741·10-s + 0.772·11-s − 0.443·12-s + 1.09·13-s − 0.928·14-s − 0.889·15-s + 0.358·16-s + 0.580·17-s + 0.160·18-s − 0.0630·19-s + 1.18·20-s − 1.11·21-s + 0.371·22-s + 0.789·23-s − 0.491·24-s + 1.37·25-s + 0.525·26-s + 0.192·27-s + 1.48·28-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 | \( 1 - 43.5T + 8.19e3T^{2} \) |
| 5 | \( 1 + 5.38e4T + 1.22e9T^{2} \) |
| 7 | \( 1 + 6.00e5T + 9.68e10T^{2} \) |
| 11 | \( 1 - 4.53e6T + 3.45e13T^{2} \) |
| 13 | \( 1 - 1.89e7T + 3.02e14T^{2} \) |
| 17 | \( 1 - 5.77e7T + 9.90e15T^{2} \) |
| 19 | \( 1 + 1.29e7T + 4.20e16T^{2} \) |
| 23 | \( 1 - 5.60e8T + 5.04e17T^{2} \) |
| 29 | \( 1 + 4.37e9T + 1.02e19T^{2} \) |
| 31 | \( 1 + 7.07e9T + 2.44e19T^{2} \) |
| 37 | \( 1 - 1.95e10T + 2.43e20T^{2} \) |
| 41 | \( 1 - 1.43e10T + 9.25e20T^{2} \) |
| 43 | \( 1 - 1.26e10T + 1.71e21T^{2} \) |
| 47 | \( 1 + 2.13e10T + 5.46e21T^{2} \) |
| 53 | \( 1 - 1.08e11T + 2.60e22T^{2} \) |
| 61 | \( 1 + 6.15e11T + 1.61e23T^{2} \) |
| 67 | \( 1 - 2.30e11T + 5.48e23T^{2} \) |
| 71 | \( 1 + 4.88e10T + 1.16e24T^{2} \) |
| 73 | \( 1 - 2.23e12T + 1.67e24T^{2} \) |
| 79 | \( 1 + 3.41e12T + 4.66e24T^{2} \) |
| 83 | \( 1 - 2.93e12T + 8.87e24T^{2} \) |
| 89 | \( 1 - 4.55e12T + 2.19e25T^{2} \) |
| 97 | \( 1 + 8.26e12T + 6.73e25T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.429710614936970497327996084571, −9.053971866194178128510190599285, −7.87861874469178273770584331926, −6.82070510116108564331708023073, −5.74608514387958499220950909854, −4.07558070213356719641529903068, −3.68418377527729489598815804381, −3.05268615533647666437304192994, −0.898884417959466176718457264691, 0,
0.898884417959466176718457264691, 3.05268615533647666437304192994, 3.68418377527729489598815804381, 4.07558070213356719641529903068, 5.74608514387958499220950909854, 6.82070510116108564331708023073, 7.87861874469178273770584331926, 9.053971866194178128510190599285, 9.429710614936970497327996084571
