| L(s) = 1 | + 177.·2-s + 729·3-s + 2.32e4·4-s + 3.79e4·5-s + 1.29e5·6-s + 2.36e5·7-s + 2.66e6·8-s + 5.31e5·9-s + 6.73e6·10-s + 1.90e6·11-s + 1.69e7·12-s − 1.09e7·13-s + 4.19e7·14-s + 2.76e7·15-s + 2.82e8·16-s + 1.36e8·17-s + 9.42e7·18-s − 3.07e8·19-s + 8.82e8·20-s + 1.72e8·21-s + 3.37e8·22-s + 5.31e8·23-s + 1.94e9·24-s + 2.22e8·25-s − 1.93e9·26-s + 3.87e8·27-s + 5.49e9·28-s + ⋯ |
| L(s) = 1 | + 1.95·2-s + 0.577·3-s + 2.83·4-s + 1.08·5-s + 1.13·6-s + 0.759·7-s + 3.59·8-s + 0.333·9-s + 2.12·10-s + 0.324·11-s + 1.63·12-s − 0.626·13-s + 1.48·14-s + 0.627·15-s + 4.21·16-s + 1.36·17-s + 0.652·18-s − 1.49·19-s + 3.08·20-s + 0.438·21-s + 0.635·22-s + 0.748·23-s + 2.07·24-s + 0.182·25-s − 1.22·26-s + 0.192·27-s + 2.15·28-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
| \(L(7)\) |
\(\approx\) |
\(16.67719948\) |
| \(L(\frac12)\) |
\(\approx\) |
\(16.67719948\) |
| \(L(\frac{15}{2})\) |
|
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 | \( 1 - 177.T + 8.19e3T^{2} \) |
| 5 | \( 1 - 3.79e4T + 1.22e9T^{2} \) |
| 7 | \( 1 - 2.36e5T + 9.68e10T^{2} \) |
| 11 | \( 1 - 1.90e6T + 3.45e13T^{2} \) |
| 13 | \( 1 + 1.09e7T + 3.02e14T^{2} \) |
| 17 | \( 1 - 1.36e8T + 9.90e15T^{2} \) |
| 19 | \( 1 + 3.07e8T + 4.20e16T^{2} \) |
| 23 | \( 1 - 5.31e8T + 5.04e17T^{2} \) |
| 29 | \( 1 + 3.71e9T + 1.02e19T^{2} \) |
| 31 | \( 1 + 8.66e9T + 2.44e19T^{2} \) |
| 37 | \( 1 - 2.18e10T + 2.43e20T^{2} \) |
| 41 | \( 1 - 5.66e9T + 9.25e20T^{2} \) |
| 43 | \( 1 + 7.28e10T + 1.71e21T^{2} \) |
| 47 | \( 1 - 5.64e10T + 5.46e21T^{2} \) |
| 53 | \( 1 + 2.16e11T + 2.60e22T^{2} \) |
| 61 | \( 1 - 1.58e11T + 1.61e23T^{2} \) |
| 67 | \( 1 - 3.98e11T + 5.48e23T^{2} \) |
| 71 | \( 1 + 9.16e11T + 1.16e24T^{2} \) |
| 73 | \( 1 - 6.04e11T + 1.67e24T^{2} \) |
| 79 | \( 1 - 5.26e11T + 4.66e24T^{2} \) |
| 83 | \( 1 + 3.86e12T + 8.87e24T^{2} \) |
| 89 | \( 1 + 7.64e12T + 2.19e25T^{2} \) |
| 97 | \( 1 - 9.71e12T + 6.73e25T^{2} \) |
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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
−10.66074267646188122184331888968, −9.573994673014499847326934284331, −7.953522140730149297890735038663, −6.98781085840492608596152142966, −5.89832464884810924122090303859, −5.16730705726993251960514511441, −4.16938000517343843411404844237, −3.11534190034159985338345902499, −2.05562658000919056986425242825, −1.55004924347812682259768381222,
1.55004924347812682259768381222, 2.05562658000919056986425242825, 3.11534190034159985338345902499, 4.16938000517343843411404844237, 5.16730705726993251960514511441, 5.89832464884810924122090303859, 6.98781085840492608596152142966, 7.953522140730149297890735038663, 9.573994673014499847326934284331, 10.66074267646188122184331888968
