| L(s) = 1 | − 19.9·2-s + 27·3-s + 270.·4-s + 505.·5-s − 538.·6-s + 222.·7-s − 2.83e3·8-s + 729·9-s − 1.00e4·10-s − 1.83e3·11-s + 7.29e3·12-s + 688.·13-s − 4.44e3·14-s + 1.36e4·15-s + 2.19e4·16-s + 2.84e3·17-s − 1.45e4·18-s + 3.44e4·19-s + 1.36e5·20-s + 6.02e3·21-s + 3.65e4·22-s − 3.27e3·23-s − 7.64e4·24-s + 1.77e5·25-s − 1.37e4·26-s + 1.96e4·27-s + 6.02e4·28-s + ⋯ |
| L(s) = 1 | − 1.76·2-s + 0.577·3-s + 2.10·4-s + 1.80·5-s − 1.01·6-s + 0.245·7-s − 1.95·8-s + 0.333·9-s − 3.19·10-s − 0.414·11-s + 1.21·12-s + 0.0869·13-s − 0.433·14-s + 1.04·15-s + 1.34·16-s + 0.140·17-s − 0.587·18-s + 1.15·19-s + 3.81·20-s + 0.141·21-s + 0.731·22-s − 0.0560·23-s − 1.12·24-s + 2.27·25-s − 0.153·26-s + 0.192·27-s + 0.518·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(1.808048211\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.808048211\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - 27T \) |
| 59 | \( 1 - 2.05e5T \) |
| good | 2 | \( 1 + 19.9T + 128T^{2} \) |
| 5 | \( 1 - 505.T + 7.81e4T^{2} \) |
| 7 | \( 1 - 222.T + 8.23e5T^{2} \) |
| 11 | \( 1 + 1.83e3T + 1.94e7T^{2} \) |
| 13 | \( 1 - 688.T + 6.27e7T^{2} \) |
| 17 | \( 1 - 2.84e3T + 4.10e8T^{2} \) |
| 19 | \( 1 - 3.44e4T + 8.93e8T^{2} \) |
| 23 | \( 1 + 3.27e3T + 3.40e9T^{2} \) |
| 29 | \( 1 - 1.28e5T + 1.72e10T^{2} \) |
| 31 | \( 1 + 1.65e4T + 2.75e10T^{2} \) |
| 37 | \( 1 + 4.24e5T + 9.49e10T^{2} \) |
| 41 | \( 1 + 1.25e4T + 1.94e11T^{2} \) |
| 43 | \( 1 - 3.38e5T + 2.71e11T^{2} \) |
| 47 | \( 1 - 5.60e5T + 5.06e11T^{2} \) |
| 53 | \( 1 - 2.04e6T + 1.17e12T^{2} \) |
| 61 | \( 1 + 2.42e6T + 3.14e12T^{2} \) |
| 67 | \( 1 + 2.63e6T + 6.06e12T^{2} \) |
| 71 | \( 1 - 2.62e6T + 9.09e12T^{2} \) |
| 73 | \( 1 - 3.21e6T + 1.10e13T^{2} \) |
| 79 | \( 1 - 3.42e6T + 1.92e13T^{2} \) |
| 83 | \( 1 - 1.20e6T + 2.71e13T^{2} \) |
| 89 | \( 1 + 2.50e6T + 4.42e13T^{2} \) |
| 97 | \( 1 - 8.73e6T + 8.07e13T^{2} \) |
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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
−10.72508664057405078546983741220, −10.11177648484387002116321858416, −9.361496285965920665696383028938, −8.663239430345910027805960744076, −7.55432094986744426523274153567, −6.51871948903833711883788512223, −5.32463033752247726613683560332, −2.83445056989129761118412378835, −1.90648746126179472177499629214, −0.990596191157991163419761036133,
0.990596191157991163419761036133, 1.90648746126179472177499629214, 2.83445056989129761118412378835, 5.32463033752247726613683560332, 6.51871948903833711883788512223, 7.55432094986744426523274153567, 8.663239430345910027805960744076, 9.361496285965920665696383028938, 10.11177648484387002116321858416, 10.72508664057405078546983741220
