| L(s) = 1 | − 13.3·2-s + 27·3-s + 49.7·4-s + 152.·5-s − 360.·6-s − 1.32e3·7-s + 1.04e3·8-s + 729·9-s − 2.02e3·10-s + 5.13e3·11-s + 1.34e3·12-s − 1.22e4·13-s + 1.77e4·14-s + 4.10e3·15-s − 2.02e4·16-s + 2.66e4·17-s − 9.72e3·18-s + 3.31e4·19-s + 7.57e3·20-s − 3.58e4·21-s − 6.84e4·22-s − 9.70e4·23-s + 2.81e4·24-s − 5.49e4·25-s + 1.63e5·26-s + 1.96e4·27-s − 6.61e4·28-s + ⋯ |
| L(s) = 1 | − 1.17·2-s + 0.577·3-s + 0.388·4-s + 0.544·5-s − 0.680·6-s − 1.46·7-s + 0.720·8-s + 0.333·9-s − 0.641·10-s + 1.16·11-s + 0.224·12-s − 1.54·13-s + 1.72·14-s + 0.314·15-s − 1.23·16-s + 1.31·17-s − 0.392·18-s + 1.11·19-s + 0.211·20-s − 0.845·21-s − 1.37·22-s − 1.66·23-s + 0.415·24-s − 0.703·25-s + 1.82·26-s + 0.192·27-s − 0.569·28-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
| \(L(4)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{9}{2})\) |
|
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 | \( 1 + 13.3T + 128T^{2} \) |
| 5 | \( 1 - 152.T + 7.81e4T^{2} \) |
| 7 | \( 1 + 1.32e3T + 8.23e5T^{2} \) |
| 11 | \( 1 - 5.13e3T + 1.94e7T^{2} \) |
| 13 | \( 1 + 1.22e4T + 6.27e7T^{2} \) |
| 17 | \( 1 - 2.66e4T + 4.10e8T^{2} \) |
| 19 | \( 1 - 3.31e4T + 8.93e8T^{2} \) |
| 23 | \( 1 + 9.70e4T + 3.40e9T^{2} \) |
| 29 | \( 1 - 2.09e5T + 1.72e10T^{2} \) |
| 31 | \( 1 + 3.08e5T + 2.75e10T^{2} \) |
| 37 | \( 1 - 2.87e5T + 9.49e10T^{2} \) |
| 41 | \( 1 - 5.81e5T + 1.94e11T^{2} \) |
| 43 | \( 1 - 5.40e5T + 2.71e11T^{2} \) |
| 47 | \( 1 + 6.50e5T + 5.06e11T^{2} \) |
| 53 | \( 1 + 4.99e4T + 1.17e12T^{2} \) |
| 61 | \( 1 + 2.00e4T + 3.14e12T^{2} \) |
| 67 | \( 1 + 1.38e6T + 6.06e12T^{2} \) |
| 71 | \( 1 - 1.24e6T + 9.09e12T^{2} \) |
| 73 | \( 1 + 4.83e6T + 1.10e13T^{2} \) |
| 79 | \( 1 + 2.94e6T + 1.92e13T^{2} \) |
| 83 | \( 1 + 8.62e6T + 2.71e13T^{2} \) |
| 89 | \( 1 + 8.79e5T + 4.42e13T^{2} \) |
| 97 | \( 1 + 1.62e7T + 8.07e13T^{2} \) |
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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
−10.17251015852899453151107777561, −9.608401673410879892210733256427, −9.357011556726651384110828800857, −7.88116001252629383530939410355, −7.08653566484324779476488342796, −5.82875043538230854705460521676, −4.04981811369366164148825535771, −2.70211298916659415480997093302, −1.33273985178876221463747688938, 0,
1.33273985178876221463747688938, 2.70211298916659415480997093302, 4.04981811369366164148825535771, 5.82875043538230854705460521676, 7.08653566484324779476488342796, 7.88116001252629383530939410355, 9.357011556726651384110828800857, 9.608401673410879892210733256427, 10.17251015852899453151107777561
