| L(s) = 1 | − 12.5·2-s − 81·3-s − 354.·4-s + 2.37e3·5-s + 1.01e3·6-s + 4.67e3·7-s + 1.08e4·8-s + 6.56e3·9-s − 2.98e4·10-s + 2.73e4·11-s + 2.87e4·12-s − 9.49e4·13-s − 5.87e4·14-s − 1.92e5·15-s + 4.51e4·16-s − 3.63e5·17-s − 8.23e4·18-s − 7.03e5·19-s − 8.43e5·20-s − 3.79e5·21-s − 3.43e5·22-s + 1.27e6·23-s − 8.80e5·24-s + 3.70e6·25-s + 1.19e6·26-s − 5.31e5·27-s − 1.65e6·28-s + ⋯ |
| L(s) = 1 | − 0.554·2-s − 0.577·3-s − 0.692·4-s + 1.70·5-s + 0.320·6-s + 0.736·7-s + 0.938·8-s + 0.333·9-s − 0.943·10-s + 0.563·11-s + 0.399·12-s − 0.922·13-s − 0.408·14-s − 0.982·15-s + 0.172·16-s − 1.05·17-s − 0.184·18-s − 1.23·19-s − 1.17·20-s − 0.425·21-s − 0.312·22-s + 0.953·23-s − 0.541·24-s + 1.89·25-s + 0.511·26-s − 0.192·27-s − 0.510·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + 81T \) |
| 59 | \( 1 + 1.21e7T \) |
| good | 2 | \( 1 + 12.5T + 512T^{2} \) |
| 5 | \( 1 - 2.37e3T + 1.95e6T^{2} \) |
| 7 | \( 1 - 4.67e3T + 4.03e7T^{2} \) |
| 11 | \( 1 - 2.73e4T + 2.35e9T^{2} \) |
| 13 | \( 1 + 9.49e4T + 1.06e10T^{2} \) |
| 17 | \( 1 + 3.63e5T + 1.18e11T^{2} \) |
| 19 | \( 1 + 7.03e5T + 3.22e11T^{2} \) |
| 23 | \( 1 - 1.27e6T + 1.80e12T^{2} \) |
| 29 | \( 1 + 6.03e6T + 1.45e13T^{2} \) |
| 31 | \( 1 + 2.91e6T + 2.64e13T^{2} \) |
| 37 | \( 1 - 8.60e6T + 1.29e14T^{2} \) |
| 41 | \( 1 + 1.66e7T + 3.27e14T^{2} \) |
| 43 | \( 1 - 3.19e7T + 5.02e14T^{2} \) |
| 47 | \( 1 - 4.06e7T + 1.11e15T^{2} \) |
| 53 | \( 1 + 6.97e7T + 3.29e15T^{2} \) |
| 61 | \( 1 + 6.34e7T + 1.16e16T^{2} \) |
| 67 | \( 1 + 1.72e8T + 2.72e16T^{2} \) |
| 71 | \( 1 + 1.91e8T + 4.58e16T^{2} \) |
| 73 | \( 1 - 1.49e8T + 5.88e16T^{2} \) |
| 79 | \( 1 + 7.22e7T + 1.19e17T^{2} \) |
| 83 | \( 1 + 1.57e8T + 1.86e17T^{2} \) |
| 89 | \( 1 - 5.53e8T + 3.50e17T^{2} \) |
| 97 | \( 1 - 6.86e8T + 7.60e17T^{2} \) |
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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
−10.44190599453182211397174946075, −9.363783484253839687427163628435, −8.941620386984793769218353077678, −7.42324763805154108975416337192, −6.23995246708810055022301454329, −5.20020693805274983831302637094, −4.38562804426193943236995788290, −2.17798823013130778416549316382, −1.36993192533978915598353718491, 0,
1.36993192533978915598353718491, 2.17798823013130778416549316382, 4.38562804426193943236995788290, 5.20020693805274983831302637094, 6.23995246708810055022301454329, 7.42324763805154108975416337192, 8.941620386984793769218353077678, 9.363783484253839687427163628435, 10.44190599453182211397174946075
