| L(s) = 1 | + 3.83·2-s + 2.25e3·3-s − 8.17e3·4-s − 3.87e4·5-s + 8.64e3·6-s − 5.47e5·7-s − 6.27e4·8-s + 3.48e6·9-s − 1.48e5·10-s − 1.77e6·11-s − 1.84e7·12-s − 7.21e6·13-s − 2.09e6·14-s − 8.73e7·15-s + 6.67e7·16-s − 1.23e8·17-s + 1.33e7·18-s + 1.60e8·19-s + 3.17e8·20-s − 1.23e9·21-s − 6.79e6·22-s + 1.39e8·23-s − 1.41e8·24-s + 2.82e8·25-s − 2.76e7·26-s + 4.26e9·27-s + 4.47e9·28-s + ⋯ |
| L(s) = 1 | + 0.0423·2-s + 1.78·3-s − 0.998·4-s − 1.10·5-s + 0.0756·6-s − 1.75·7-s − 0.0846·8-s + 2.18·9-s − 0.0470·10-s − 0.301·11-s − 1.78·12-s − 0.414·13-s − 0.0744·14-s − 1.98·15-s + 0.994·16-s − 1.23·17-s + 0.0926·18-s + 0.784·19-s + 1.10·20-s − 3.13·21-s − 0.0127·22-s + 0.196·23-s − 0.151·24-s + 0.231·25-s − 0.0175·26-s + 2.11·27-s + 1.75·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 11 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(14-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11 ^{s/2} \, \Gamma_{\C}(s+13/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 11 | \( 1 + 1.77e6T \) |
| good | 2 | \( 1 - 3.83T + 8.19e3T^{2} \) |
| 3 | \( 1 - 2.25e3T + 1.59e6T^{2} \) |
| 5 | \( 1 + 3.87e4T + 1.22e9T^{2} \) |
| 7 | \( 1 + 5.47e5T + 9.68e10T^{2} \) |
| 13 | \( 1 + 7.21e6T + 3.02e14T^{2} \) |
| 17 | \( 1 + 1.23e8T + 9.90e15T^{2} \) |
| 19 | \( 1 - 1.60e8T + 4.20e16T^{2} \) |
| 23 | \( 1 - 1.39e8T + 5.04e17T^{2} \) |
| 29 | \( 1 + 5.47e8T + 1.02e19T^{2} \) |
| 31 | \( 1 + 5.05e9T + 2.44e19T^{2} \) |
| 37 | \( 1 - 5.97e9T + 2.43e20T^{2} \) |
| 41 | \( 1 + 8.53e9T + 9.25e20T^{2} \) |
| 43 | \( 1 - 2.90e10T + 1.71e21T^{2} \) |
| 47 | \( 1 + 1.01e11T + 5.46e21T^{2} \) |
| 53 | \( 1 - 2.46e11T + 2.60e22T^{2} \) |
| 59 | \( 1 + 2.51e11T + 1.04e23T^{2} \) |
| 61 | \( 1 + 3.46e11T + 1.61e23T^{2} \) |
| 67 | \( 1 - 5.98e11T + 5.48e23T^{2} \) |
| 71 | \( 1 - 2.96e11T + 1.16e24T^{2} \) |
| 73 | \( 1 + 2.52e12T + 1.67e24T^{2} \) |
| 79 | \( 1 + 1.34e12T + 4.66e24T^{2} \) |
| 83 | \( 1 + 2.88e12T + 8.87e24T^{2} \) |
| 89 | \( 1 - 7.16e11T + 2.19e25T^{2} \) |
| 97 | \( 1 + 2.38e12T + 6.73e25T^{2} \) |
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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
−16.04615629132939190864417407745, −15.03327629626478231283535377201, −13.56093606830744295851770065536, −12.73159432420794729568053575941, −9.758975601402302747241374336491, −8.804482240001268229157234513930, −7.38404150983480921923092180577, −4.04555556760300556346583373584, −3.03482525132290781034712952445, 0,
3.03482525132290781034712952445, 4.04555556760300556346583373584, 7.38404150983480921923092180577, 8.804482240001268229157234513930, 9.758975601402302747241374336491, 12.73159432420794729568053575941, 13.56093606830744295851770065536, 15.03327629626478231283535377201, 16.04615629132939190864417407745
