| L(s) = 1 | − 124.·2-s − 1.75e3·3-s + 7.43e3·4-s + 2.73e4·5-s + 2.18e5·6-s + 7.52e4·7-s + 9.49e4·8-s + 1.46e6·9-s − 3.42e6·10-s − 1.77e6·11-s − 1.30e7·12-s + 1.16e7·13-s − 9.40e6·14-s − 4.79e7·15-s − 7.27e7·16-s − 1.25e8·17-s − 1.83e8·18-s + 2.18e8·19-s + 2.03e8·20-s − 1.31e8·21-s + 2.21e8·22-s + 9.99e8·23-s − 1.66e8·24-s − 4.71e8·25-s − 1.45e9·26-s + 2.18e8·27-s + 5.58e8·28-s + ⋯ |
| L(s) = 1 | − 1.38·2-s − 1.38·3-s + 0.907·4-s + 0.783·5-s + 1.91·6-s + 0.241·7-s + 0.128·8-s + 0.921·9-s − 1.08·10-s − 0.301·11-s − 1.25·12-s + 0.671·13-s − 0.333·14-s − 1.08·15-s − 1.08·16-s − 1.26·17-s − 1.27·18-s + 1.06·19-s + 0.710·20-s − 0.334·21-s + 0.416·22-s + 1.40·23-s − 0.177·24-s − 0.386·25-s − 0.926·26-s + 0.108·27-s + 0.219·28-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 | \( 1 + 124.T + 8.19e3T^{2} \) |
| 3 | \( 1 + 1.75e3T + 1.59e6T^{2} \) |
| 5 | \( 1 - 2.73e4T + 1.22e9T^{2} \) |
| 7 | \( 1 - 7.52e4T + 9.68e10T^{2} \) |
| 13 | \( 1 - 1.16e7T + 3.02e14T^{2} \) |
| 17 | \( 1 + 1.25e8T + 9.90e15T^{2} \) |
| 19 | \( 1 - 2.18e8T + 4.20e16T^{2} \) |
| 23 | \( 1 - 9.99e8T + 5.04e17T^{2} \) |
| 29 | \( 1 + 1.29e9T + 1.02e19T^{2} \) |
| 31 | \( 1 + 6.48e9T + 2.44e19T^{2} \) |
| 37 | \( 1 + 2.98e10T + 2.43e20T^{2} \) |
| 41 | \( 1 + 2.41e10T + 9.25e20T^{2} \) |
| 43 | \( 1 + 3.77e10T + 1.71e21T^{2} \) |
| 47 | \( 1 - 8.20e10T + 5.46e21T^{2} \) |
| 53 | \( 1 - 1.37e11T + 2.60e22T^{2} \) |
| 59 | \( 1 + 2.62e11T + 1.04e23T^{2} \) |
| 61 | \( 1 - 7.10e10T + 1.61e23T^{2} \) |
| 67 | \( 1 - 8.31e11T + 5.48e23T^{2} \) |
| 71 | \( 1 + 1.74e11T + 1.16e24T^{2} \) |
| 73 | \( 1 + 2.21e12T + 1.67e24T^{2} \) |
| 79 | \( 1 + 3.44e12T + 4.66e24T^{2} \) |
| 83 | \( 1 - 1.50e12T + 8.87e24T^{2} \) |
| 89 | \( 1 - 2.74e12T + 2.19e25T^{2} \) |
| 97 | \( 1 - 4.96e12T + 6.73e25T^{2} \) |
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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
−17.08042722846052150931371502251, −15.87705577122075826912949738202, −13.39347161859646720962417068972, −11.38918322214334395898317332921, −10.45334061861574183526918956826, −8.968669119891794363502089843787, −6.93469958466916536339152157830, −5.29342614602785183353818983280, −1.50809579182413042561356067714, 0,
1.50809579182413042561356067714, 5.29342614602785183353818983280, 6.93469958466916536339152157830, 8.968669119891794363502089843787, 10.45334061861574183526918956826, 11.38918322214334395898317332921, 13.39347161859646720962417068972, 15.87705577122075826912949738202, 17.08042722846052150931371502251
