| L(s) = 1 | + 43.1·2-s − 694.·3-s − 3.09e4·4-s + 5.25e4·5-s − 2.99e4·6-s + 2.43e6·7-s − 2.74e6·8-s − 1.38e7·9-s + 2.26e6·10-s − 1.14e7·11-s + 2.14e7·12-s + 3.67e8·13-s + 1.05e8·14-s − 3.64e7·15-s + 8.94e8·16-s − 7.57e8·17-s − 5.98e8·18-s + 4.84e7·19-s − 1.62e9·20-s − 1.69e9·21-s − 4.94e8·22-s − 1.31e9·23-s + 1.90e9·24-s − 2.77e10·25-s + 1.58e10·26-s + 1.95e10·27-s − 7.52e10·28-s + ⋯ |
| L(s) = 1 | + 0.238·2-s − 0.183·3-s − 0.943·4-s + 0.300·5-s − 0.0437·6-s + 1.11·7-s − 0.463·8-s − 0.966·9-s + 0.0717·10-s − 0.177·11-s + 0.172·12-s + 1.62·13-s + 0.266·14-s − 0.0551·15-s + 0.832·16-s − 0.447·17-s − 0.230·18-s + 0.0124·19-s − 0.283·20-s − 0.204·21-s − 0.0422·22-s − 0.0805·23-s + 0.0849·24-s − 0.909·25-s + 0.387·26-s + 0.360·27-s − 1.05·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(16-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s+15/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(8)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{17}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 29 | \( 1 - 1.72e10T \) |
| good | 2 | \( 1 - 43.1T + 3.27e4T^{2} \) |
| 3 | \( 1 + 694.T + 1.43e7T^{2} \) |
| 5 | \( 1 - 5.25e4T + 3.05e10T^{2} \) |
| 7 | \( 1 - 2.43e6T + 4.74e12T^{2} \) |
| 11 | \( 1 + 1.14e7T + 4.17e15T^{2} \) |
| 13 | \( 1 - 3.67e8T + 5.11e16T^{2} \) |
| 17 | \( 1 + 7.57e8T + 2.86e18T^{2} \) |
| 19 | \( 1 - 4.84e7T + 1.51e19T^{2} \) |
| 23 | \( 1 + 1.31e9T + 2.66e20T^{2} \) |
| 31 | \( 1 + 2.04e11T + 2.34e22T^{2} \) |
| 37 | \( 1 + 2.18e11T + 3.33e23T^{2} \) |
| 41 | \( 1 + 2.07e12T + 1.55e24T^{2} \) |
| 43 | \( 1 + 2.63e12T + 3.17e24T^{2} \) |
| 47 | \( 1 - 2.71e12T + 1.20e25T^{2} \) |
| 53 | \( 1 - 1.25e12T + 7.31e25T^{2} \) |
| 59 | \( 1 + 2.79e13T + 3.65e26T^{2} \) |
| 61 | \( 1 + 2.57e13T + 6.02e26T^{2} \) |
| 67 | \( 1 + 2.02e13T + 2.46e27T^{2} \) |
| 71 | \( 1 + 1.74e13T + 5.87e27T^{2} \) |
| 73 | \( 1 - 2.24e13T + 8.90e27T^{2} \) |
| 79 | \( 1 + 1.06e13T + 2.91e28T^{2} \) |
| 83 | \( 1 - 2.91e14T + 6.11e28T^{2} \) |
| 89 | \( 1 + 1.94e13T + 1.74e29T^{2} \) |
| 97 | \( 1 + 8.54e14T + 6.33e29T^{2} \) |
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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
−13.39110338927418247446635406917, −11.78444174727482631539651504264, −10.69534368669706302738667293535, −8.974013870732748610803823356804, −8.162556751100513672494724767761, −5.99928454729764559570878430496, −4.98213347443858887455877270768, −3.54790746928865192557930009995, −1.58603747267720177768843128604, 0,
1.58603747267720177768843128604, 3.54790746928865192557930009995, 4.98213347443858887455877270768, 5.99928454729764559570878430496, 8.162556751100513672494724767761, 8.974013870732748610803823356804, 10.69534368669706302738667293535, 11.78444174727482631539651504264, 13.39110338927418247446635406917
