| L(s) = 1 | + 263.·2-s − 7.37e3·3-s + 3.67e4·4-s + 1.91e5·5-s − 1.94e6·6-s − 9.06e5·7-s + 1.04e6·8-s + 4.00e7·9-s + 5.03e7·10-s − 3.10e7·11-s − 2.71e8·12-s + 2.22e8·13-s − 2.39e8·14-s − 1.40e9·15-s − 9.27e8·16-s − 5.52e8·17-s + 1.05e10·18-s + 3.42e9·19-s + 7.01e9·20-s + 6.68e9·21-s − 8.17e9·22-s − 2.92e10·23-s − 7.73e9·24-s + 5.96e9·25-s + 5.86e10·26-s − 1.89e11·27-s − 3.33e10·28-s + ⋯ |
| L(s) = 1 | + 1.45·2-s − 1.94·3-s + 1.12·4-s + 1.09·5-s − 2.83·6-s − 0.416·7-s + 0.176·8-s + 2.79·9-s + 1.59·10-s − 0.479·11-s − 2.18·12-s + 0.982·13-s − 0.606·14-s − 2.12·15-s − 0.863·16-s − 0.326·17-s + 4.06·18-s + 0.880·19-s + 1.22·20-s + 0.810·21-s − 0.698·22-s − 1.79·23-s − 0.344·24-s + 0.195·25-s + 1.43·26-s − 3.48·27-s − 0.466·28-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 | \( 1 - 263.T + 3.27e4T^{2} \) |
| 3 | \( 1 + 7.37e3T + 1.43e7T^{2} \) |
| 5 | \( 1 - 1.91e5T + 3.05e10T^{2} \) |
| 7 | \( 1 + 9.06e5T + 4.74e12T^{2} \) |
| 11 | \( 1 + 3.10e7T + 4.17e15T^{2} \) |
| 13 | \( 1 - 2.22e8T + 5.11e16T^{2} \) |
| 17 | \( 1 + 5.52e8T + 2.86e18T^{2} \) |
| 19 | \( 1 - 3.42e9T + 1.51e19T^{2} \) |
| 23 | \( 1 + 2.92e10T + 2.66e20T^{2} \) |
| 31 | \( 1 + 2.60e11T + 2.34e22T^{2} \) |
| 37 | \( 1 + 7.43e11T + 3.33e23T^{2} \) |
| 41 | \( 1 + 4.05e11T + 1.55e24T^{2} \) |
| 43 | \( 1 - 1.92e11T + 3.17e24T^{2} \) |
| 47 | \( 1 + 1.47e11T + 1.20e25T^{2} \) |
| 53 | \( 1 - 7.47e12T + 7.31e25T^{2} \) |
| 59 | \( 1 - 2.62e12T + 3.65e26T^{2} \) |
| 61 | \( 1 - 3.24e12T + 6.02e26T^{2} \) |
| 67 | \( 1 - 6.15e13T + 2.46e27T^{2} \) |
| 71 | \( 1 + 7.16e13T + 5.87e27T^{2} \) |
| 73 | \( 1 - 1.02e14T + 8.90e27T^{2} \) |
| 79 | \( 1 + 2.23e14T + 2.91e28T^{2} \) |
| 83 | \( 1 - 2.57e14T + 6.11e28T^{2} \) |
| 89 | \( 1 + 5.73e14T + 1.74e29T^{2} \) |
| 97 | \( 1 + 1.10e15T + 6.33e29T^{2} \) |
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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
−13.01923235705353792644677260434, −12.07474788974917725547158132352, −10.97261894639342910643010386866, −9.815701038958533454768764972171, −6.78114724705448459226209688727, −5.82718124417471639299062474464, −5.34186826159007463754759846526, −3.88275150068296368754742163616, −1.75837404201288544568615534902, 0,
1.75837404201288544568615534902, 3.88275150068296368754742163616, 5.34186826159007463754759846526, 5.82718124417471639299062474464, 6.78114724705448459226209688727, 9.815701038958533454768764972171, 10.97261894639342910643010386866, 12.07474788974917725547158132352, 13.01923235705353792644677260434
