| L(s) = 1 | + 238.·2-s + 2.50e3·3-s + 2.39e4·4-s − 2.10e5·5-s + 5.96e5·6-s + 1.58e6·7-s − 2.10e6·8-s − 8.08e6·9-s − 5.01e7·10-s + 2.79e7·11-s + 5.99e7·12-s + 3.45e7·13-s + 3.78e8·14-s − 5.27e8·15-s − 1.28e9·16-s − 4.80e8·17-s − 1.92e9·18-s − 6.32e9·19-s − 5.04e9·20-s + 3.97e9·21-s + 6.66e9·22-s − 2.01e10·23-s − 5.25e9·24-s + 1.38e10·25-s + 8.23e9·26-s − 5.61e10·27-s + 3.80e10·28-s + ⋯ |
| L(s) = 1 | + 1.31·2-s + 0.660·3-s + 0.730·4-s − 1.20·5-s + 0.869·6-s + 0.728·7-s − 0.354·8-s − 0.563·9-s − 1.58·10-s + 0.432·11-s + 0.482·12-s + 0.152·13-s + 0.958·14-s − 0.796·15-s − 1.19·16-s − 0.283·17-s − 0.741·18-s − 1.62·19-s − 0.881·20-s + 0.481·21-s + 0.569·22-s − 1.23·23-s − 0.234·24-s + 0.453·25-s + 0.201·26-s − 1.03·27-s + 0.532·28-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 | \( 1 - 238.T + 3.27e4T^{2} \) |
| 3 | \( 1 - 2.50e3T + 1.43e7T^{2} \) |
| 5 | \( 1 + 2.10e5T + 3.05e10T^{2} \) |
| 7 | \( 1 - 1.58e6T + 4.74e12T^{2} \) |
| 11 | \( 1 - 2.79e7T + 4.17e15T^{2} \) |
| 13 | \( 1 - 3.45e7T + 5.11e16T^{2} \) |
| 17 | \( 1 + 4.80e8T + 2.86e18T^{2} \) |
| 19 | \( 1 + 6.32e9T + 1.51e19T^{2} \) |
| 23 | \( 1 + 2.01e10T + 2.66e20T^{2} \) |
| 31 | \( 1 + 1.24e11T + 2.34e22T^{2} \) |
| 37 | \( 1 - 7.71e11T + 3.33e23T^{2} \) |
| 41 | \( 1 - 8.80e11T + 1.55e24T^{2} \) |
| 43 | \( 1 - 1.75e12T + 3.17e24T^{2} \) |
| 47 | \( 1 + 3.79e12T + 1.20e25T^{2} \) |
| 53 | \( 1 - 6.67e12T + 7.31e25T^{2} \) |
| 59 | \( 1 - 1.01e13T + 3.65e26T^{2} \) |
| 61 | \( 1 - 7.05e11T + 6.02e26T^{2} \) |
| 67 | \( 1 - 2.02e12T + 2.46e27T^{2} \) |
| 71 | \( 1 + 1.16e14T + 5.87e27T^{2} \) |
| 73 | \( 1 + 1.66e14T + 8.90e27T^{2} \) |
| 79 | \( 1 - 1.77e14T + 2.91e28T^{2} \) |
| 83 | \( 1 - 3.58e14T + 6.11e28T^{2} \) |
| 89 | \( 1 - 4.84e14T + 1.74e29T^{2} \) |
| 97 | \( 1 - 5.88e14T + 6.33e29T^{2} \) |
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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
−13.21742891599631016434558320264, −11.98094185545148566681832577558, −11.15397605576095865328129249433, −8.854978979933887890502068503774, −7.84848910448213648775957595409, −6.10240151493451975549230888260, −4.44909389363120911906073033612, −3.71822178188455105275964397510, −2.29484429622119934708450768141, 0,
2.29484429622119934708450768141, 3.71822178188455105275964397510, 4.44909389363120911906073033612, 6.10240151493451975549230888260, 7.84848910448213648775957595409, 8.854978979933887890502068503774, 11.15397605576095865328129249433, 11.98094185545148566681832577558, 13.21742891599631016434558320264
