| L(s) = 1 | − 8·2-s + 77.5·3-s + 64·4-s − 471.·5-s − 620.·6-s − 401.·7-s − 512·8-s + 3.82e3·9-s + 3.77e3·10-s + 2.59e3·11-s + 4.96e3·12-s + 1.35e4·13-s + 3.20e3·14-s − 3.65e4·15-s + 4.09e3·16-s + 1.19e4·17-s − 3.05e4·18-s + 2.67e4·19-s − 3.01e4·20-s − 3.10e4·21-s − 2.07e4·22-s − 1.58e4·23-s − 3.96e4·24-s + 1.44e5·25-s − 1.08e5·26-s + 1.26e5·27-s − 2.56e4·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.65·3-s + 0.5·4-s − 1.68·5-s − 1.17·6-s − 0.441·7-s − 0.353·8-s + 1.74·9-s + 1.19·10-s + 0.586·11-s + 0.828·12-s + 1.70·13-s + 0.312·14-s − 2.79·15-s + 0.250·16-s + 0.590·17-s − 1.23·18-s + 0.895·19-s − 0.843·20-s − 0.732·21-s − 0.414·22-s − 0.272·23-s − 0.585·24-s + 1.84·25-s − 1.20·26-s + 1.23·27-s − 0.220·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 62 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 62 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(1.958058148\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.958058148\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 8T \) |
| 31 | \( 1 + 2.97e4T \) |
| good | 3 | \( 1 - 77.5T + 2.18e3T^{2} \) |
| 5 | \( 1 + 471.T + 7.81e4T^{2} \) |
| 7 | \( 1 + 401.T + 8.23e5T^{2} \) |
| 11 | \( 1 - 2.59e3T + 1.94e7T^{2} \) |
| 13 | \( 1 - 1.35e4T + 6.27e7T^{2} \) |
| 17 | \( 1 - 1.19e4T + 4.10e8T^{2} \) |
| 19 | \( 1 - 2.67e4T + 8.93e8T^{2} \) |
| 23 | \( 1 + 1.58e4T + 3.40e9T^{2} \) |
| 29 | \( 1 - 2.09e5T + 1.72e10T^{2} \) |
| 37 | \( 1 - 2.62e4T + 9.49e10T^{2} \) |
| 41 | \( 1 + 4.77e5T + 1.94e11T^{2} \) |
| 43 | \( 1 - 9.44e5T + 2.71e11T^{2} \) |
| 47 | \( 1 + 1.22e6T + 5.06e11T^{2} \) |
| 53 | \( 1 - 3.90e5T + 1.17e12T^{2} \) |
| 59 | \( 1 - 2.25e6T + 2.48e12T^{2} \) |
| 61 | \( 1 + 4.04e5T + 3.14e12T^{2} \) |
| 67 | \( 1 - 2.32e6T + 6.06e12T^{2} \) |
| 71 | \( 1 + 7.76e5T + 9.09e12T^{2} \) |
| 73 | \( 1 - 5.50e6T + 1.10e13T^{2} \) |
| 79 | \( 1 - 1.48e6T + 1.92e13T^{2} \) |
| 83 | \( 1 + 2.04e6T + 2.71e13T^{2} \) |
| 89 | \( 1 + 2.19e6T + 4.42e13T^{2} \) |
| 97 | \( 1 + 1.71e7T + 8.07e13T^{2} \) |
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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
−13.73671737646815897867589990354, −12.34334245086670939133336574122, −11.19745481170308546562347516970, −9.698956004120989844600033876347, −8.515307687427896070573862216489, −8.053009029877443457917126173521, −6.83764982337566587207023323332, −3.87307327377466591970747836605, −3.15777117520131615378624195888, −1.07289814588735869390369453757,
1.07289814588735869390369453757, 3.15777117520131615378624195888, 3.87307327377466591970747836605, 6.83764982337566587207023323332, 8.053009029877443457917126173521, 8.515307687427896070573862216489, 9.698956004120989844600033876347, 11.19745481170308546562347516970, 12.34334245086670939133336574122, 13.73671737646815897867589990354
