L(s) = 1 | + 4.50·2-s − 3·3-s + 12.2·4-s + 17.0·5-s − 13.5·6-s + 15.6·7-s + 19.1·8-s + 9·9-s + 76.6·10-s + 68.3·11-s − 36.7·12-s − 72.1·13-s + 70.5·14-s − 51.0·15-s − 11.9·16-s − 87.4·17-s + 40.5·18-s + 105.·19-s + 208.·20-s − 47.0·21-s + 307.·22-s + 74.4·23-s − 57.4·24-s + 164.·25-s − 324.·26-s − 27·27-s + 192.·28-s + ⋯ |
L(s) = 1 | + 1.59·2-s − 0.577·3-s + 1.53·4-s + 1.52·5-s − 0.918·6-s + 0.846·7-s + 0.845·8-s + 0.333·9-s + 2.42·10-s + 1.87·11-s − 0.884·12-s − 1.53·13-s + 1.34·14-s − 0.878·15-s − 0.186·16-s − 1.24·17-s + 0.530·18-s + 1.27·19-s + 2.33·20-s − 0.488·21-s + 2.98·22-s + 0.675·23-s − 0.488·24-s + 1.31·25-s − 2.45·26-s − 0.192·27-s + 1.29·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 471 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 471 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(5.787339822\) |
\(L(\frac12)\) |
\(\approx\) |
\(5.787339822\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + 3T \) |
| 157 | \( 1 + 157T \) |
good | 2 | \( 1 - 4.50T + 8T^{2} \) |
| 5 | \( 1 - 17.0T + 125T^{2} \) |
| 7 | \( 1 - 15.6T + 343T^{2} \) |
| 11 | \( 1 - 68.3T + 1.33e3T^{2} \) |
| 13 | \( 1 + 72.1T + 2.19e3T^{2} \) |
| 17 | \( 1 + 87.4T + 4.91e3T^{2} \) |
| 19 | \( 1 - 105.T + 6.85e3T^{2} \) |
| 23 | \( 1 - 74.4T + 1.21e4T^{2} \) |
| 29 | \( 1 + 209.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 85.9T + 2.97e4T^{2} \) |
| 37 | \( 1 - 361.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 301.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 295.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 369.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 474.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 22.1T + 2.05e5T^{2} \) |
| 61 | \( 1 - 765.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 18.4T + 3.00e5T^{2} \) |
| 71 | \( 1 + 314.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 935.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 903.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 270.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.28e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 195.T + 9.12e5T^{2} \) |
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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
−11.18596946212448669505684405513, −9.643615075914290981663500639576, −9.220652674773136784405123336874, −7.25236580460699722133170833251, −6.53536858719859584109477242540, −5.64857015797604183909940584286, −4.97928509667526153209688110443, −4.11575256825564312960747708846, −2.50875232655755050656068902022, −1.50760179566963151956423050757,
1.50760179566963151956423050757, 2.50875232655755050656068902022, 4.11575256825564312960747708846, 4.97928509667526153209688110443, 5.64857015797604183909940584286, 6.53536858719859584109477242540, 7.25236580460699722133170833251, 9.220652674773136784405123336874, 9.643615075914290981663500639576, 11.18596946212448669505684405513