| L(s) = 1 | + 2.35·2-s − 1.20·3-s − 2.45·4-s + 5·5-s − 2.83·6-s + 20.7·7-s − 24.6·8-s − 25.5·9-s + 11.7·10-s + 11·11-s + 2.95·12-s − 64.1·13-s + 48.8·14-s − 6.02·15-s − 38.3·16-s + 41.0·17-s − 60.1·18-s − 19·19-s − 12.2·20-s − 25.0·21-s + 25.9·22-s + 122.·23-s + 29.6·24-s + 25·25-s − 151.·26-s + 63.3·27-s − 50.9·28-s + ⋯ |
| L(s) = 1 | + 0.832·2-s − 0.232·3-s − 0.306·4-s + 0.447·5-s − 0.193·6-s + 1.12·7-s − 1.08·8-s − 0.946·9-s + 0.372·10-s + 0.301·11-s + 0.0712·12-s − 1.36·13-s + 0.933·14-s − 0.103·15-s − 0.599·16-s + 0.585·17-s − 0.787·18-s − 0.229·19-s − 0.137·20-s − 0.260·21-s + 0.251·22-s + 1.10·23-s + 0.252·24-s + 0.200·25-s − 1.13·26-s + 0.451·27-s − 0.343·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.613956135\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.613956135\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 - 5T \) |
| 11 | \( 1 - 11T \) |
| 19 | \( 1 + 19T \) |
| good | 2 | \( 1 - 2.35T + 8T^{2} \) |
| 3 | \( 1 + 1.20T + 27T^{2} \) |
| 7 | \( 1 - 20.7T + 343T^{2} \) |
| 13 | \( 1 + 64.1T + 2.19e3T^{2} \) |
| 17 | \( 1 - 41.0T + 4.91e3T^{2} \) |
| 23 | \( 1 - 122.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 194.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 41.0T + 2.97e4T^{2} \) |
| 37 | \( 1 - 179.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 57.1T + 6.89e4T^{2} \) |
| 43 | \( 1 + 149.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 81.0T + 1.03e5T^{2} \) |
| 53 | \( 1 + 47.4T + 1.48e5T^{2} \) |
| 59 | \( 1 - 338.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 345.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 412.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 1.08e3T + 3.57e5T^{2} \) |
| 73 | \( 1 - 98.3T + 3.89e5T^{2} \) |
| 79 | \( 1 - 1.18e3T + 4.93e5T^{2} \) |
| 83 | \( 1 - 672.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 187.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 544.T + 9.12e5T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.515766077986292970691417948312, −8.692600830553736618802917294596, −7.971145792297742194571809130703, −6.78373308357984944680666363876, −5.84465691047977424850792843386, −4.99265192576979071220733061636, −4.69103473681941461138630542218, −3.25904306175143552767256704947, −2.32902698386702909885720800766, −0.76304875380864664611197449971,
0.76304875380864664611197449971, 2.32902698386702909885720800766, 3.25904306175143552767256704947, 4.69103473681941461138630542218, 4.99265192576979071220733061636, 5.84465691047977424850792843386, 6.78373308357984944680666363876, 7.971145792297742194571809130703, 8.692600830553736618802917294596, 9.515766077986292970691417948312
