| L(s) = 1 | + 1.92·2-s − 3-s + 1.69·4-s + 2.85·5-s − 1.92·6-s − 7-s − 0.584·8-s + 9-s + 5.49·10-s + 3.16·11-s − 1.69·12-s + 4.79·13-s − 1.92·14-s − 2.85·15-s − 4.51·16-s − 7.60·17-s + 1.92·18-s + 7.66·19-s + 4.84·20-s + 21-s + 6.07·22-s + 7.15·23-s + 0.584·24-s + 3.16·25-s + 9.22·26-s − 27-s − 1.69·28-s + ⋯ |
| L(s) = 1 | + 1.35·2-s − 0.577·3-s + 0.848·4-s + 1.27·5-s − 0.784·6-s − 0.377·7-s − 0.206·8-s + 0.333·9-s + 1.73·10-s + 0.952·11-s − 0.489·12-s + 1.33·13-s − 0.513·14-s − 0.737·15-s − 1.12·16-s − 1.84·17-s + 0.453·18-s + 1.75·19-s + 1.08·20-s + 0.218·21-s + 1.29·22-s + 1.49·23-s + 0.119·24-s + 0.632·25-s + 1.80·26-s − 0.192·27-s − 0.320·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 861 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 861 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.207813478\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.207813478\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 41 | \( 1 - T \) |
| good | 2 | \( 1 - 1.92T + 2T^{2} \) |
| 5 | \( 1 - 2.85T + 5T^{2} \) |
| 11 | \( 1 - 3.16T + 11T^{2} \) |
| 13 | \( 1 - 4.79T + 13T^{2} \) |
| 17 | \( 1 + 7.60T + 17T^{2} \) |
| 19 | \( 1 - 7.66T + 19T^{2} \) |
| 23 | \( 1 - 7.15T + 23T^{2} \) |
| 29 | \( 1 - 4.61T + 29T^{2} \) |
| 31 | \( 1 + 1.05T + 31T^{2} \) |
| 37 | \( 1 + 9.40T + 37T^{2} \) |
| 43 | \( 1 + 9.58T + 43T^{2} \) |
| 47 | \( 1 - 9.47T + 47T^{2} \) |
| 53 | \( 1 + 10.3T + 53T^{2} \) |
| 59 | \( 1 - 1.88T + 59T^{2} \) |
| 61 | \( 1 - 10.6T + 61T^{2} \) |
| 67 | \( 1 + 1.82T + 67T^{2} \) |
| 71 | \( 1 + 4.82T + 71T^{2} \) |
| 73 | \( 1 + 8.67T + 73T^{2} \) |
| 79 | \( 1 + 2.54T + 79T^{2} \) |
| 83 | \( 1 + 2.72T + 83T^{2} \) |
| 89 | \( 1 + 3.03T + 89T^{2} \) |
| 97 | \( 1 - 10.4T + 97T^{2} \) |
| 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
−10.35514137924454897260145643173, −9.230085336083084548147429792244, −8.872013143091754155604506959648, −6.80344270688411647433214734166, −6.57235496287069193760176221226, −5.65583765661268358895019467585, −5.00855085654481737087368489819, −3.92390019623929265558280167471, −2.91236529834929583523984320634, −1.45549726046569311120119494402,
1.45549726046569311120119494402, 2.91236529834929583523984320634, 3.92390019623929265558280167471, 5.00855085654481737087368489819, 5.65583765661268358895019467585, 6.57235496287069193760176221226, 6.80344270688411647433214734166, 8.872013143091754155604506959648, 9.230085336083084548147429792244, 10.35514137924454897260145643173
