| L(s) = 1 | − 1.30·2-s − 0.302·4-s + 5-s + 7-s + 3·8-s − 1.30·10-s + 3·11-s − 2.30·13-s − 1.30·14-s − 3.30·16-s + 1.30·17-s + 0.302·19-s − 0.302·20-s − 3.90·22-s + 4.30·23-s + 25-s + 3·26-s − 0.302·28-s + 1.69·29-s − 9.60·31-s − 1.69·32-s − 1.69·34-s + 35-s − 3.60·37-s − 0.394·38-s + 3·40-s + 3.90·41-s + ⋯ |
| L(s) = 1 | − 0.921·2-s − 0.151·4-s + 0.447·5-s + 0.377·7-s + 1.06·8-s − 0.411·10-s + 0.904·11-s − 0.638·13-s − 0.348·14-s − 0.825·16-s + 0.315·17-s + 0.0694·19-s − 0.0677·20-s − 0.833·22-s + 0.897·23-s + 0.200·25-s + 0.588·26-s − 0.0572·28-s + 0.315·29-s − 1.72·31-s − 0.300·32-s − 0.291·34-s + 0.169·35-s − 0.592·37-s − 0.0639·38-s + 0.474·40-s + 0.610·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 945 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 945 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.033869831\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.033869831\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| good | 2 | \( 1 + 1.30T + 2T^{2} \) |
| 11 | \( 1 - 3T + 11T^{2} \) |
| 13 | \( 1 + 2.30T + 13T^{2} \) |
| 17 | \( 1 - 1.30T + 17T^{2} \) |
| 19 | \( 1 - 0.302T + 19T^{2} \) |
| 23 | \( 1 - 4.30T + 23T^{2} \) |
| 29 | \( 1 - 1.69T + 29T^{2} \) |
| 31 | \( 1 + 9.60T + 31T^{2} \) |
| 37 | \( 1 + 3.60T + 37T^{2} \) |
| 41 | \( 1 - 3.90T + 41T^{2} \) |
| 43 | \( 1 - 10.2T + 43T^{2} \) |
| 47 | \( 1 - 0.394T + 47T^{2} \) |
| 53 | \( 1 - 12.5T + 53T^{2} \) |
| 59 | \( 1 + 8.21T + 59T^{2} \) |
| 61 | \( 1 - 14.1T + 61T^{2} \) |
| 67 | \( 1 + 4.51T + 67T^{2} \) |
| 71 | \( 1 - 12.9T + 71T^{2} \) |
| 73 | \( 1 - 4.21T + 73T^{2} \) |
| 79 | \( 1 + 6.09T + 79T^{2} \) |
| 83 | \( 1 - 2.21T + 83T^{2} \) |
| 89 | \( 1 + 10.8T + 89T^{2} \) |
| 97 | \( 1 - 8.51T + 97T^{2} \) |
| 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
−9.861698593090997841796095439340, −9.157573622441697866832827370646, −8.671805266307598352259911950727, −7.54840750603366428197244184178, −6.99399019176353753734224880731, −5.69420788193719833106852752689, −4.80938751146346007031929738094, −3.76206591111235032074363160548, −2.16773271275343725080656488029, −0.978012379483523488823020879447,
0.978012379483523488823020879447, 2.16773271275343725080656488029, 3.76206591111235032074363160548, 4.80938751146346007031929738094, 5.69420788193719833106852752689, 6.99399019176353753734224880731, 7.54840750603366428197244184178, 8.671805266307598352259911950727, 9.157573622441697866832827370646, 9.861698593090997841796095439340
