| L(s) = 1 | + 2.54·2-s − 3-s + 4.48·4-s − 2.54·6-s − 4.63·7-s + 6.31·8-s + 9-s + 4.61·11-s − 4.48·12-s − 0.863·13-s − 11.7·14-s + 7.11·16-s + 6.02·17-s + 2.54·18-s + 7.67·19-s + 4.63·21-s + 11.7·22-s − 3.18·23-s − 6.31·24-s − 2.19·26-s − 27-s − 20.7·28-s + 0.501·29-s + 31-s + 5.47·32-s − 4.61·33-s + 15.3·34-s + ⋯ |
| L(s) = 1 | + 1.80·2-s − 0.577·3-s + 2.24·4-s − 1.03·6-s − 1.75·7-s + 2.23·8-s + 0.333·9-s + 1.39·11-s − 1.29·12-s − 0.239·13-s − 3.15·14-s + 1.77·16-s + 1.46·17-s + 0.600·18-s + 1.76·19-s + 1.01·21-s + 2.50·22-s − 0.663·23-s − 1.28·24-s − 0.431·26-s − 0.192·27-s − 3.92·28-s + 0.0930·29-s + 0.179·31-s + 0.968·32-s − 0.803·33-s + 2.63·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2325 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2325 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.369058235\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.369058235\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| 31 | \( 1 - T \) |
| good | 2 | \( 1 - 2.54T + 2T^{2} \) |
| 7 | \( 1 + 4.63T + 7T^{2} \) |
| 11 | \( 1 - 4.61T + 11T^{2} \) |
| 13 | \( 1 + 0.863T + 13T^{2} \) |
| 17 | \( 1 - 6.02T + 17T^{2} \) |
| 19 | \( 1 - 7.67T + 19T^{2} \) |
| 23 | \( 1 + 3.18T + 23T^{2} \) |
| 29 | \( 1 - 0.501T + 29T^{2} \) |
| 37 | \( 1 - 3.39T + 37T^{2} \) |
| 41 | \( 1 - 4.36T + 41T^{2} \) |
| 43 | \( 1 - 0.151T + 43T^{2} \) |
| 47 | \( 1 - 6.57T + 47T^{2} \) |
| 53 | \( 1 + 12.0T + 53T^{2} \) |
| 59 | \( 1 - 5.62T + 59T^{2} \) |
| 61 | \( 1 + 4.69T + 61T^{2} \) |
| 67 | \( 1 - 13.5T + 67T^{2} \) |
| 71 | \( 1 + 10.3T + 71T^{2} \) |
| 73 | \( 1 - 6.13T + 73T^{2} \) |
| 79 | \( 1 - 15.0T + 79T^{2} \) |
| 83 | \( 1 - 6.90T + 83T^{2} \) |
| 89 | \( 1 + 10.9T + 89T^{2} \) |
| 97 | \( 1 + 10.5T + 97T^{2} \) |
| 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
−9.493480841136095646522036182252, −7.73450359307080062415041347934, −6.99429640203123113387075684112, −6.35132050785193554583900016116, −5.84406926600834241957182668608, −5.14132831141110336828118935497, −3.98148021789484049501809843588, −3.51918195155553988758375159060, −2.72856241436799944906867876262, −1.13631072852140083701705421931,
1.13631072852140083701705421931, 2.72856241436799944906867876262, 3.51918195155553988758375159060, 3.98148021789484049501809843588, 5.14132831141110336828118935497, 5.84406926600834241957182668608, 6.35132050785193554583900016116, 6.99429640203123113387075684112, 7.73450359307080062415041347934, 9.493480841136095646522036182252
