| L(s) = 1 | + 1.55·2-s − 3-s + 0.414·4-s + 5-s − 1.55·6-s + 3.94·7-s − 2.46·8-s + 9-s + 1.55·10-s − 1.46·11-s − 0.414·12-s − 4.70·13-s + 6.12·14-s − 15-s − 4.65·16-s + 4.00·17-s + 1.55·18-s − 1.13·19-s + 0.414·20-s − 3.94·21-s − 2.26·22-s + 4.26·23-s + 2.46·24-s + 25-s − 7.31·26-s − 27-s + 1.63·28-s + ⋯ |
| L(s) = 1 | + 1.09·2-s − 0.577·3-s + 0.207·4-s + 0.447·5-s − 0.634·6-s + 1.48·7-s − 0.870·8-s + 0.333·9-s + 0.491·10-s − 0.440·11-s − 0.119·12-s − 1.30·13-s + 1.63·14-s − 0.258·15-s − 1.16·16-s + 0.971·17-s + 0.366·18-s − 0.260·19-s + 0.0927·20-s − 0.860·21-s − 0.483·22-s + 0.889·23-s + 0.502·24-s + 0.200·25-s − 1.43·26-s − 0.192·27-s + 0.309·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6015 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.101456276\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.101456276\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 401 | \( 1 + T \) |
| good | 2 | \( 1 - 1.55T + 2T^{2} \) |
| 7 | \( 1 - 3.94T + 7T^{2} \) |
| 11 | \( 1 + 1.46T + 11T^{2} \) |
| 13 | \( 1 + 4.70T + 13T^{2} \) |
| 17 | \( 1 - 4.00T + 17T^{2} \) |
| 19 | \( 1 + 1.13T + 19T^{2} \) |
| 23 | \( 1 - 4.26T + 23T^{2} \) |
| 29 | \( 1 + 2.71T + 29T^{2} \) |
| 31 | \( 1 + 3.79T + 31T^{2} \) |
| 37 | \( 1 - 3.33T + 37T^{2} \) |
| 41 | \( 1 - 8.91T + 41T^{2} \) |
| 43 | \( 1 - 7.86T + 43T^{2} \) |
| 47 | \( 1 - 7.04T + 47T^{2} \) |
| 53 | \( 1 + 0.667T + 53T^{2} \) |
| 59 | \( 1 + 5.36T + 59T^{2} \) |
| 61 | \( 1 - 8.15T + 61T^{2} \) |
| 67 | \( 1 - 8.69T + 67T^{2} \) |
| 71 | \( 1 + 1.33T + 71T^{2} \) |
| 73 | \( 1 + 0.419T + 73T^{2} \) |
| 79 | \( 1 - 5.20T + 79T^{2} \) |
| 83 | \( 1 + 10.0T + 83T^{2} \) |
| 89 | \( 1 - 14.3T + 89T^{2} \) |
| 97 | \( 1 - 9.11T + 97T^{2} \) |
| 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
−7.72032255930459621602634591663, −7.45202304834888712200446502625, −6.38558424673170244879282713436, −5.45882455485205605549067442573, −5.32979039800878710134422683285, −4.61979636207404457881729419722, −3.97895306072619836979601404414, −2.78786728578415330431406268194, −2.07954495047521559094404628509, −0.814198595363353127520760516680,
0.814198595363353127520760516680, 2.07954495047521559094404628509, 2.78786728578415330431406268194, 3.97895306072619836979601404414, 4.61979636207404457881729419722, 5.32979039800878710134422683285, 5.45882455485205605549067442573, 6.38558424673170244879282713436, 7.45202304834888712200446502625, 7.72032255930459621602634591663
