| L(s) = 1 | + 1.57·2-s + 0.483·4-s + 0.593·5-s − 2.68·7-s − 2.39·8-s + 0.934·10-s + 11-s + 3.82·13-s − 4.23·14-s − 4.73·16-s + 2.31·17-s − 0.611·19-s + 0.286·20-s + 1.57·22-s + 1.73·23-s − 4.64·25-s + 6.02·26-s − 1.29·28-s + 1.70·29-s − 2.23·31-s − 2.67·32-s + 3.64·34-s − 1.59·35-s − 10.8·37-s − 0.964·38-s − 1.41·40-s − 2.81·41-s + ⋯ |
| L(s) = 1 | + 1.11·2-s + 0.241·4-s + 0.265·5-s − 1.01·7-s − 0.845·8-s + 0.295·10-s + 0.301·11-s + 1.06·13-s − 1.13·14-s − 1.18·16-s + 0.560·17-s − 0.140·19-s + 0.0640·20-s + 0.335·22-s + 0.362·23-s − 0.929·25-s + 1.18·26-s − 0.245·28-s + 0.316·29-s − 0.400·31-s − 0.473·32-s + 0.624·34-s − 0.269·35-s − 1.78·37-s − 0.156·38-s − 0.224·40-s − 0.439·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6039 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6039 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 11 | \( 1 - T \) |
| 61 | \( 1 + T \) |
| good | 2 | \( 1 - 1.57T + 2T^{2} \) |
| 5 | \( 1 - 0.593T + 5T^{2} \) |
| 7 | \( 1 + 2.68T + 7T^{2} \) |
| 13 | \( 1 - 3.82T + 13T^{2} \) |
| 17 | \( 1 - 2.31T + 17T^{2} \) |
| 19 | \( 1 + 0.611T + 19T^{2} \) |
| 23 | \( 1 - 1.73T + 23T^{2} \) |
| 29 | \( 1 - 1.70T + 29T^{2} \) |
| 31 | \( 1 + 2.23T + 31T^{2} \) |
| 37 | \( 1 + 10.8T + 37T^{2} \) |
| 41 | \( 1 + 2.81T + 41T^{2} \) |
| 43 | \( 1 - 2.17T + 43T^{2} \) |
| 47 | \( 1 - 2.48T + 47T^{2} \) |
| 53 | \( 1 + 0.251T + 53T^{2} \) |
| 59 | \( 1 + 2.34T + 59T^{2} \) |
| 67 | \( 1 + 5.63T + 67T^{2} \) |
| 71 | \( 1 + 2.24T + 71T^{2} \) |
| 73 | \( 1 + 9.37T + 73T^{2} \) |
| 79 | \( 1 + 2.23T + 79T^{2} \) |
| 83 | \( 1 + 5.01T + 83T^{2} \) |
| 89 | \( 1 - 5.69T + 89T^{2} \) |
| 97 | \( 1 + 10.7T + 97T^{2} \) |
| 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
−7.51920356062064013383275554186, −6.68652503911520368975287472690, −6.12904918416286550906498654070, −5.63469773358214843317187146429, −4.84154383154812950503556633630, −3.85400481724648941187597471583, −3.49859145517078826885985004747, −2.69976934683057247539526544628, −1.47145355832176816279359871695, 0,
1.47145355832176816279359871695, 2.69976934683057247539526544628, 3.49859145517078826885985004747, 3.85400481724648941187597471583, 4.84154383154812950503556633630, 5.63469773358214843317187146429, 6.12904918416286550906498654070, 6.68652503911520368975287472690, 7.51920356062064013383275554186
