| L(s) = 1 | − 2.23·2-s + 3.00·4-s − 2·7-s − 2.23·8-s + 2·11-s − 2·13-s + 4.47·14-s − 0.999·16-s + 4.47·17-s + 2·19-s − 4.47·22-s − 2·23-s + 4.47·26-s − 6.00·28-s + 29-s − 2·31-s + 6.70·32-s − 10.0·34-s − 8.47·37-s − 4.47·38-s − 2·41-s − 4·43-s + 6.00·44-s + 4.47·46-s + 12.9·47-s − 3·49-s − 6.00·52-s + ⋯ |
| L(s) = 1 | − 1.58·2-s + 1.50·4-s − 0.755·7-s − 0.790·8-s + 0.603·11-s − 0.554·13-s + 1.19·14-s − 0.249·16-s + 1.08·17-s + 0.458·19-s − 0.953·22-s − 0.417·23-s + 0.877·26-s − 1.13·28-s + 0.185·29-s − 0.359·31-s + 1.18·32-s − 1.71·34-s − 1.39·37-s − 0.725·38-s − 0.312·41-s − 0.609·43-s + 0.904·44-s + 0.659·46-s + 1.88·47-s − 0.428·49-s − 0.832·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| 29 | \( 1 - T \) |
| good | 2 | \( 1 + 2.23T + 2T^{2} \) |
| 7 | \( 1 + 2T + 7T^{2} \) |
| 11 | \( 1 - 2T + 11T^{2} \) |
| 13 | \( 1 + 2T + 13T^{2} \) |
| 17 | \( 1 - 4.47T + 17T^{2} \) |
| 19 | \( 1 - 2T + 19T^{2} \) |
| 23 | \( 1 + 2T + 23T^{2} \) |
| 31 | \( 1 + 2T + 31T^{2} \) |
| 37 | \( 1 + 8.47T + 37T^{2} \) |
| 41 | \( 1 + 2T + 41T^{2} \) |
| 43 | \( 1 + 4T + 43T^{2} \) |
| 47 | \( 1 - 12.9T + 47T^{2} \) |
| 53 | \( 1 - 2T + 53T^{2} \) |
| 59 | \( 1 + 8T + 59T^{2} \) |
| 61 | \( 1 - 6.94T + 61T^{2} \) |
| 67 | \( 1 + 2.94T + 67T^{2} \) |
| 71 | \( 1 + 4T + 71T^{2} \) |
| 73 | \( 1 + 3.52T + 73T^{2} \) |
| 79 | \( 1 + 2.94T + 79T^{2} \) |
| 83 | \( 1 + 14.9T + 83T^{2} \) |
| 89 | \( 1 - 6T + 89T^{2} \) |
| 97 | \( 1 - 17.4T + 97T^{2} \) |
| 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
−7.53515489131485993955253361034, −7.33907012272636896905536836678, −6.50601033978516139779880443693, −5.81021479695041887186574864687, −4.86450749633793195285726173166, −3.76135615106878143798858614516, −2.99387677399965553313537251335, −1.97316435317603932964279383144, −1.07303161613340445794747158450, 0,
1.07303161613340445794747158450, 1.97316435317603932964279383144, 2.99387677399965553313537251335, 3.76135615106878143798858614516, 4.86450749633793195285726173166, 5.81021479695041887186574864687, 6.50601033978516139779880443693, 7.33907012272636896905536836678, 7.53515489131485993955253361034
