| L(s) = 1 | − 0.239·2-s − 1.94·4-s + 1.18·5-s − 7-s + 0.942·8-s − 0.282·10-s − 3.70·11-s + 13-s + 0.239·14-s + 3.66·16-s − 6.94·17-s + 1.94·19-s − 2.29·20-s + 0.885·22-s − 5.60·23-s − 3.60·25-s − 0.239·26-s + 1.94·28-s + 0.239·29-s + 1.66·31-s − 2.76·32-s + 1.66·34-s − 1.18·35-s − 9.54·37-s − 0.464·38-s + 1.11·40-s − 10.1·41-s + ⋯ |
| L(s) = 1 | − 0.169·2-s − 0.971·4-s + 0.528·5-s − 0.377·7-s + 0.333·8-s − 0.0893·10-s − 1.11·11-s + 0.277·13-s + 0.0639·14-s + 0.915·16-s − 1.68·17-s + 0.445·19-s − 0.513·20-s + 0.188·22-s − 1.16·23-s − 0.720·25-s − 0.0468·26-s + 0.367·28-s + 0.0444·29-s + 0.298·31-s − 0.488·32-s + 0.284·34-s − 0.199·35-s − 1.56·37-s − 0.0753·38-s + 0.176·40-s − 1.59·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 + T \) |
| good | 2 | \( 1 + 0.239T + 2T^{2} \) |
| 5 | \( 1 - 1.18T + 5T^{2} \) |
| 11 | \( 1 + 3.70T + 11T^{2} \) |
| 13 | \( 1 - T + 13T^{2} \) |
| 17 | \( 1 + 6.94T + 17T^{2} \) |
| 19 | \( 1 - 1.94T + 19T^{2} \) |
| 23 | \( 1 + 5.60T + 23T^{2} \) |
| 29 | \( 1 - 0.239T + 29T^{2} \) |
| 31 | \( 1 - 1.66T + 31T^{2} \) |
| 37 | \( 1 + 9.54T + 37T^{2} \) |
| 41 | \( 1 + 10.1T + 41T^{2} \) |
| 43 | \( 1 - 2.22T + 43T^{2} \) |
| 47 | \( 1 - 5.82T + 47T^{2} \) |
| 53 | \( 1 + 11.6T + 53T^{2} \) |
| 59 | \( 1 - 2.60T + 59T^{2} \) |
| 61 | \( 1 + 7.60T + 61T^{2} \) |
| 67 | \( 1 - 3.50T + 67T^{2} \) |
| 71 | \( 1 - 8.60T + 71T^{2} \) |
| 73 | \( 1 - 15.1T + 73T^{2} \) |
| 79 | \( 1 - 7.37T + 79T^{2} \) |
| 83 | \( 1 + 6.94T + 83T^{2} \) |
| 89 | \( 1 - 2.74T + 89T^{2} \) |
| 97 | \( 1 - 7.16T + 97T^{2} \) |
| 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
−10.16751574014947657595814419137, −9.478926114819564418637865086677, −8.611894876496353828936787676140, −7.85211843023724863786930854203, −6.60980171432291819252986663250, −5.58760781409360007139260328445, −4.71364151121870565129588476827, −3.55591448621153857250248519602, −2.07072477954214546651559024617, 0,
2.07072477954214546651559024617, 3.55591448621153857250248519602, 4.71364151121870565129588476827, 5.58760781409360007139260328445, 6.60980171432291819252986663250, 7.85211843023724863786930854203, 8.611894876496353828936787676140, 9.478926114819564418637865086677, 10.16751574014947657595814419137
