| L(s) = 1 | − 0.539·2-s − 1.70·4-s − 3.70·7-s + 2·8-s − 1.87·11-s + 6.58·13-s + 2·14-s + 2.34·16-s − 3.09·17-s − 7.34·19-s + 1.01·22-s + 7.29·23-s − 3.55·26-s + 6.34·28-s − 0.170·29-s − 31-s − 5.26·32-s + 1.66·34-s − 6.82·37-s + 3.95·38-s + 7.95·41-s − 4.18·43-s + 3.21·44-s − 3.93·46-s + 11.7·47-s + 6.75·49-s − 11.2·52-s + ⋯ |
| L(s) = 1 | − 0.381·2-s − 0.854·4-s − 1.40·7-s + 0.707·8-s − 0.566·11-s + 1.82·13-s + 0.534·14-s + 0.585·16-s − 0.749·17-s − 1.68·19-s + 0.216·22-s + 1.52·23-s − 0.696·26-s + 1.19·28-s − 0.0315·29-s − 0.179·31-s − 0.930·32-s + 0.285·34-s − 1.12·37-s + 0.642·38-s + 1.24·41-s − 0.637·43-s + 0.484·44-s − 0.580·46-s + 1.71·47-s + 0.965·49-s − 1.56·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6975 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6975 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 31 | \( 1 + T \) |
| good | 2 | \( 1 + 0.539T + 2T^{2} \) |
| 7 | \( 1 + 3.70T + 7T^{2} \) |
| 11 | \( 1 + 1.87T + 11T^{2} \) |
| 13 | \( 1 - 6.58T + 13T^{2} \) |
| 17 | \( 1 + 3.09T + 17T^{2} \) |
| 19 | \( 1 + 7.34T + 19T^{2} \) |
| 23 | \( 1 - 7.29T + 23T^{2} \) |
| 29 | \( 1 + 0.170T + 29T^{2} \) |
| 37 | \( 1 + 6.82T + 37T^{2} \) |
| 41 | \( 1 - 7.95T + 41T^{2} \) |
| 43 | \( 1 + 4.18T + 43T^{2} \) |
| 47 | \( 1 - 11.7T + 47T^{2} \) |
| 53 | \( 1 + 13.9T + 53T^{2} \) |
| 59 | \( 1 - 6.72T + 59T^{2} \) |
| 61 | \( 1 - 1.84T + 61T^{2} \) |
| 67 | \( 1 - 1.81T + 67T^{2} \) |
| 71 | \( 1 - 11.2T + 71T^{2} \) |
| 73 | \( 1 + 2.92T + 73T^{2} \) |
| 79 | \( 1 - 6.24T + 79T^{2} \) |
| 83 | \( 1 - 6.21T + 83T^{2} \) |
| 89 | \( 1 - 12.4T + 89T^{2} \) |
| 97 | \( 1 + 3.94T + 97T^{2} \) |
| 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
−7.74829403040151654082862336657, −6.74330396197272121037271470402, −6.35719696215255263520131377460, −5.54756519287934772306552593218, −4.67574232913596602907931002610, −3.85192186316182830073683047689, −3.35499738909050301764921963201, −2.25924458085128941666757207611, −0.984899974874516840922938564223, 0,
0.984899974874516840922938564223, 2.25924458085128941666757207611, 3.35499738909050301764921963201, 3.85192186316182830073683047689, 4.67574232913596602907931002610, 5.54756519287934772306552593218, 6.35719696215255263520131377460, 6.74330396197272121037271470402, 7.74829403040151654082862336657
