| L(s) = 1 | − 2-s − 3-s − 4-s − 5-s + 6-s + 3·8-s + 9-s + 10-s + 12-s + 6·13-s + 15-s − 16-s − 6·17-s − 18-s + 4·19-s + 20-s − 3·24-s + 25-s − 6·26-s − 27-s + 29-s − 30-s + 4·31-s − 5·32-s + 6·34-s − 36-s + 2·37-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s − 1/2·4-s − 0.447·5-s + 0.408·6-s + 1.06·8-s + 1/3·9-s + 0.316·10-s + 0.288·12-s + 1.66·13-s + 0.258·15-s − 1/4·16-s − 1.45·17-s − 0.235·18-s + 0.917·19-s + 0.223·20-s − 0.612·24-s + 1/5·25-s − 1.17·26-s − 0.192·27-s + 0.185·29-s − 0.182·30-s + 0.718·31-s − 0.883·32-s + 1.02·34-s − 1/6·36-s + 0.328·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 230115 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 230115 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 23 | \( 1 \) |
| 29 | \( 1 - T \) |
| good | 2 | \( 1 + T + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + 2 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
| 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
−13.17769893305594, −12.75942403073825, −12.15003435070349, −11.57129230541166, −11.24822226642971, −10.81271922034342, −10.46955894316406, −9.870875716070268, −9.323722393521879, −9.006097100392801, −8.430713354036605, −8.103065677392287, −7.651927779055251, −6.928257063712359, −6.567383459616298, −6.097787812714740, −5.369719118709297, −4.961799329851697, −4.347806634686174, −3.975394547816299, −3.430188394735805, −2.703844229851172, −1.804388433102846, −1.251663736063579, −0.7120462535296708, 0,
0.7120462535296708, 1.251663736063579, 1.804388433102846, 2.703844229851172, 3.430188394735805, 3.975394547816299, 4.347806634686174, 4.961799329851697, 5.369719118709297, 6.097787812714740, 6.567383459616298, 6.928257063712359, 7.651927779055251, 8.103065677392287, 8.430713354036605, 9.006097100392801, 9.323722393521879, 9.870875716070268, 10.46955894316406, 10.81271922034342, 11.24822226642971, 11.57129230541166, 12.15003435070349, 12.75942403073825, 13.17769893305594
