| L(s) = 1 | − 2-s + 2·3-s + 4-s − 2·6-s − 5·7-s − 8-s + 9-s − 3·11-s + 2·12-s − 13-s + 5·14-s + 16-s − 3·17-s − 18-s − 4·19-s − 10·21-s + 3·22-s − 6·23-s − 2·24-s + 26-s − 4·27-s − 5·28-s + 9·29-s + 5·31-s − 32-s − 6·33-s + 3·34-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.15·3-s + 1/2·4-s − 0.816·6-s − 1.88·7-s − 0.353·8-s + 1/3·9-s − 0.904·11-s + 0.577·12-s − 0.277·13-s + 1.33·14-s + 1/4·16-s − 0.727·17-s − 0.235·18-s − 0.917·19-s − 2.18·21-s + 0.639·22-s − 1.25·23-s − 0.408·24-s + 0.196·26-s − 0.769·27-s − 0.944·28-s + 1.67·29-s + 0.898·31-s − 0.176·32-s − 1.04·33-s + 0.514·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 650 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 650 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 + T \) |
| 5 | \( 1 \) |
| 13 | \( 1 + T \) |
| good | 3 | \( 1 - 2 T + p T^{2} \) |
| 7 | \( 1 + 5 T + p T^{2} \) |
| 11 | \( 1 + 3 T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 - 9 T + p T^{2} \) |
| 31 | \( 1 - 5 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 - 9 T + p T^{2} \) |
| 53 | \( 1 - 9 T + p T^{2} \) |
| 59 | \( 1 + 9 T + p T^{2} \) |
| 61 | \( 1 + T + p T^{2} \) |
| 67 | \( 1 + 5 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 + 16 T + p T^{2} \) |
| 83 | \( 1 - 15 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 8 T + p T^{2} \) |
| 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
−10.20402423859091916997783692335, −9.103611083397007748425818392285, −8.588436424001057736781809699325, −7.68139369219990412183321078107, −6.70376515467910657944845071259, −5.93987983462099200069259985801, −4.16564611607850973391924581761, −2.95720673678180523275331210977, −2.41480506675707795537505240460, 0,
2.41480506675707795537505240460, 2.95720673678180523275331210977, 4.16564611607850973391924581761, 5.93987983462099200069259985801, 6.70376515467910657944845071259, 7.68139369219990412183321078107, 8.588436424001057736781809699325, 9.103611083397007748425818392285, 10.20402423859091916997783692335
