| L(s) = 1 | + 2-s + 3-s + 4-s − 5-s + 6-s − 2·7-s + 8-s + 9-s − 10-s + 11-s + 12-s − 4·13-s − 2·14-s − 15-s + 16-s − 2·17-s + 18-s + 6·19-s − 20-s − 2·21-s + 22-s + 24-s + 25-s − 4·26-s + 27-s − 2·28-s − 30-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.447·5-s + 0.408·6-s − 0.755·7-s + 0.353·8-s + 1/3·9-s − 0.316·10-s + 0.301·11-s + 0.288·12-s − 1.10·13-s − 0.534·14-s − 0.258·15-s + 1/4·16-s − 0.485·17-s + 0.235·18-s + 1.37·19-s − 0.223·20-s − 0.436·21-s + 0.213·22-s + 0.204·24-s + 1/5·25-s − 0.784·26-s + 0.192·27-s − 0.377·28-s − 0.182·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 29370 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 29370 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | Isogeny Class over $\mathbf{F}_p$ |
|---|
| bad | 2 | \( 1 - T \) | |
| 3 | \( 1 - T \) | |
| 5 | \( 1 + T \) | |
| 11 | \( 1 - T \) | |
| 89 | \( 1 - T \) | |
| good | 7 | \( 1 + 2 T + p T^{2} \) | 1.7.c |
| 13 | \( 1 + 4 T + p T^{2} \) | 1.13.e |
| 17 | \( 1 + 2 T + p T^{2} \) | 1.17.c |
| 19 | \( 1 - 6 T + p T^{2} \) | 1.19.ag |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 + p T^{2} \) | 1.29.a |
| 31 | \( 1 + 4 T + p T^{2} \) | 1.31.e |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 + p T^{2} \) | 1.41.a |
| 43 | \( 1 - 6 T + p T^{2} \) | 1.43.ag |
| 47 | \( 1 - 8 T + p T^{2} \) | 1.47.ai |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 + 8 T + p T^{2} \) | 1.61.i |
| 67 | \( 1 - 12 T + p T^{2} \) | 1.67.am |
| 71 | \( 1 + 8 T + p T^{2} \) | 1.71.i |
| 73 | \( 1 + 2 T + p T^{2} \) | 1.73.c |
| 79 | \( 1 - 4 T + p T^{2} \) | 1.79.ae |
| 83 | \( 1 - 2 T + p T^{2} \) | 1.83.ac |
| 97 | \( 1 + 2 T + p T^{2} \) | 1.97.c |
| 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
−15.51615151452420, −14.73284058000783, −14.43143360309870, −13.83457824713881, −13.40041138538641, −12.73418966435290, −12.32082341330981, −11.92763799397803, −11.22060791621031, −10.67886608583099, −9.971086682363010, −9.394580112556434, −9.116960172526809, −8.194284098947696, −7.576099280696287, −7.177499315682531, −6.660101452800547, −5.895357410005417, −5.235927493043114, −4.610428090258812, −3.954831443956339, −3.357038223260514, −2.817626975511151, −2.154661528374574, −1.157778402307285, 0,
1.157778402307285, 2.154661528374574, 2.817626975511151, 3.357038223260514, 3.954831443956339, 4.610428090258812, 5.235927493043114, 5.895357410005417, 6.660101452800547, 7.177499315682531, 7.576099280696287, 8.194284098947696, 9.116960172526809, 9.394580112556434, 9.971086682363010, 10.67886608583099, 11.22060791621031, 11.92763799397803, 12.32082341330981, 12.73418966435290, 13.40041138538641, 13.83457824713881, 14.43143360309870, 14.73284058000783, 15.51615151452420
