| L(s) = 1 | − 3-s + 5-s + 2·7-s + 9-s − 2·11-s − 13-s − 15-s − 7·17-s − 5·19-s − 2·21-s + 6·23-s − 4·25-s − 27-s − 7·31-s + 2·33-s + 2·35-s − 2·37-s + 39-s + 41-s − 4·43-s + 45-s + 12·47-s − 3·49-s + 7·51-s − 6·53-s − 2·55-s + 5·57-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.447·5-s + 0.755·7-s + 1/3·9-s − 0.603·11-s − 0.277·13-s − 0.258·15-s − 1.69·17-s − 1.14·19-s − 0.436·21-s + 1.25·23-s − 4/5·25-s − 0.192·27-s − 1.25·31-s + 0.348·33-s + 0.338·35-s − 0.328·37-s + 0.160·39-s + 0.156·41-s − 0.609·43-s + 0.149·45-s + 1.75·47-s − 3/7·49-s + 0.980·51-s − 0.824·53-s − 0.269·55-s + 0.662·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1968 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1968 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 3 | \( 1 + T \) | |
| 41 | \( 1 - T \) | |
| good | 5 | \( 1 - T + p T^{2} \) | 1.5.ab |
| 7 | \( 1 - 2 T + p T^{2} \) | 1.7.ac |
| 11 | \( 1 + 2 T + p T^{2} \) | 1.11.c |
| 13 | \( 1 + T + p T^{2} \) | 1.13.b |
| 17 | \( 1 + 7 T + p T^{2} \) | 1.17.h |
| 19 | \( 1 + 5 T + p T^{2} \) | 1.19.f |
| 23 | \( 1 - 6 T + p T^{2} \) | 1.23.ag |
| 29 | \( 1 + p T^{2} \) | 1.29.a |
| 31 | \( 1 + 7 T + p T^{2} \) | 1.31.h |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 - 12 T + p T^{2} \) | 1.47.am |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 + 5 T + p T^{2} \) | 1.59.f |
| 61 | \( 1 - 2 T + p T^{2} \) | 1.61.ac |
| 67 | \( 1 + 3 T + p T^{2} \) | 1.67.d |
| 71 | \( 1 - 3 T + p T^{2} \) | 1.71.ad |
| 73 | \( 1 - 9 T + p T^{2} \) | 1.73.aj |
| 79 | \( 1 + p T^{2} \) | 1.79.a |
| 83 | \( 1 + 9 T + p T^{2} \) | 1.83.j |
| 89 | \( 1 - 5 T + p T^{2} \) | 1.89.af |
| 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
−8.854254836977639607550866347797, −8.003031325092420449682750698421, −7.10616315740238742017999604006, −6.42016684201542349324991768856, −5.47955107858208539285719708514, −4.83070569253156695749738552314, −4.03107537860149562958636706208, −2.52375865447385408837459108701, −1.70296665492210811677280010233, 0,
1.70296665492210811677280010233, 2.52375865447385408837459108701, 4.03107537860149562958636706208, 4.83070569253156695749738552314, 5.47955107858208539285719708514, 6.42016684201542349324991768856, 7.10616315740238742017999604006, 8.003031325092420449682750698421, 8.854254836977639607550866347797
