| L(s) = 1 | − 7-s + 2·11-s + 4·13-s − 7·17-s + 2·19-s − 8·23-s − 6·29-s + 4·31-s + 2·37-s + 41-s + 4·43-s + 7·47-s − 6·49-s − 6·53-s + 4·59-s − 4·61-s + 2·67-s + 8·71-s + 3·73-s − 2·77-s − 5·79-s + 6·83-s − 5·89-s − 4·91-s − 3·97-s + 101-s + 103-s + ⋯ |
| L(s) = 1 | − 0.377·7-s + 0.603·11-s + 1.10·13-s − 1.69·17-s + 0.458·19-s − 1.66·23-s − 1.11·29-s + 0.718·31-s + 0.328·37-s + 0.156·41-s + 0.609·43-s + 1.02·47-s − 6/7·49-s − 0.824·53-s + 0.520·59-s − 0.512·61-s + 0.244·67-s + 0.949·71-s + 0.351·73-s − 0.227·77-s − 0.562·79-s + 0.658·83-s − 0.529·89-s − 0.419·91-s − 0.304·97-s + 0.0995·101-s + 0.0985·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 259200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 259200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 5 | \( 1 \) | |
| good | 7 | \( 1 + T + p T^{2} \) | 1.7.b |
| 11 | \( 1 - 2 T + p T^{2} \) | 1.11.ac |
| 13 | \( 1 - 4 T + p T^{2} \) | 1.13.ae |
| 17 | \( 1 + 7 T + p T^{2} \) | 1.17.h |
| 19 | \( 1 - 2 T + p T^{2} \) | 1.19.ac |
| 23 | \( 1 + 8 T + p T^{2} \) | 1.23.i |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 - 4 T + p T^{2} \) | 1.31.ae |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 41 | \( 1 - T + p T^{2} \) | 1.41.ab |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 - 7 T + p T^{2} \) | 1.47.ah |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 - 4 T + p T^{2} \) | 1.59.ae |
| 61 | \( 1 + 4 T + p T^{2} \) | 1.61.e |
| 67 | \( 1 - 2 T + p T^{2} \) | 1.67.ac |
| 71 | \( 1 - 8 T + p T^{2} \) | 1.71.ai |
| 73 | \( 1 - 3 T + p T^{2} \) | 1.73.ad |
| 79 | \( 1 + 5 T + p T^{2} \) | 1.79.f |
| 83 | \( 1 - 6 T + p T^{2} \) | 1.83.ag |
| 89 | \( 1 + 5 T + p T^{2} \) | 1.89.f |
| 97 | \( 1 + 3 T + p T^{2} \) | 1.97.d |
| 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
−13.14518888339537, −12.60220162505484, −12.12559120314450, −11.57563530084191, −11.26346976302785, −10.78390893991878, −10.37719754625143, −9.607426816925703, −9.408393409006226, −8.950807807198105, −8.326995967979635, −8.032789080236596, −7.405107663994653, −6.750418992144026, −6.463203029347760, −5.956252225241631, −5.594619601039276, −4.773539793702642, −4.206623943570278, −3.910002266544416, −3.373047841476412, −2.632864331968732, −2.061354809727737, −1.530657950065923, −0.7438415519023910, 0,
0.7438415519023910, 1.530657950065923, 2.061354809727737, 2.632864331968732, 3.373047841476412, 3.910002266544416, 4.206623943570278, 4.773539793702642, 5.594619601039276, 5.956252225241631, 6.463203029347760, 6.750418992144026, 7.405107663994653, 8.032789080236596, 8.326995967979635, 8.950807807198105, 9.408393409006226, 9.607426816925703, 10.37719754625143, 10.78390893991878, 11.26346976302785, 11.57563530084191, 12.12559120314450, 12.60220162505484, 13.14518888339537
