| L(s) = 1 | − 2-s + 4-s − 5-s + 3·7-s − 8-s + 10-s − 11-s + 3·13-s − 3·14-s + 16-s + 4·17-s − 7·19-s − 20-s + 22-s + 3·23-s + 25-s − 3·26-s + 3·28-s − 8·29-s − 6·31-s − 32-s − 4·34-s − 3·35-s + 6·37-s + 7·38-s + 40-s − 5·41-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.447·5-s + 1.13·7-s − 0.353·8-s + 0.316·10-s − 0.301·11-s + 0.832·13-s − 0.801·14-s + 1/4·16-s + 0.970·17-s − 1.60·19-s − 0.223·20-s + 0.213·22-s + 0.625·23-s + 1/5·25-s − 0.588·26-s + 0.566·28-s − 1.48·29-s − 1.07·31-s − 0.176·32-s − 0.685·34-s − 0.507·35-s + 0.986·37-s + 1.13·38-s + 0.158·40-s − 0.780·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8910 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8910 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 5 | \( 1 + T \) | |
| 11 | \( 1 + T \) | |
| good | 7 | \( 1 - 3 T + p T^{2} \) | 1.7.ad |
| 13 | \( 1 - 3 T + p T^{2} \) | 1.13.ad |
| 17 | \( 1 - 4 T + p T^{2} \) | 1.17.ae |
| 19 | \( 1 + 7 T + p T^{2} \) | 1.19.h |
| 23 | \( 1 - 3 T + p T^{2} \) | 1.23.ad |
| 29 | \( 1 + 8 T + p T^{2} \) | 1.29.i |
| 31 | \( 1 + 6 T + p T^{2} \) | 1.31.g |
| 37 | \( 1 - 6 T + p T^{2} \) | 1.37.ag |
| 41 | \( 1 + 5 T + p T^{2} \) | 1.41.f |
| 43 | \( 1 - 10 T + p T^{2} \) | 1.43.ak |
| 47 | \( 1 + 7 T + p T^{2} \) | 1.47.h |
| 53 | \( 1 - T + p T^{2} \) | 1.53.ab |
| 59 | \( 1 + 5 T + p T^{2} \) | 1.59.f |
| 61 | \( 1 + 8 T + p T^{2} \) | 1.61.i |
| 67 | \( 1 - 2 T + p T^{2} \) | 1.67.ac |
| 71 | \( 1 + 16 T + p T^{2} \) | 1.71.q |
| 73 | \( 1 + 4 T + p T^{2} \) | 1.73.e |
| 79 | \( 1 - 4 T + p T^{2} \) | 1.79.ae |
| 83 | \( 1 + 6 T + p T^{2} \) | 1.83.g |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 97 | \( 1 + 6 T + p T^{2} \) | 1.97.g |
| 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
−7.80032683986574847045829707423, −6.91460149772637398409112696601, −6.03786386801792676264452866006, −5.45638161792961286871195968236, −4.54914721032101987670703272401, −3.86195209789198512358521755523, −2.97379744233495498715842664976, −1.92314566421283105102627918247, −1.27461233905345186237923820791, 0,
1.27461233905345186237923820791, 1.92314566421283105102627918247, 2.97379744233495498715842664976, 3.86195209789198512358521755523, 4.54914721032101987670703272401, 5.45638161792961286871195968236, 6.03786386801792676264452866006, 6.91460149772637398409112696601, 7.80032683986574847045829707423
