| L(s) = 1 | − 2-s + 4-s − 4·7-s − 8-s − 3·11-s − 2·13-s + 4·14-s + 16-s + 6·17-s + 3·22-s + 6·23-s − 5·25-s + 2·26-s − 4·28-s − 2·31-s − 32-s − 6·34-s + 10·37-s + 9·41-s − 4·43-s − 3·44-s − 6·46-s + 9·49-s + 5·50-s − 2·52-s + 6·53-s + 4·56-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 1.51·7-s − 0.353·8-s − 0.904·11-s − 0.554·13-s + 1.06·14-s + 1/4·16-s + 1.45·17-s + 0.639·22-s + 1.25·23-s − 25-s + 0.392·26-s − 0.755·28-s − 0.359·31-s − 0.176·32-s − 1.02·34-s + 1.64·37-s + 1.40·41-s − 0.609·43-s − 0.452·44-s − 0.884·46-s + 9/7·49-s + 0.707·50-s − 0.277·52-s + 0.824·53-s + 0.534·56-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6498 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6498 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 19 | \( 1 \) | |
| good | 5 | \( 1 + p T^{2} \) | 1.5.a |
| 7 | \( 1 + 4 T + p T^{2} \) | 1.7.e |
| 11 | \( 1 + 3 T + p T^{2} \) | 1.11.d |
| 13 | \( 1 + 2 T + p T^{2} \) | 1.13.c |
| 17 | \( 1 - 6 T + p T^{2} \) | 1.17.ag |
| 23 | \( 1 - 6 T + p T^{2} \) | 1.23.ag |
| 29 | \( 1 + p T^{2} \) | 1.29.a |
| 31 | \( 1 + 2 T + p T^{2} \) | 1.31.c |
| 37 | \( 1 - 10 T + p T^{2} \) | 1.37.ak |
| 41 | \( 1 - 9 T + p T^{2} \) | 1.41.aj |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 + 9 T + p T^{2} \) | 1.59.j |
| 61 | \( 1 + 4 T + p T^{2} \) | 1.61.e |
| 67 | \( 1 - 7 T + p T^{2} \) | 1.67.ah |
| 71 | \( 1 + 6 T + p T^{2} \) | 1.71.g |
| 73 | \( 1 + T + p T^{2} \) | 1.73.b |
| 79 | \( 1 - 4 T + p T^{2} \) | 1.79.ae |
| 83 | \( 1 + 3 T + p T^{2} \) | 1.83.d |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 97 | \( 1 + 17 T + p T^{2} \) | 1.97.r |
| 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
−7.52661242074846556034272909440, −7.26148883879091309597772692140, −6.21193272331988358393092187886, −5.79223845167588020740444905018, −4.91356351664481277876221686524, −3.75312605283684034529829190309, −2.99811094189296323989805262826, −2.44703263969237924080657596698, −1.03579620913104356707609348611, 0,
1.03579620913104356707609348611, 2.44703263969237924080657596698, 2.99811094189296323989805262826, 3.75312605283684034529829190309, 4.91356351664481277876221686524, 5.79223845167588020740444905018, 6.21193272331988358393092187886, 7.26148883879091309597772692140, 7.52661242074846556034272909440
