| L(s) = 1 | − 5-s − 3·9-s − 2·13-s + 17-s − 4·19-s − 8·23-s + 25-s + 2·29-s − 8·31-s + 2·37-s + 2·41-s − 4·43-s + 3·45-s − 7·49-s + 6·53-s − 4·59-s − 6·61-s + 2·65-s + 4·67-s − 8·71-s + 2·73-s + 9·81-s + 4·83-s − 85-s − 6·89-s + 4·95-s + 18·97-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 9-s − 0.554·13-s + 0.242·17-s − 0.917·19-s − 1.66·23-s + 1/5·25-s + 0.371·29-s − 1.43·31-s + 0.328·37-s + 0.312·41-s − 0.609·43-s + 0.447·45-s − 49-s + 0.824·53-s − 0.520·59-s − 0.768·61-s + 0.248·65-s + 0.488·67-s − 0.949·71-s + 0.234·73-s + 81-s + 0.439·83-s − 0.108·85-s − 0.635·89-s + 0.410·95-s + 1.82·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 680 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 5 | \( 1 + T \) | |
| 17 | \( 1 - T \) | |
| good | 3 | \( 1 + p T^{2} \) | 1.3.a |
| 7 | \( 1 + p T^{2} \) | 1.7.a |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 13 | \( 1 + 2 T + p T^{2} \) | 1.13.c |
| 19 | \( 1 + 4 T + p T^{2} \) | 1.19.e |
| 23 | \( 1 + 8 T + p T^{2} \) | 1.23.i |
| 29 | \( 1 - 2 T + p T^{2} \) | 1.29.ac |
| 31 | \( 1 + 8 T + p T^{2} \) | 1.31.i |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 41 | \( 1 - 2 T + p T^{2} \) | 1.41.ac |
| 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 | \( 1 + 4 T + p T^{2} \) | 1.59.e |
| 61 | \( 1 + 6 T + p T^{2} \) | 1.61.g |
| 67 | \( 1 - 4 T + p T^{2} \) | 1.67.ae |
| 71 | \( 1 + 8 T + p T^{2} \) | 1.71.i |
| 73 | \( 1 - 2 T + p T^{2} \) | 1.73.ac |
| 79 | \( 1 + p T^{2} \) | 1.79.a |
| 83 | \( 1 - 4 T + p T^{2} \) | 1.83.ae |
| 89 | \( 1 + 6 T + p T^{2} \) | 1.89.g |
| 97 | \( 1 - 18 T + p T^{2} \) | 1.97.as |
| 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
−10.11412498419995635064768169571, −9.120593456645095048374168582985, −8.277492610756201606277035581204, −7.58297430387033033142097041666, −6.42605354962456456940109508006, −5.57397530040962827098114178875, −4.44860264411923876548469666329, −3.37622543093299548354865598128, −2.13823702677682119982697431133, 0,
2.13823702677682119982697431133, 3.37622543093299548354865598128, 4.44860264411923876548469666329, 5.57397530040962827098114178875, 6.42605354962456456940109508006, 7.58297430387033033142097041666, 8.277492610756201606277035581204, 9.120593456645095048374168582985, 10.11412498419995635064768169571
