| L(s) = 1 | − 3-s + 2·7-s + 9-s + 6·11-s − 2·13-s + 2·17-s + 2·19-s − 2·21-s − 6·23-s − 27-s − 10·29-s − 4·31-s − 6·33-s + 2·37-s + 2·39-s + 6·41-s − 6·47-s − 3·49-s − 2·51-s − 10·53-s − 2·57-s − 6·59-s − 6·61-s + 2·63-s − 16·67-s + 6·69-s + 8·71-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.755·7-s + 1/3·9-s + 1.80·11-s − 0.554·13-s + 0.485·17-s + 0.458·19-s − 0.436·21-s − 1.25·23-s − 0.192·27-s − 1.85·29-s − 0.718·31-s − 1.04·33-s + 0.328·37-s + 0.320·39-s + 0.937·41-s − 0.875·47-s − 3/7·49-s − 0.280·51-s − 1.37·53-s − 0.264·57-s − 0.781·59-s − 0.768·61-s + 0.251·63-s − 1.95·67-s + 0.722·69-s + 0.949·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) | |
| 5 | \( 1 \) | |
| good | 7 | \( 1 - 2 T + p T^{2} \) | 1.7.ac |
| 11 | \( 1 - 6 T + p T^{2} \) | 1.11.ag |
| 13 | \( 1 + 2 T + p T^{2} \) | 1.13.c |
| 17 | \( 1 - 2 T + p T^{2} \) | 1.17.ac |
| 19 | \( 1 - 2 T + p T^{2} \) | 1.19.ac |
| 23 | \( 1 + 6 T + p T^{2} \) | 1.23.g |
| 29 | \( 1 + 10 T + p T^{2} \) | 1.29.k |
| 31 | \( 1 + 4 T + p T^{2} \) | 1.31.e |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 + p T^{2} \) | 1.43.a |
| 47 | \( 1 + 6 T + p T^{2} \) | 1.47.g |
| 53 | \( 1 + 10 T + p T^{2} \) | 1.53.k |
| 59 | \( 1 + 6 T + p T^{2} \) | 1.59.g |
| 61 | \( 1 + 6 T + p T^{2} \) | 1.61.g |
| 67 | \( 1 + 16 T + p T^{2} \) | 1.67.q |
| 71 | \( 1 - 8 T + p T^{2} \) | 1.71.ai |
| 73 | \( 1 + 10 T + p T^{2} \) | 1.73.k |
| 79 | \( 1 + p T^{2} \) | 1.79.a |
| 83 | \( 1 + 12 T + p T^{2} \) | 1.83.m |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 97 | \( 1 + 10 T + p T^{2} \) | 1.97.k |
| 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
−7.50278501731735086893323428056, −6.56679046914786382102179396707, −5.97177849751939319800915132925, −5.38003376477027826305941448413, −4.48257256210332541274860588392, −4.01479343359877635382137143741, −3.15403930439004724341196027840, −1.79257246735232709799363831166, −1.40704939112635752986610876796, 0,
1.40704939112635752986610876796, 1.79257246735232709799363831166, 3.15403930439004724341196027840, 4.01479343359877635382137143741, 4.48257256210332541274860588392, 5.38003376477027826305941448413, 5.97177849751939319800915132925, 6.56679046914786382102179396707, 7.50278501731735086893323428056
