| L(s) = 1 | − 3-s − 5-s + 9-s − 2·13-s + 15-s − 2·17-s + 4·19-s + 25-s − 27-s + 6·29-s − 31-s − 2·37-s + 2·39-s − 6·41-s − 4·43-s − 45-s − 7·49-s + 2·51-s − 6·53-s − 4·57-s + 12·59-s − 2·61-s + 2·65-s + 8·67-s + 12·71-s − 2·73-s − 75-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s + 1/3·9-s − 0.554·13-s + 0.258·15-s − 0.485·17-s + 0.917·19-s + 1/5·25-s − 0.192·27-s + 1.11·29-s − 0.179·31-s − 0.328·37-s + 0.320·39-s − 0.937·41-s − 0.609·43-s − 0.149·45-s − 49-s + 0.280·51-s − 0.824·53-s − 0.529·57-s + 1.56·59-s − 0.256·61-s + 0.248·65-s + 0.977·67-s + 1.42·71-s − 0.234·73-s − 0.115·75-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7440 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7440 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) | |
| 31 | \( 1 + T \) | |
| good | 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 |
| 17 | \( 1 + 2 T + p T^{2} \) | 1.17.c |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 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.g |
| 59 | \( 1 - 12 T + p T^{2} \) | 1.59.am |
| 61 | \( 1 + 2 T + p T^{2} \) | 1.61.c |
| 67 | \( 1 - 8 T + p T^{2} \) | 1.67.ai |
| 71 | \( 1 - 12 T + p T^{2} \) | 1.71.am |
| 73 | \( 1 + 2 T + p T^{2} \) | 1.73.c |
| 79 | \( 1 - 8 T + p T^{2} \) | 1.79.ai |
| 83 | \( 1 + 4 T + p T^{2} \) | 1.83.e |
| 89 | \( 1 - 14 T + p T^{2} \) | 1.89.ao |
| 97 | \( 1 + 6 T + p T^{2} \) | 1.97.g |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.50918321403516735776031309243, −6.76420988929176273992540665535, −6.33217062584634179998573384580, −5.12910768980703652613080611587, −5.00451054368126818122030808013, −3.96122049970010105742578826449, −3.22239648754543709287730855815, −2.23463613205613376831866305419, −1.11614195366998383379618542092, 0,
1.11614195366998383379618542092, 2.23463613205613376831866305419, 3.22239648754543709287730855815, 3.96122049970010105742578826449, 5.00451054368126818122030808013, 5.12910768980703652613080611587, 6.33217062584634179998573384580, 6.76420988929176273992540665535, 7.50918321403516735776031309243
