| L(s) = 1 | + 2-s + 4-s + 5-s + 7-s + 8-s + 10-s − 2·11-s + 14-s + 16-s − 17-s − 6·19-s + 20-s − 2·22-s + 25-s + 28-s + 32-s − 34-s + 35-s − 8·37-s − 6·38-s + 40-s − 6·41-s − 12·43-s − 2·44-s + 4·47-s + 49-s + 50-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.447·5-s + 0.377·7-s + 0.353·8-s + 0.316·10-s − 0.603·11-s + 0.267·14-s + 1/4·16-s − 0.242·17-s − 1.37·19-s + 0.223·20-s − 0.426·22-s + 1/5·25-s + 0.188·28-s + 0.176·32-s − 0.171·34-s + 0.169·35-s − 1.31·37-s − 0.973·38-s + 0.158·40-s − 0.937·41-s − 1.82·43-s − 0.301·44-s + 0.583·47-s + 1/7·49-s + 0.141·50-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 10710 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10710 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) | |
| 7 | \( 1 - T \) | |
| 17 | \( 1 + T \) | |
| good | 11 | \( 1 + 2 T + p T^{2} \) | 1.11.c |
| 13 | \( 1 + p T^{2} \) | 1.13.a |
| 19 | \( 1 + 6 T + p T^{2} \) | 1.19.g |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 + p T^{2} \) | 1.29.a |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 + 8 T + p T^{2} \) | 1.37.i |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 + 12 T + p T^{2} \) | 1.43.m |
| 47 | \( 1 - 4 T + p T^{2} \) | 1.47.ae |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 + 6 T + p T^{2} \) | 1.59.g |
| 61 | \( 1 + 2 T + p T^{2} \) | 1.61.c |
| 67 | \( 1 + 8 T + p T^{2} \) | 1.67.i |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 + 6 T + p T^{2} \) | 1.73.g |
| 79 | \( 1 + p T^{2} \) | 1.79.a |
| 83 | \( 1 - 6 T + p T^{2} \) | 1.83.ag |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 97 | \( 1 + 2 T + p T^{2} \) | 1.97.c |
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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
−16.86346131486438, −16.20489164651376, −15.50108921079080, −15.02541798620612, −14.64852135273440, −13.77668244355863, −13.48274952259801, −12.94138807470506, −12.22300939291525, −11.80304414867102, −10.96443959713718, −10.48002002156851, −10.08575230977257, −9.068365913775920, −8.523343240994656, −7.908156008177674, −7.051581787786150, −6.552832023115394, −5.829472594232651, −5.149795795675348, −4.636726749524019, −3.826866408131391, −3.002930845692430, −2.175500639328194, −1.547874938738485, 0,
1.547874938738485, 2.175500639328194, 3.002930845692430, 3.826866408131391, 4.636726749524019, 5.149795795675348, 5.829472594232651, 6.552832023115394, 7.051581787786150, 7.908156008177674, 8.523343240994656, 9.068365913775920, 10.08575230977257, 10.48002002156851, 10.96443959713718, 11.80304414867102, 12.22300939291525, 12.94138807470506, 13.48274952259801, 13.77668244355863, 14.64852135273440, 15.02541798620612, 15.50108921079080, 16.20489164651376, 16.86346131486438
