| L(s) = 1 | + 2-s + 4-s − 5-s + 7-s + 8-s − 10-s + 3·11-s + 13-s + 14-s + 16-s − 6·17-s + 2·19-s − 20-s + 3·22-s − 9·23-s + 25-s + 26-s + 28-s − 6·29-s + 5·31-s + 32-s − 6·34-s − 35-s − 7·37-s + 2·38-s − 40-s − 9·41-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 + 0.904·11-s + 0.277·13-s + 0.267·14-s + 1/4·16-s − 1.45·17-s + 0.458·19-s − 0.223·20-s + 0.639·22-s − 1.87·23-s + 1/5·25-s + 0.196·26-s + 0.188·28-s − 1.11·29-s + 0.898·31-s + 0.176·32-s − 1.02·34-s − 0.169·35-s − 1.15·37-s + 0.324·38-s − 0.158·40-s − 1.40·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8190 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8190 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) | |
| 13 | \( 1 - T \) | |
| good | 11 | \( 1 - 3 T + p T^{2} \) | 1.11.ad |
| 17 | \( 1 + 6 T + p T^{2} \) | 1.17.g |
| 19 | \( 1 - 2 T + p T^{2} \) | 1.19.ac |
| 23 | \( 1 + 9 T + p T^{2} \) | 1.23.j |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 - 5 T + p T^{2} \) | 1.31.af |
| 37 | \( 1 + 7 T + p T^{2} \) | 1.37.h |
| 41 | \( 1 + 9 T + p T^{2} \) | 1.41.j |
| 43 | \( 1 + 10 T + p T^{2} \) | 1.43.k |
| 47 | \( 1 + 3 T + p T^{2} \) | 1.47.d |
| 53 | \( 1 + p T^{2} \) | 1.53.a |
| 59 | \( 1 + 12 T + p T^{2} \) | 1.59.m |
| 61 | \( 1 + T + p T^{2} \) | 1.61.b |
| 67 | \( 1 + 13 T + p T^{2} \) | 1.67.n |
| 71 | \( 1 - 12 T + p T^{2} \) | 1.71.am |
| 73 | \( 1 - 11 T + p T^{2} \) | 1.73.al |
| 79 | \( 1 - 17 T + p T^{2} \) | 1.79.ar |
| 83 | \( 1 - 6 T + p T^{2} \) | 1.83.ag |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 97 | \( 1 + 19 T + p T^{2} \) | 1.97.t |
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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
−7.40763354077839180383777566032, −6.49706965430865153113993271148, −6.32078817545384183719404055896, −5.19051988984154532318023526779, −4.65309669947823058501964831138, −3.83257242655015074719387362363, −3.43058769171146650699367646612, −2.16633758076795978441090467118, −1.54930151612806125462070897382, 0,
1.54930151612806125462070897382, 2.16633758076795978441090467118, 3.43058769171146650699367646612, 3.83257242655015074719387362363, 4.65309669947823058501964831138, 5.19051988984154532318023526779, 6.32078817545384183719404055896, 6.49706965430865153113993271148, 7.40763354077839180383777566032
