| L(s) = 1 | + 2-s + 4-s + 5-s − 2·7-s + 8-s + 10-s − 2·11-s + 2·13-s − 2·14-s + 16-s − 8·19-s + 20-s − 2·22-s − 8·23-s + 25-s + 2·26-s − 2·28-s + 2·29-s + 32-s − 2·35-s − 8·37-s − 8·38-s + 40-s − 8·41-s − 43-s − 2·44-s − 8·46-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.447·5-s − 0.755·7-s + 0.353·8-s + 0.316·10-s − 0.603·11-s + 0.554·13-s − 0.534·14-s + 1/4·16-s − 1.83·19-s + 0.223·20-s − 0.426·22-s − 1.66·23-s + 1/5·25-s + 0.392·26-s − 0.377·28-s + 0.371·29-s + 0.176·32-s − 0.338·35-s − 1.31·37-s − 1.29·38-s + 0.158·40-s − 1.24·41-s − 0.152·43-s − 0.301·44-s − 1.17·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3870 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3870 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) | |
| 43 | \( 1 + T \) | |
| good | 7 | \( 1 + 2 T + p T^{2} \) | 1.7.c |
| 11 | \( 1 + 2 T + p T^{2} \) | 1.11.c |
| 13 | \( 1 - 2 T + p T^{2} \) | 1.13.ac |
| 17 | \( 1 + p T^{2} \) | 1.17.a |
| 19 | \( 1 + 8 T + p T^{2} \) | 1.19.i |
| 23 | \( 1 + 8 T + p T^{2} \) | 1.23.i |
| 29 | \( 1 - 2 T + p T^{2} \) | 1.29.ac |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 + 8 T + p T^{2} \) | 1.37.i |
| 41 | \( 1 + 8 T + p T^{2} \) | 1.41.i |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 - 2 T + p T^{2} \) | 1.53.ac |
| 59 | \( 1 - 6 T + p T^{2} \) | 1.59.ag |
| 61 | \( 1 - 8 T + p T^{2} \) | 1.61.ai |
| 67 | \( 1 - 12 T + p T^{2} \) | 1.67.am |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 + 2 T + p T^{2} \) | 1.73.c |
| 79 | \( 1 + 12 T + p T^{2} \) | 1.79.m |
| 83 | \( 1 + 18 T + p T^{2} \) | 1.83.s |
| 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
−8.319032662008985682304769683964, −7.07971684083888780573394487818, −6.49731751142243162145368747002, −5.91693749124977648487288049255, −5.17604758376517246625086087791, −4.19644283660445282828933934104, −3.54247184883496116287275373735, −2.53098797024157302197884676883, −1.76967905770166065871918754646, 0,
1.76967905770166065871918754646, 2.53098797024157302197884676883, 3.54247184883496116287275373735, 4.19644283660445282828933934104, 5.17604758376517246625086087791, 5.91693749124977648487288049255, 6.49731751142243162145368747002, 7.07971684083888780573394487818, 8.319032662008985682304769683964
