| L(s) = 1 | + 2-s + 4-s − 2·5-s + 8-s − 2·10-s − 4·11-s + 16-s − 17-s − 4·19-s − 2·20-s − 4·22-s − 25-s + 10·29-s − 8·31-s + 32-s − 34-s + 2·37-s − 4·38-s − 2·40-s + 10·41-s + 12·43-s − 4·44-s − 7·49-s − 50-s − 6·53-s + 8·55-s + 10·58-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s − 0.894·5-s + 0.353·8-s − 0.632·10-s − 1.20·11-s + 1/4·16-s − 0.242·17-s − 0.917·19-s − 0.447·20-s − 0.852·22-s − 1/5·25-s + 1.85·29-s − 1.43·31-s + 0.176·32-s − 0.171·34-s + 0.328·37-s − 0.648·38-s − 0.316·40-s + 1.56·41-s + 1.82·43-s − 0.603·44-s − 49-s − 0.141·50-s − 0.824·53-s + 1.07·55-s + 1.31·58-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 51714 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 51714 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 13 | \( 1 \) | |
| 17 | \( 1 + T \) | |
| good | 5 | \( 1 + 2 T + p T^{2} \) | 1.5.c |
| 7 | \( 1 + p T^{2} \) | 1.7.a |
| 11 | \( 1 + 4 T + p T^{2} \) | 1.11.e |
| 19 | \( 1 + 4 T + p T^{2} \) | 1.19.e |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 - 10 T + p T^{2} \) | 1.29.ak |
| 31 | \( 1 + 8 T + p T^{2} \) | 1.31.i |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 41 | \( 1 - 10 T + p T^{2} \) | 1.41.ak |
| 43 | \( 1 - 12 T + p T^{2} \) | 1.43.am |
| 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 | \( 1 + 10 T + p T^{2} \) | 1.61.k |
| 67 | \( 1 - 12 T + p T^{2} \) | 1.67.am |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 + 10 T + p T^{2} \) | 1.73.k |
| 79 | \( 1 + 8 T + p T^{2} \) | 1.79.i |
| 83 | \( 1 - 4 T + p T^{2} \) | 1.83.ae |
| 89 | \( 1 + 6 T + p T^{2} \) | 1.89.g |
| 97 | \( 1 - 14 T + p T^{2} \) | 1.97.ao |
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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
−14.69268085396058, −14.26846429412790, −13.77620632924828, −13.00018758906546, −12.68674642599943, −12.45870774419958, −11.59381322679460, −11.24649908747256, −10.73849104847743, −10.32021020134128, −9.631099639307153, −8.885312661675985, −8.348828446398061, −7.751178174736247, −7.474328176120557, −6.773622833459754, −6.092162601048353, −5.661704755974815, −4.859097576296529, −4.436792992210715, −3.930229296556916, −3.187226993238680, −2.580529433080218, −2.028475772612476, −0.8779421951309576, 0,
0.8779421951309576, 2.028475772612476, 2.580529433080218, 3.187226993238680, 3.930229296556916, 4.436792992210715, 4.859097576296529, 5.661704755974815, 6.092162601048353, 6.773622833459754, 7.474328176120557, 7.751178174736247, 8.348828446398061, 8.885312661675985, 9.631099639307153, 10.32021020134128, 10.73849104847743, 11.24649908747256, 11.59381322679460, 12.45870774419958, 12.68674642599943, 13.00018758906546, 13.77620632924828, 14.26846429412790, 14.69268085396058
