| L(s) = 1 | + 2-s − 3-s + 4-s − 5-s − 6-s + 7-s + 8-s − 2·9-s − 10-s − 5·11-s − 12-s − 13-s + 14-s + 15-s + 16-s + 2·17-s − 2·18-s − 6·19-s − 20-s − 21-s − 5·22-s − 3·23-s − 24-s + 25-s − 26-s + 5·27-s + 28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.447·5-s − 0.408·6-s + 0.377·7-s + 0.353·8-s − 2/3·9-s − 0.316·10-s − 1.50·11-s − 0.288·12-s − 0.277·13-s + 0.267·14-s + 0.258·15-s + 1/4·16-s + 0.485·17-s − 0.471·18-s − 1.37·19-s − 0.223·20-s − 0.218·21-s − 1.06·22-s − 0.625·23-s − 0.204·24-s + 1/5·25-s − 0.196·26-s + 0.962·27-s + 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 910 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 910 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) | |
| 5 | \( 1 + T \) | |
| 7 | \( 1 - T \) | |
| 13 | \( 1 + T \) | |
| good | 3 | \( 1 + T + p T^{2} \) | 1.3.b |
| 11 | \( 1 + 5 T + p T^{2} \) | 1.11.f |
| 17 | \( 1 - 2 T + p T^{2} \) | 1.17.ac |
| 19 | \( 1 + 6 T + p T^{2} \) | 1.19.g |
| 23 | \( 1 + 3 T + p T^{2} \) | 1.23.d |
| 29 | \( 1 + 10 T + p T^{2} \) | 1.29.k |
| 31 | \( 1 - 9 T + p T^{2} \) | 1.31.aj |
| 37 | \( 1 + 5 T + p T^{2} \) | 1.37.f |
| 41 | \( 1 - T + p T^{2} \) | 1.41.ab |
| 43 | \( 1 + 10 T + p T^{2} \) | 1.43.k |
| 47 | \( 1 + 9 T + p T^{2} \) | 1.47.j |
| 53 | \( 1 - 8 T + p T^{2} \) | 1.53.ai |
| 59 | \( 1 - 4 T + p T^{2} \) | 1.59.ae |
| 61 | \( 1 - 5 T + p T^{2} \) | 1.61.af |
| 67 | \( 1 + 3 T + p T^{2} \) | 1.67.d |
| 71 | \( 1 + 8 T + p T^{2} \) | 1.71.i |
| 73 | \( 1 + 5 T + p T^{2} \) | 1.73.f |
| 79 | \( 1 - 5 T + p T^{2} \) | 1.79.af |
| 83 | \( 1 + 18 T + p T^{2} \) | 1.83.s |
| 89 | \( 1 - 10 T + p T^{2} \) | 1.89.ak |
| 97 | \( 1 - 5 T + p T^{2} \) | 1.97.af |
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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
−10.08118361209269534739122723327, −8.517979701972612992710112116129, −7.993331686971697038775733351604, −7.00318815334225711564307306201, −5.95542821103104987285543734461, −5.26575472203688448064136784067, −4.48757382051147212534675136997, −3.25841080774946936203711237253, −2.15831730349145946196247078896, 0,
2.15831730349145946196247078896, 3.25841080774946936203711237253, 4.48757382051147212534675136997, 5.26575472203688448064136784067, 5.95542821103104987285543734461, 7.00318815334225711564307306201, 7.993331686971697038775733351604, 8.517979701972612992710112116129, 10.08118361209269534739122723327
