| L(s) = 1 | − 5-s − 3·9-s − 2·11-s − 2·13-s − 2·19-s − 8·23-s + 25-s + 29-s + 2·31-s − 4·37-s − 10·41-s + 4·43-s + 3·45-s + 12·47-s − 7·49-s − 6·53-s + 2·55-s − 12·59-s − 10·61-s + 2·65-s + 12·67-s + 12·71-s + 12·73-s + 2·79-s + 9·81-s − 4·83-s − 10·89-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 9-s − 0.603·11-s − 0.554·13-s − 0.458·19-s − 1.66·23-s + 1/5·25-s + 0.185·29-s + 0.359·31-s − 0.657·37-s − 1.56·41-s + 0.609·43-s + 0.447·45-s + 1.75·47-s − 49-s − 0.824·53-s + 0.269·55-s − 1.56·59-s − 1.28·61-s + 0.248·65-s + 1.46·67-s + 1.42·71-s + 1.40·73-s + 0.225·79-s + 81-s − 0.439·83-s − 1.05·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 580 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 580 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 5 | \( 1 + T \) | |
| 29 | \( 1 - T \) | |
| good | 3 | \( 1 + p T^{2} \) | 1.3.a |
| 7 | \( 1 + p T^{2} \) | 1.7.a |
| 11 | \( 1 + 2 T + p T^{2} \) | 1.11.c |
| 13 | \( 1 + 2 T + p T^{2} \) | 1.13.c |
| 17 | \( 1 + p T^{2} \) | 1.17.a |
| 19 | \( 1 + 2 T + p T^{2} \) | 1.19.c |
| 23 | \( 1 + 8 T + p T^{2} \) | 1.23.i |
| 31 | \( 1 - 2 T + p T^{2} \) | 1.31.ac |
| 37 | \( 1 + 4 T + p T^{2} \) | 1.37.e |
| 41 | \( 1 + 10 T + p T^{2} \) | 1.41.k |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 - 12 T + p T^{2} \) | 1.47.am |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 + 12 T + p T^{2} \) | 1.59.m |
| 61 | \( 1 + 10 T + p T^{2} \) | 1.61.k |
| 67 | \( 1 - 12 T + p T^{2} \) | 1.67.am |
| 71 | \( 1 - 12 T + p T^{2} \) | 1.71.am |
| 73 | \( 1 - 12 T + p T^{2} \) | 1.73.am |
| 79 | \( 1 - 2 T + p T^{2} \) | 1.79.ac |
| 83 | \( 1 + 4 T + p T^{2} \) | 1.83.e |
| 89 | \( 1 + 10 T + p T^{2} \) | 1.89.k |
| 97 | \( 1 - 8 T + p T^{2} \) | 1.97.ai |
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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
−10.36899301525826491834627347814, −9.397617347972410257430780166137, −8.329116501837670063664994005112, −7.83015685450328496110225583107, −6.62261929390285823343425035466, −5.64822775290293109104559559401, −4.64564724150104580498645753154, −3.41814948518614960282736687706, −2.25208308434029396683925627401, 0,
2.25208308434029396683925627401, 3.41814948518614960282736687706, 4.64564724150104580498645753154, 5.64822775290293109104559559401, 6.62261929390285823343425035466, 7.83015685450328496110225583107, 8.329116501837670063664994005112, 9.397617347972410257430780166137, 10.36899301525826491834627347814
