| L(s) = 1 | + 3-s + 2·5-s + 2·7-s + 9-s + 6·13-s + 2·15-s + 2·21-s − 6·23-s − 25-s + 27-s − 6·29-s − 10·31-s + 4·35-s − 2·37-s + 6·39-s − 4·43-s + 2·45-s − 8·47-s − 3·49-s − 6·53-s − 10·61-s + 2·63-s + 12·65-s + 8·67-s − 6·69-s − 10·71-s − 16·73-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.894·5-s + 0.755·7-s + 1/3·9-s + 1.66·13-s + 0.516·15-s + 0.436·21-s − 1.25·23-s − 1/5·25-s + 0.192·27-s − 1.11·29-s − 1.79·31-s + 0.676·35-s − 0.328·37-s + 0.960·39-s − 0.609·43-s + 0.298·45-s − 1.16·47-s − 3/7·49-s − 0.824·53-s − 1.28·61-s + 0.251·63-s + 1.48·65-s + 0.977·67-s − 0.722·69-s − 1.18·71-s − 1.87·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 55488 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 55488 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 3 | \( 1 - T \) | |
| 17 | \( 1 \) | |
| good | 5 | \( 1 - 2 T + p T^{2} \) | 1.5.ac |
| 7 | \( 1 - 2 T + p T^{2} \) | 1.7.ac |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 13 | \( 1 - 6 T + p T^{2} \) | 1.13.ag |
| 19 | \( 1 + p T^{2} \) | 1.19.a |
| 23 | \( 1 + 6 T + p T^{2} \) | 1.23.g |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 + 10 T + p T^{2} \) | 1.31.k |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 + p T^{2} \) | 1.41.a |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 + 8 T + p T^{2} \) | 1.47.i |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 + 10 T + p T^{2} \) | 1.61.k |
| 67 | \( 1 - 8 T + p T^{2} \) | 1.67.ai |
| 71 | \( 1 + 10 T + p T^{2} \) | 1.71.k |
| 73 | \( 1 + 16 T + p T^{2} \) | 1.73.q |
| 79 | \( 1 - 6 T + p T^{2} \) | 1.79.ag |
| 83 | \( 1 - 16 T + p T^{2} \) | 1.83.aq |
| 89 | \( 1 - 10 T + p T^{2} \) | 1.89.ak |
| 97 | \( 1 - 12 T + p T^{2} \) | 1.97.am |
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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
−14.53940765584642, −14.20755231106981, −13.61158296168224, −13.17089942411105, −12.97398726774504, −12.06459196329132, −11.53180754033149, −11.05223295777539, −10.51862085645427, −10.06210350795819, −9.284927558655438, −9.087405322708048, −8.449042907380102, −7.844076771645922, −7.567333860226776, −6.627552011460213, −6.111668465269029, −5.707802411329676, −5.037713930964686, −4.365614132737453, −3.530649606753152, −3.399840967787306, −2.137008802169747, −1.803593044004954, −1.341154403894468, 0,
1.341154403894468, 1.803593044004954, 2.137008802169747, 3.399840967787306, 3.530649606753152, 4.365614132737453, 5.037713930964686, 5.707802411329676, 6.111668465269029, 6.627552011460213, 7.567333860226776, 7.844076771645922, 8.449042907380102, 9.087405322708048, 9.284927558655438, 10.06210350795819, 10.51862085645427, 11.05223295777539, 11.53180754033149, 12.06459196329132, 12.97398726774504, 13.17089942411105, 13.61158296168224, 14.20755231106981, 14.53940765584642
