| L(s) = 1 | + 2-s − 4-s − 5-s + 7-s − 3·8-s − 10-s + 4·11-s + 14-s − 16-s − 6·17-s − 8·19-s + 20-s + 4·22-s + 4·23-s + 25-s − 28-s + 2·29-s + 4·31-s + 5·32-s − 6·34-s − 35-s − 10·37-s − 8·38-s + 3·40-s − 2·41-s + 8·43-s − 4·44-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1/2·4-s − 0.447·5-s + 0.377·7-s − 1.06·8-s − 0.316·10-s + 1.20·11-s + 0.267·14-s − 1/4·16-s − 1.45·17-s − 1.83·19-s + 0.223·20-s + 0.852·22-s + 0.834·23-s + 1/5·25-s − 0.188·28-s + 0.371·29-s + 0.718·31-s + 0.883·32-s − 1.02·34-s − 0.169·35-s − 1.64·37-s − 1.29·38-s + 0.474·40-s − 0.312·41-s + 1.21·43-s − 0.603·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 53235 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 53235 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) | |
| 5 | \( 1 + T \) | |
| 7 | \( 1 - T \) | |
| 13 | \( 1 \) | |
| good | 2 | \( 1 - T + p T^{2} \) | 1.2.ab |
| 11 | \( 1 - 4 T + p T^{2} \) | 1.11.ae |
| 17 | \( 1 + 6 T + p T^{2} \) | 1.17.g |
| 19 | \( 1 + 8 T + p T^{2} \) | 1.19.i |
| 23 | \( 1 - 4 T + p T^{2} \) | 1.23.ae |
| 29 | \( 1 - 2 T + p T^{2} \) | 1.29.ac |
| 31 | \( 1 - 4 T + p T^{2} \) | 1.31.ae |
| 37 | \( 1 + 10 T + p T^{2} \) | 1.37.k |
| 41 | \( 1 + 2 T + p T^{2} \) | 1.41.c |
| 43 | \( 1 - 8 T + p T^{2} \) | 1.43.ai |
| 47 | \( 1 - 8 T + p T^{2} \) | 1.47.ai |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 - 8 T + p T^{2} \) | 1.59.ai |
| 61 | \( 1 + 10 T + p T^{2} \) | 1.61.k |
| 67 | \( 1 - 12 T + p T^{2} \) | 1.67.am |
| 71 | \( 1 + 8 T + p T^{2} \) | 1.71.i |
| 73 | \( 1 + 6 T + p T^{2} \) | 1.73.g |
| 79 | \( 1 - 8 T + p T^{2} \) | 1.79.ai |
| 83 | \( 1 + 4 T + p T^{2} \) | 1.83.e |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 97 | \( 1 - 2 T + p T^{2} \) | 1.97.ac |
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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
−14.50574858678673, −14.32590374222989, −13.64232567583773, −13.27224067928531, −12.61390063609577, −12.29315757069830, −11.77050052824713, −11.18647377533501, −10.72963925841582, −10.19429807511595, −9.236958315785313, −8.950941779606126, −8.605897595208677, −8.049475880313802, −7.132408644215556, −6.607480941253157, −6.300715391105631, −5.508210810179620, −4.803611581940835, −4.313476555718261, −4.082586653864797, −3.344041607509224, −2.561563185586372, −1.861552202237150, −0.8722399294079397, 0,
0.8722399294079397, 1.861552202237150, 2.561563185586372, 3.344041607509224, 4.082586653864797, 4.313476555718261, 4.803611581940835, 5.508210810179620, 6.300715391105631, 6.607480941253157, 7.132408644215556, 8.049475880313802, 8.605897595208677, 8.950941779606126, 9.236958315785313, 10.19429807511595, 10.72963925841582, 11.18647377533501, 11.77050052824713, 12.29315757069830, 12.61390063609577, 13.27224067928531, 13.64232567583773, 14.32590374222989, 14.50574858678673
