| L(s) = 1 | − 2-s + 2·3-s + 4-s − 2·6-s + 5·7-s − 8-s + 9-s − 3·11-s + 2·12-s − 5·14-s + 16-s − 17-s − 18-s − 2·19-s + 10·21-s + 3·22-s + 9·23-s − 2·24-s − 4·27-s + 5·28-s + 6·29-s − 8·31-s − 32-s − 6·33-s + 34-s + 36-s + 2·37-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.15·3-s + 1/2·4-s − 0.816·6-s + 1.88·7-s − 0.353·8-s + 1/3·9-s − 0.904·11-s + 0.577·12-s − 1.33·14-s + 1/4·16-s − 0.242·17-s − 0.235·18-s − 0.458·19-s + 2.18·21-s + 0.639·22-s + 1.87·23-s − 0.408·24-s − 0.769·27-s + 0.944·28-s + 1.11·29-s − 1.43·31-s − 0.176·32-s − 1.04·33-s + 0.171·34-s + 1/6·36-s + 0.328·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 143650 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 143650 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 13 | \( 1 \) | |
| 17 | \( 1 + T \) | |
| good | 3 | \( 1 - 2 T + p T^{2} \) | 1.3.ac |
| 7 | \( 1 - 5 T + p T^{2} \) | 1.7.af |
| 11 | \( 1 + 3 T + p T^{2} \) | 1.11.d |
| 19 | \( 1 + 2 T + p T^{2} \) | 1.19.c |
| 23 | \( 1 - 9 T + p T^{2} \) | 1.23.aj |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 + 8 T + p T^{2} \) | 1.31.i |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 41 | \( 1 + 12 T + p T^{2} \) | 1.41.m |
| 43 | \( 1 - T + p T^{2} \) | 1.43.ab |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 + 12 T + p T^{2} \) | 1.59.m |
| 61 | \( 1 + 13 T + p T^{2} \) | 1.61.n |
| 67 | \( 1 - 5 T + p T^{2} \) | 1.67.af |
| 71 | \( 1 + 12 T + p T^{2} \) | 1.71.m |
| 73 | \( 1 - 5 T + p T^{2} \) | 1.73.af |
| 79 | \( 1 - 2 T + p T^{2} \) | 1.79.ac |
| 83 | \( 1 + 15 T + p T^{2} \) | 1.83.p |
| 89 | \( 1 - 9 T + p T^{2} \) | 1.89.aj |
| 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
−13.49983008271386, −13.44231205482268, −12.68519702486529, −12.12184329803861, −11.54701212950764, −11.13165518912797, −10.61405344741047, −10.46408191762047, −9.609035451487889, −9.024863069693427, −8.673313594060079, −8.447917857688285, −7.821983019663635, −7.520814142451815, −7.095779785914075, −6.352103546962742, −5.544728456490369, −5.086110168198841, −4.649672366035039, −3.988621163316347, −3.093841735210082, −2.838703810589421, −2.056048602411006, −1.710194727110118, −1.031813593967525, 0,
1.031813593967525, 1.710194727110118, 2.056048602411006, 2.838703810589421, 3.093841735210082, 3.988621163316347, 4.649672366035039, 5.086110168198841, 5.544728456490369, 6.352103546962742, 7.095779785914075, 7.520814142451815, 7.821983019663635, 8.447917857688285, 8.673313594060079, 9.024863069693427, 9.609035451487889, 10.46408191762047, 10.61405344741047, 11.13165518912797, 11.54701212950764, 12.12184329803861, 12.68519702486529, 13.44231205482268, 13.49983008271386
