| L(s) = 1 | + 2-s + 3-s + 4-s − 5-s + 6-s − 7-s + 8-s − 2·9-s − 10-s + 12-s + 3·13-s − 14-s − 15-s + 16-s − 7·17-s − 2·18-s − 20-s − 21-s − 5·23-s + 24-s + 25-s + 3·26-s − 5·27-s − 28-s + 5·29-s − 30-s − 10·31-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 + 0.288·12-s + 0.832·13-s − 0.267·14-s − 0.258·15-s + 1/4·16-s − 1.69·17-s − 0.471·18-s − 0.223·20-s − 0.218·21-s − 1.04·23-s + 0.204·24-s + 1/5·25-s + 0.588·26-s − 0.962·27-s − 0.188·28-s + 0.928·29-s − 0.182·30-s − 1.79·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3610 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3610 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) | |
| 19 | \( 1 \) | |
| good | 3 | \( 1 - T + p T^{2} \) | 1.3.ab |
| 7 | \( 1 + T + p T^{2} \) | 1.7.b |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 13 | \( 1 - 3 T + p T^{2} \) | 1.13.ad |
| 17 | \( 1 + 7 T + p T^{2} \) | 1.17.h |
| 23 | \( 1 + 5 T + p T^{2} \) | 1.23.f |
| 29 | \( 1 - 5 T + p T^{2} \) | 1.29.af |
| 31 | \( 1 + 10 T + p T^{2} \) | 1.31.k |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 + 2 T + p T^{2} \) | 1.41.c |
| 43 | \( 1 - 6 T + p T^{2} \) | 1.43.ag |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 + 9 T + p T^{2} \) | 1.53.j |
| 59 | \( 1 - 7 T + p T^{2} \) | 1.59.ah |
| 61 | \( 1 + 4 T + p T^{2} \) | 1.61.e |
| 67 | \( 1 + 7 T + p T^{2} \) | 1.67.h |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 + 9 T + p T^{2} \) | 1.73.j |
| 79 | \( 1 - 10 T + p T^{2} \) | 1.79.ak |
| 83 | \( 1 + 2 T + p T^{2} \) | 1.83.c |
| 89 | \( 1 - 10 T + p T^{2} \) | 1.89.ak |
| 97 | \( 1 - 18 T + p T^{2} \) | 1.97.as |
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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
−8.191868264089508253256338295144, −7.44535643802257450278537419964, −6.51993119387254589452561824735, −6.04316355352539573674489533411, −5.05715563726441276240225133967, −4.12437728160403676428154736833, −3.55039547730181048275638828106, −2.70581739593662064650976028533, −1.81036261835437996152074082764, 0,
1.81036261835437996152074082764, 2.70581739593662064650976028533, 3.55039547730181048275638828106, 4.12437728160403676428154736833, 5.05715563726441276240225133967, 6.04316355352539573674489533411, 6.51993119387254589452561824735, 7.44535643802257450278537419964, 8.191868264089508253256338295144
