| L(s) = 1 | − 2-s − 3-s + 4-s − 2·5-s + 6-s + 7-s − 8-s + 9-s + 2·10-s − 2·11-s − 12-s + 4·13-s − 14-s + 2·15-s + 16-s + 17-s − 18-s − 2·19-s − 2·20-s − 21-s + 2·22-s − 4·23-s + 24-s − 25-s − 4·26-s − 27-s + 28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.894·5-s + 0.408·6-s + 0.377·7-s − 0.353·8-s + 1/3·9-s + 0.632·10-s − 0.603·11-s − 0.288·12-s + 1.10·13-s − 0.267·14-s + 0.516·15-s + 1/4·16-s + 0.242·17-s − 0.235·18-s − 0.458·19-s − 0.447·20-s − 0.218·21-s + 0.426·22-s − 0.834·23-s + 0.204·24-s − 1/5·25-s − 0.784·26-s − 0.192·27-s + 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 714 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 714 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) | |
| 3 | \( 1 + T \) | |
| 7 | \( 1 - T \) | |
| 17 | \( 1 - T \) | |
| good | 5 | \( 1 + 2 T + p T^{2} \) | 1.5.c |
| 11 | \( 1 + 2 T + p T^{2} \) | 1.11.c |
| 13 | \( 1 - 4 T + p T^{2} \) | 1.13.ae |
| 19 | \( 1 + 2 T + p T^{2} \) | 1.19.c |
| 23 | \( 1 + 4 T + p T^{2} \) | 1.23.e |
| 29 | \( 1 + p T^{2} \) | 1.29.a |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 + 8 T + p T^{2} \) | 1.37.i |
| 41 | \( 1 + 2 T + p T^{2} \) | 1.41.c |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 + 10 T + p T^{2} \) | 1.59.k |
| 61 | \( 1 + 6 T + p T^{2} \) | 1.61.g |
| 67 | \( 1 + p T^{2} \) | 1.67.a |
| 71 | \( 1 + 8 T + p T^{2} \) | 1.71.i |
| 73 | \( 1 + 6 T + p T^{2} \) | 1.73.g |
| 79 | \( 1 + 12 T + p T^{2} \) | 1.79.m |
| 83 | \( 1 - 6 T + p T^{2} \) | 1.83.ag |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 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
−10.24806755681582212164444761454, −9.011731510253748412404240731210, −8.168533204068992686254287894534, −7.61191197369836842681142112998, −6.53839129510517023307217018082, −5.63367551876896838319289589237, −4.43544805921715351911578987483, −3.35161964452269235625257955776, −1.65281106742929297340572396932, 0,
1.65281106742929297340572396932, 3.35161964452269235625257955776, 4.43544805921715351911578987483, 5.63367551876896838319289589237, 6.53839129510517023307217018082, 7.61191197369836842681142112998, 8.168533204068992686254287894534, 9.011731510253748412404240731210, 10.24806755681582212164444761454
