| L(s) = 1 | − 2-s − 3-s − 4-s − 5-s + 6-s + 7-s + 3·8-s − 2·9-s + 10-s + 4·11-s + 12-s − 14-s + 15-s − 16-s − 17-s + 2·18-s + 19-s + 20-s − 21-s − 4·22-s − 3·23-s − 3·24-s + 25-s + 5·27-s − 28-s − 6·29-s − 30-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 + 1.06·8-s − 2/3·9-s + 0.316·10-s + 1.20·11-s + 0.288·12-s − 0.267·14-s + 0.258·15-s − 1/4·16-s − 0.242·17-s + 0.471·18-s + 0.229·19-s + 0.223·20-s − 0.218·21-s − 0.852·22-s − 0.625·23-s − 0.612·24-s + 1/5·25-s + 0.962·27-s − 0.188·28-s − 1.11·29-s − 0.182·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 665 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 665 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 + T \) | |
| 7 | \( 1 - T \) | |
| 19 | \( 1 - T \) | |
| good | 2 | \( 1 + T + p T^{2} \) | 1.2.b |
| 3 | \( 1 + T + p T^{2} \) | 1.3.b |
| 11 | \( 1 - 4 T + p T^{2} \) | 1.11.ae |
| 13 | \( 1 + p T^{2} \) | 1.13.a |
| 17 | \( 1 + T + p T^{2} \) | 1.17.b |
| 23 | \( 1 + 3 T + p T^{2} \) | 1.23.d |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 + 11 T + p T^{2} \) | 1.37.l |
| 41 | \( 1 - 5 T + p T^{2} \) | 1.41.af |
| 43 | \( 1 + 7 T + p T^{2} \) | 1.43.h |
| 47 | \( 1 + 8 T + p T^{2} \) | 1.47.i |
| 53 | \( 1 + 3 T + p T^{2} \) | 1.53.d |
| 59 | \( 1 - 2 T + p T^{2} \) | 1.59.ac |
| 61 | \( 1 + 8 T + p T^{2} \) | 1.61.i |
| 67 | \( 1 - 2 T + p T^{2} \) | 1.67.ac |
| 71 | \( 1 - T + p T^{2} \) | 1.71.ab |
| 73 | \( 1 + 11 T + p T^{2} \) | 1.73.l |
| 79 | \( 1 + 12 T + p T^{2} \) | 1.79.m |
| 83 | \( 1 + p T^{2} \) | 1.83.a |
| 89 | \( 1 - 10 T + p T^{2} \) | 1.89.ak |
| 97 | \( 1 + 2 T + p T^{2} \) | 1.97.c |
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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
−10.04201122052202230625551273309, −9.089511133962875088821967125514, −8.525739829803107516492723214102, −7.62048335838313796019554431286, −6.62284643852099624859279541625, −5.50776315197719585679607520825, −4.55439283051130886075090275413, −3.54513003562805481623977464655, −1.55803189478422709123567323036, 0,
1.55803189478422709123567323036, 3.54513003562805481623977464655, 4.55439283051130886075090275413, 5.50776315197719585679607520825, 6.62284643852099624859279541625, 7.62048335838313796019554431286, 8.525739829803107516492723214102, 9.089511133962875088821967125514, 10.04201122052202230625551273309
