| L(s) = 1 | + 2-s − 3-s + 4-s − 3·5-s − 6-s − 3·7-s + 8-s − 2·9-s − 3·10-s − 12-s − 3·14-s + 3·15-s + 16-s − 3·17-s − 2·18-s − 6·19-s − 3·20-s + 3·21-s + 6·23-s − 24-s + 4·25-s + 5·27-s − 3·28-s + 3·30-s + 32-s − 3·34-s + 9·35-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s − 1.34·5-s − 0.408·6-s − 1.13·7-s + 0.353·8-s − 2/3·9-s − 0.948·10-s − 0.288·12-s − 0.801·14-s + 0.774·15-s + 1/4·16-s − 0.727·17-s − 0.471·18-s − 1.37·19-s − 0.670·20-s + 0.654·21-s + 1.25·23-s − 0.204·24-s + 4/5·25-s + 0.962·27-s − 0.566·28-s + 0.547·30-s + 0.176·32-s − 0.514·34-s + 1.52·35-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 338 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 338 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) | |
| 13 | \( 1 \) | |
| good | 3 | \( 1 + T + p T^{2} \) | 1.3.b |
| 5 | \( 1 + 3 T + p T^{2} \) | 1.5.d |
| 7 | \( 1 + 3 T + p T^{2} \) | 1.7.d |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 17 | \( 1 + 3 T + p T^{2} \) | 1.17.d |
| 19 | \( 1 + 6 T + p T^{2} \) | 1.19.g |
| 23 | \( 1 - 6 T + p T^{2} \) | 1.23.ag |
| 29 | \( 1 + p T^{2} \) | 1.29.a |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 + 3 T + p T^{2} \) | 1.37.d |
| 41 | \( 1 + p T^{2} \) | 1.41.a |
| 43 | \( 1 - T + p T^{2} \) | 1.43.ab |
| 47 | \( 1 + 3 T + p T^{2} \) | 1.47.d |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 - 6 T + p T^{2} \) | 1.59.ag |
| 61 | \( 1 + 8 T + p T^{2} \) | 1.61.i |
| 67 | \( 1 + 12 T + p T^{2} \) | 1.67.m |
| 71 | \( 1 - 15 T + p T^{2} \) | 1.71.ap |
| 73 | \( 1 + 6 T + p T^{2} \) | 1.73.g |
| 79 | \( 1 - 10 T + p T^{2} \) | 1.79.ak |
| 83 | \( 1 - 6 T + p T^{2} \) | 1.83.ag |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 97 | \( 1 + 12 T + p T^{2} \) | 1.97.m |
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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
−11.17340990769020288918868897736, −10.65715903774738937428432927887, −9.146762478880660559641766391878, −8.145771200216828229562409340065, −6.89586188102753730335166773104, −6.28038385958604892687331420872, −4.97317601947872031747497490464, −3.91117219162115692903174376516, −2.87766967614925167756596691984, 0,
2.87766967614925167756596691984, 3.91117219162115692903174376516, 4.97317601947872031747497490464, 6.28038385958604892687331420872, 6.89586188102753730335166773104, 8.145771200216828229562409340065, 9.146762478880660559641766391878, 10.65715903774738937428432927887, 11.17340990769020288918868897736
