| L(s) = 1 | + 2-s + 4-s + 5-s + 7-s + 8-s + 10-s − 11-s − 6·13-s + 14-s + 16-s − 2·17-s − 6·19-s + 20-s − 22-s − 23-s + 25-s − 6·26-s + 28-s + 7·29-s + 32-s − 2·34-s + 35-s − 2·37-s − 6·38-s + 40-s − 6·41-s + 2·43-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.447·5-s + 0.377·7-s + 0.353·8-s + 0.316·10-s − 0.301·11-s − 1.66·13-s + 0.267·14-s + 1/4·16-s − 0.485·17-s − 1.37·19-s + 0.223·20-s − 0.213·22-s − 0.208·23-s + 1/5·25-s − 1.17·26-s + 0.188·28-s + 1.29·29-s + 0.176·32-s − 0.342·34-s + 0.169·35-s − 0.328·37-s − 0.973·38-s + 0.158·40-s − 0.937·41-s + 0.304·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6210 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 5 | \( 1 - T \) | |
| 23 | \( 1 + T \) | |
| good | 7 | \( 1 - T + p T^{2} \) | 1.7.ab |
| 11 | \( 1 + T + p T^{2} \) | 1.11.b |
| 13 | \( 1 + 6 T + p T^{2} \) | 1.13.g |
| 17 | \( 1 + 2 T + p T^{2} \) | 1.17.c |
| 19 | \( 1 + 6 T + p T^{2} \) | 1.19.g |
| 29 | \( 1 - 7 T + p T^{2} \) | 1.29.ah |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 - 2 T + p T^{2} \) | 1.43.ac |
| 47 | \( 1 + 3 T + p T^{2} \) | 1.47.d |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 - T + p T^{2} \) | 1.61.ab |
| 67 | \( 1 + 10 T + p T^{2} \) | 1.67.k |
| 71 | \( 1 + 13 T + p T^{2} \) | 1.71.n |
| 73 | \( 1 - 6 T + p T^{2} \) | 1.73.ag |
| 79 | \( 1 + 4 T + p T^{2} \) | 1.79.e |
| 83 | \( 1 - 4 T + p T^{2} \) | 1.83.ae |
| 89 | \( 1 + 7 T + p T^{2} \) | 1.89.h |
| 97 | \( 1 - 6 T + p T^{2} \) | 1.97.ag |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.62049549965919591404158224202, −6.79896923241648647833509554503, −6.33079932214340623922196767951, −5.40130024458059010048162245929, −4.74199772798651453374853346578, −4.33811674090356828550985286797, −3.08797240706266296139631090562, −2.39905437978870241941973946650, −1.67349036542362161280137807725, 0,
1.67349036542362161280137807725, 2.39905437978870241941973946650, 3.08797240706266296139631090562, 4.33811674090356828550985286797, 4.74199772798651453374853346578, 5.40130024458059010048162245929, 6.33079932214340623922196767951, 6.79896923241648647833509554503, 7.62049549965919591404158224202
