| L(s) = 1 | + 2·7-s − 6·11-s + 13-s + 3·17-s + 2·19-s − 9·23-s − 5·25-s + 6·29-s + 8·31-s − 10·37-s − 6·41-s + 11·43-s − 6·47-s − 3·49-s − 9·53-s − 61-s − 4·67-s + 12·71-s − 16·73-s − 12·77-s − 79-s + 6·83-s − 6·89-s + 2·91-s + 8·97-s − 3·101-s − 16·103-s + ⋯ |
| L(s) = 1 | + 0.755·7-s − 1.80·11-s + 0.277·13-s + 0.727·17-s + 0.458·19-s − 1.87·23-s − 25-s + 1.11·29-s + 1.43·31-s − 1.64·37-s − 0.937·41-s + 1.67·43-s − 0.875·47-s − 3/7·49-s − 1.23·53-s − 0.128·61-s − 0.488·67-s + 1.42·71-s − 1.87·73-s − 1.36·77-s − 0.112·79-s + 0.658·83-s − 0.635·89-s + 0.209·91-s + 0.812·97-s − 0.298·101-s − 1.57·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4212 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4212 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 3 | \( 1 \) | |
| 13 | \( 1 - T \) | |
| good | 5 | \( 1 + p T^{2} \) | 1.5.a |
| 7 | \( 1 - 2 T + p T^{2} \) | 1.7.ac |
| 11 | \( 1 + 6 T + p T^{2} \) | 1.11.g |
| 17 | \( 1 - 3 T + p T^{2} \) | 1.17.ad |
| 19 | \( 1 - 2 T + p T^{2} \) | 1.19.ac |
| 23 | \( 1 + 9 T + p T^{2} \) | 1.23.j |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 - 8 T + p T^{2} \) | 1.31.ai |
| 37 | \( 1 + 10 T + p T^{2} \) | 1.37.k |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 - 11 T + p T^{2} \) | 1.43.al |
| 47 | \( 1 + 6 T + p T^{2} \) | 1.47.g |
| 53 | \( 1 + 9 T + p T^{2} \) | 1.53.j |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 + T + p T^{2} \) | 1.61.b |
| 67 | \( 1 + 4 T + p T^{2} \) | 1.67.e |
| 71 | \( 1 - 12 T + p T^{2} \) | 1.71.am |
| 73 | \( 1 + 16 T + p T^{2} \) | 1.73.q |
| 79 | \( 1 + T + p T^{2} \) | 1.79.b |
| 83 | \( 1 - 6 T + p T^{2} \) | 1.83.ag |
| 89 | \( 1 + 6 T + p T^{2} \) | 1.89.g |
| 97 | \( 1 - 8 T + p T^{2} \) | 1.97.ai |
| 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
−8.011485516047432859802092989700, −7.63917040215018426929877996228, −6.50735049473854513344086448375, −5.69982767426692867170226691618, −5.10656866705605840759876110111, −4.35594400132373968398710701366, −3.30803258045840683268212768301, −2.44746961539795065405643763958, −1.48225262084884234583906972441, 0,
1.48225262084884234583906972441, 2.44746961539795065405643763958, 3.30803258045840683268212768301, 4.35594400132373968398710701366, 5.10656866705605840759876110111, 5.69982767426692867170226691618, 6.50735049473854513344086448375, 7.63917040215018426929877996228, 8.011485516047432859802092989700
