| L(s) = 1 | + 3-s − 5-s − 2·9-s − 4·11-s + 13-s − 15-s + 3·17-s + 5·23-s − 4·25-s − 5·27-s − 7·29-s − 4·31-s − 4·33-s − 10·37-s + 39-s + 5·41-s − 5·43-s + 2·45-s − 7·47-s − 7·49-s + 3·51-s − 11·53-s + 4·55-s − 3·59-s + 11·61-s − 65-s + 3·67-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.447·5-s − 2/3·9-s − 1.20·11-s + 0.277·13-s − 0.258·15-s + 0.727·17-s + 1.04·23-s − 4/5·25-s − 0.962·27-s − 1.29·29-s − 0.718·31-s − 0.696·33-s − 1.64·37-s + 0.160·39-s + 0.780·41-s − 0.762·43-s + 0.298·45-s − 1.02·47-s − 49-s + 0.420·51-s − 1.51·53-s + 0.539·55-s − 0.390·59-s + 1.40·61-s − 0.124·65-s + 0.366·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1444 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1444 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 19 | \( 1 \) | |
| good | 3 | \( 1 - T + p T^{2} \) | 1.3.ab |
| 5 | \( 1 + T + p T^{2} \) | 1.5.b |
| 7 | \( 1 + p T^{2} \) | 1.7.a |
| 11 | \( 1 + 4 T + p T^{2} \) | 1.11.e |
| 13 | \( 1 - T + p T^{2} \) | 1.13.ab |
| 17 | \( 1 - 3 T + p T^{2} \) | 1.17.ad |
| 23 | \( 1 - 5 T + p T^{2} \) | 1.23.af |
| 29 | \( 1 + 7 T + p T^{2} \) | 1.29.h |
| 31 | \( 1 + 4 T + p T^{2} \) | 1.31.e |
| 37 | \( 1 + 10 T + p T^{2} \) | 1.37.k |
| 41 | \( 1 - 5 T + p T^{2} \) | 1.41.af |
| 43 | \( 1 + 5 T + p T^{2} \) | 1.43.f |
| 47 | \( 1 + 7 T + p T^{2} \) | 1.47.h |
| 53 | \( 1 + 11 T + p T^{2} \) | 1.53.l |
| 59 | \( 1 + 3 T + p T^{2} \) | 1.59.d |
| 61 | \( 1 - 11 T + p T^{2} \) | 1.61.al |
| 67 | \( 1 - 3 T + p T^{2} \) | 1.67.ad |
| 71 | \( 1 + 11 T + p T^{2} \) | 1.71.l |
| 73 | \( 1 - 15 T + p T^{2} \) | 1.73.ap |
| 79 | \( 1 - 13 T + p T^{2} \) | 1.79.an |
| 83 | \( 1 + p T^{2} \) | 1.83.a |
| 89 | \( 1 + 3 T + p T^{2} \) | 1.89.d |
| 97 | \( 1 - 5 T + p T^{2} \) | 1.97.af |
| 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
−9.105008113833057026857036839363, −8.100558399299494778884078521035, −7.84988968865075234633713051253, −6.83040517436078033519471877597, −5.63050469033946262906588250604, −5.06364783637482652474882917253, −3.66113815902587852745351634340, −3.09017486549877239651441604378, −1.89788337754159354249824502826, 0,
1.89788337754159354249824502826, 3.09017486549877239651441604378, 3.66113815902587852745351634340, 5.06364783637482652474882917253, 5.63050469033946262906588250604, 6.83040517436078033519471877597, 7.84988968865075234633713051253, 8.100558399299494778884078521035, 9.105008113833057026857036839363
