| L(s) = 1 | + 2-s + 4-s + 2·5-s − 7-s + 8-s + 2·10-s + 11-s − 14-s + 16-s + 2·20-s + 22-s − 5·23-s − 25-s − 28-s + 29-s + 5·31-s + 32-s − 2·35-s + 3·37-s + 2·40-s + 6·41-s + 43-s + 44-s − 5·46-s + 4·47-s − 6·49-s − 50-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.894·5-s − 0.377·7-s + 0.353·8-s + 0.632·10-s + 0.301·11-s − 0.267·14-s + 1/4·16-s + 0.447·20-s + 0.213·22-s − 1.04·23-s − 1/5·25-s − 0.188·28-s + 0.185·29-s + 0.898·31-s + 0.176·32-s − 0.338·35-s + 0.493·37-s + 0.316·40-s + 0.937·41-s + 0.152·43-s + 0.150·44-s − 0.737·46-s + 0.583·47-s − 6/7·49-s − 0.141·50-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9126 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9126 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.018086842\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.018086842\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 13 | \( 1 \) | |
| good | 5 | \( 1 - 2 T + p T^{2} \) | 1.5.ac |
| 7 | \( 1 + T + p T^{2} \) | 1.7.b |
| 11 | \( 1 - T + p T^{2} \) | 1.11.ab |
| 17 | \( 1 + p T^{2} \) | 1.17.a |
| 19 | \( 1 + p T^{2} \) | 1.19.a |
| 23 | \( 1 + 5 T + p T^{2} \) | 1.23.f |
| 29 | \( 1 - T + p T^{2} \) | 1.29.ab |
| 31 | \( 1 - 5 T + p T^{2} \) | 1.31.af |
| 37 | \( 1 - 3 T + p T^{2} \) | 1.37.ad |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 - T + p T^{2} \) | 1.43.ab |
| 47 | \( 1 - 4 T + p T^{2} \) | 1.47.ae |
| 53 | \( 1 - 3 T + p T^{2} \) | 1.53.ad |
| 59 | \( 1 - 11 T + p T^{2} \) | 1.59.al |
| 61 | \( 1 - 4 T + p T^{2} \) | 1.61.ae |
| 67 | \( 1 - 8 T + p T^{2} \) | 1.67.ai |
| 71 | \( 1 + 8 T + p T^{2} \) | 1.71.i |
| 73 | \( 1 - 16 T + p T^{2} \) | 1.73.aq |
| 79 | \( 1 - 8 T + p T^{2} \) | 1.79.ai |
| 83 | \( 1 - 9 T + p T^{2} \) | 1.83.aj |
| 89 | \( 1 + 9 T + p T^{2} \) | 1.89.j |
| 97 | \( 1 - 4 T + p T^{2} \) | 1.97.ae |
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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
−7.65251858417770489815548669600, −6.75951072189175211374634589368, −6.25707492356602596076031387323, −5.74597584272867140878874894185, −5.02546003404257615140957509429, −4.18375699409982458853189619571, −3.56044527638303425192510885631, −2.54990619069967869651192670906, −2.02445069549739722127519956246, −0.871515025639793957514033854437,
0.871515025639793957514033854437, 2.02445069549739722127519956246, 2.54990619069967869651192670906, 3.56044527638303425192510885631, 4.18375699409982458853189619571, 5.02546003404257615140957509429, 5.74597584272867140878874894185, 6.25707492356602596076031387323, 6.75951072189175211374634589368, 7.65251858417770489815548669600
