| L(s) = 1 | + 3-s + 4·5-s + 9-s + 2·11-s − 13-s + 4·15-s + 2·17-s − 8·19-s − 4·23-s + 11·25-s + 27-s − 6·29-s + 4·31-s + 2·33-s + 6·37-s − 39-s − 12·41-s − 4·43-s + 4·45-s + 6·47-s − 7·49-s + 2·51-s − 2·53-s + 8·55-s − 8·57-s + 14·59-s + 10·61-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1.78·5-s + 1/3·9-s + 0.603·11-s − 0.277·13-s + 1.03·15-s + 0.485·17-s − 1.83·19-s − 0.834·23-s + 11/5·25-s + 0.192·27-s − 1.11·29-s + 0.718·31-s + 0.348·33-s + 0.986·37-s − 0.160·39-s − 1.87·41-s − 0.609·43-s + 0.596·45-s + 0.875·47-s − 49-s + 0.280·51-s − 0.274·53-s + 1.07·55-s − 1.05·57-s + 1.82·59-s + 1.28·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 624 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 624 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.372685272\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.372685272\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) | |
| 13 | \( 1 + T \) | |
| good | 5 | \( 1 - 4 T + p T^{2} \) | 1.5.ae |
| 7 | \( 1 + p T^{2} \) | 1.7.a |
| 11 | \( 1 - 2 T + p T^{2} \) | 1.11.ac |
| 17 | \( 1 - 2 T + p T^{2} \) | 1.17.ac |
| 19 | \( 1 + 8 T + p T^{2} \) | 1.19.i |
| 23 | \( 1 + 4 T + p T^{2} \) | 1.23.e |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 - 4 T + p T^{2} \) | 1.31.ae |
| 37 | \( 1 - 6 T + p T^{2} \) | 1.37.ag |
| 41 | \( 1 + 12 T + p T^{2} \) | 1.41.m |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 - 6 T + p T^{2} \) | 1.47.ag |
| 53 | \( 1 + 2 T + p T^{2} \) | 1.53.c |
| 59 | \( 1 - 14 T + p T^{2} \) | 1.59.ao |
| 61 | \( 1 - 10 T + p T^{2} \) | 1.61.ak |
| 67 | \( 1 - 4 T + p T^{2} \) | 1.67.ae |
| 71 | \( 1 + 2 T + p T^{2} \) | 1.71.c |
| 73 | \( 1 + 2 T + p T^{2} \) | 1.73.c |
| 79 | \( 1 - 8 T + p T^{2} \) | 1.79.ai |
| 83 | \( 1 + 14 T + p T^{2} \) | 1.83.o |
| 89 | \( 1 + p T^{2} \) | 1.89.a |
| 97 | \( 1 + 10 T + p T^{2} \) | 1.97.k |
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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
−10.15794269434022573867939029827, −9.892497274915361873535985179047, −8.943402214761282290578247197554, −8.217111615393858677135023040041, −6.82576336993450810735269740334, −6.18574778583849864438867610565, −5.19839034221395538345933157261, −3.94825004599146973995971761943, −2.49937121389233024134417293034, −1.67948869184017392713797049524,
1.67948869184017392713797049524, 2.49937121389233024134417293034, 3.94825004599146973995971761943, 5.19839034221395538345933157261, 6.18574778583849864438867610565, 6.82576336993450810735269740334, 8.217111615393858677135023040041, 8.943402214761282290578247197554, 9.892497274915361873535985179047, 10.15794269434022573867939029827
