| L(s) = 1 | + 2·5-s + 7-s − 13-s + 4·17-s + 4·19-s + 6·23-s − 25-s − 6·29-s + 2·35-s + 6·37-s + 6·41-s − 4·43-s + 2·47-s + 49-s − 6·53-s − 6·59-s + 14·61-s − 2·65-s + 4·67-s + 4·71-s + 6·73-s + 4·79-s − 6·83-s + 8·85-s + 2·89-s − 91-s + 8·95-s + ⋯ |
| L(s) = 1 | + 0.894·5-s + 0.377·7-s − 0.277·13-s + 0.970·17-s + 0.917·19-s + 1.25·23-s − 1/5·25-s − 1.11·29-s + 0.338·35-s + 0.986·37-s + 0.937·41-s − 0.609·43-s + 0.291·47-s + 1/7·49-s − 0.824·53-s − 0.781·59-s + 1.79·61-s − 0.248·65-s + 0.488·67-s + 0.474·71-s + 0.702·73-s + 0.450·79-s − 0.658·83-s + 0.867·85-s + 0.211·89-s − 0.104·91-s + 0.820·95-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 52416 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 52416 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.685509176\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.685509176\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 7 | \( 1 - T \) | |
| 13 | \( 1 + T \) | |
| good | 5 | \( 1 - 2 T + p T^{2} \) | 1.5.ac |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 17 | \( 1 - 4 T + p T^{2} \) | 1.17.ae |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 23 | \( 1 - 6 T + p T^{2} \) | 1.23.ag |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 - 6 T + p T^{2} \) | 1.37.ag |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 - 2 T + p T^{2} \) | 1.47.ac |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 + 6 T + p T^{2} \) | 1.59.g |
| 61 | \( 1 - 14 T + p T^{2} \) | 1.61.ao |
| 67 | \( 1 - 4 T + p T^{2} \) | 1.67.ae |
| 71 | \( 1 - 4 T + p T^{2} \) | 1.71.ae |
| 73 | \( 1 - 6 T + p T^{2} \) | 1.73.ag |
| 79 | \( 1 - 4 T + p T^{2} \) | 1.79.ae |
| 83 | \( 1 + 6 T + p T^{2} \) | 1.83.g |
| 89 | \( 1 - 2 T + p T^{2} \) | 1.89.ac |
| 97 | \( 1 + 6 T + p T^{2} \) | 1.97.g |
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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
−14.39997044481942, −14.04323973897766, −13.47178647751335, −12.89246036262728, −12.62299654625178, −11.83340647415933, −11.34001438512000, −10.98014253433784, −10.19843852666830, −9.775615759212259, −9.370902038435256, −8.898101692973244, −8.067116735480143, −7.631303056942774, −7.162392552949088, −6.406559931826635, −5.864645840871777, −5.255539266415419, −5.015332146625843, −4.083483529572088, −3.425289765830213, −2.759082033407990, −2.096650584731705, −1.368645737077701, −0.7098445272670277,
0.7098445272670277, 1.368645737077701, 2.096650584731705, 2.759082033407990, 3.425289765830213, 4.083483529572088, 5.015332146625843, 5.255539266415419, 5.864645840871777, 6.406559931826635, 7.162392552949088, 7.631303056942774, 8.067116735480143, 8.898101692973244, 9.370902038435256, 9.775615759212259, 10.19843852666830, 10.98014253433784, 11.34001438512000, 11.83340647415933, 12.62299654625178, 12.89246036262728, 13.47178647751335, 14.04323973897766, 14.39997044481942
