| L(s) = 1 | − 5-s − 7-s − 3·9-s − 4·11-s + 2·13-s + 6·17-s + 4·19-s + 8·23-s + 25-s + 6·29-s + 35-s + 10·37-s + 10·41-s − 12·43-s + 3·45-s + 49-s − 6·53-s + 4·55-s − 12·59-s + 2·61-s + 3·63-s − 2·65-s − 4·67-s + 12·71-s − 10·73-s + 4·77-s + 4·79-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 0.377·7-s − 9-s − 1.20·11-s + 0.554·13-s + 1.45·17-s + 0.917·19-s + 1.66·23-s + 1/5·25-s + 1.11·29-s + 0.169·35-s + 1.64·37-s + 1.56·41-s − 1.82·43-s + 0.447·45-s + 1/7·49-s − 0.824·53-s + 0.539·55-s − 1.56·59-s + 0.256·61-s + 0.377·63-s − 0.248·65-s − 0.488·67-s + 1.42·71-s − 1.17·73-s + 0.455·77-s + 0.450·79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1120 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.289333231\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.289333231\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 5 | \( 1 + T \) | |
| 7 | \( 1 + T \) | |
| good | 3 | \( 1 + p T^{2} \) | 1.3.a |
| 11 | \( 1 + 4 T + p T^{2} \) | 1.11.e |
| 13 | \( 1 - 2 T + p T^{2} \) | 1.13.ac |
| 17 | \( 1 - 6 T + p T^{2} \) | 1.17.ag |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 23 | \( 1 - 8 T + p T^{2} \) | 1.23.ai |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 - 10 T + p T^{2} \) | 1.37.ak |
| 41 | \( 1 - 10 T + p T^{2} \) | 1.41.ak |
| 43 | \( 1 + 12 T + p T^{2} \) | 1.43.m |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 + 12 T + p T^{2} \) | 1.59.m |
| 61 | \( 1 - 2 T + p T^{2} \) | 1.61.ac |
| 67 | \( 1 + 4 T + p T^{2} \) | 1.67.e |
| 71 | \( 1 - 12 T + p T^{2} \) | 1.71.am |
| 73 | \( 1 + 10 T + p T^{2} \) | 1.73.k |
| 79 | \( 1 - 4 T + p T^{2} \) | 1.79.ae |
| 83 | \( 1 + p T^{2} \) | 1.83.a |
| 89 | \( 1 - 18 T + p T^{2} \) | 1.89.as |
| 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
−9.820989038241779128425202531231, −8.996859442069574446012296119347, −8.037900348473534138033816450738, −7.60311258326382760044136016719, −6.39846001733426673553936791813, −5.53669328084318604320858748149, −4.78800294574407855270905770249, −3.26872294384490062221080825193, −2.87145921957628391924790945512, −0.866149926275300885195766074991,
0.866149926275300885195766074991, 2.87145921957628391924790945512, 3.26872294384490062221080825193, 4.78800294574407855270905770249, 5.53669328084318604320858748149, 6.39846001733426673553936791813, 7.60311258326382760044136016719, 8.037900348473534138033816450738, 8.996859442069574446012296119347, 9.820989038241779128425202531231
