| L(s) = 1 | + 4·5-s − 3·7-s + 4·11-s − 13-s + 4·17-s + 19-s − 4·23-s + 11·25-s − 4·31-s − 12·35-s + 9·37-s + 8·43-s + 12·47-s + 2·49-s − 8·53-s + 16·55-s + 4·59-s + 5·61-s − 4·65-s − 11·67-s − 8·71-s + 73-s − 12·77-s − 5·79-s + 8·83-s + 16·85-s − 12·89-s + ⋯ |
| L(s) = 1 | + 1.78·5-s − 1.13·7-s + 1.20·11-s − 0.277·13-s + 0.970·17-s + 0.229·19-s − 0.834·23-s + 11/5·25-s − 0.718·31-s − 2.02·35-s + 1.47·37-s + 1.21·43-s + 1.75·47-s + 2/7·49-s − 1.09·53-s + 2.15·55-s + 0.520·59-s + 0.640·61-s − 0.496·65-s − 1.34·67-s − 0.949·71-s + 0.117·73-s − 1.36·77-s − 0.562·79-s + 0.878·83-s + 1.73·85-s − 1.27·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.338102627\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.338102627\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| good | 5 | \( 1 - 4 T + p T^{2} \) | 1.5.ae |
| 7 | \( 1 + 3 T + p T^{2} \) | 1.7.d |
| 11 | \( 1 - 4 T + p T^{2} \) | 1.11.ae |
| 13 | \( 1 + T + p T^{2} \) | 1.13.b |
| 17 | \( 1 - 4 T + p T^{2} \) | 1.17.ae |
| 19 | \( 1 - T + p T^{2} \) | 1.19.ab |
| 23 | \( 1 + 4 T + p T^{2} \) | 1.23.e |
| 29 | \( 1 + p T^{2} \) | 1.29.a |
| 31 | \( 1 + 4 T + p T^{2} \) | 1.31.e |
| 37 | \( 1 - 9 T + p T^{2} \) | 1.37.aj |
| 41 | \( 1 + p T^{2} \) | 1.41.a |
| 43 | \( 1 - 8 T + p T^{2} \) | 1.43.ai |
| 47 | \( 1 - 12 T + p T^{2} \) | 1.47.am |
| 53 | \( 1 + 8 T + p T^{2} \) | 1.53.i |
| 59 | \( 1 - 4 T + p T^{2} \) | 1.59.ae |
| 61 | \( 1 - 5 T + p T^{2} \) | 1.61.af |
| 67 | \( 1 + 11 T + p T^{2} \) | 1.67.l |
| 71 | \( 1 + 8 T + p T^{2} \) | 1.71.i |
| 73 | \( 1 - T + p T^{2} \) | 1.73.ab |
| 79 | \( 1 + 5 T + p T^{2} \) | 1.79.f |
| 83 | \( 1 - 8 T + p T^{2} \) | 1.83.ai |
| 89 | \( 1 + 12 T + p T^{2} \) | 1.89.m |
| 97 | \( 1 - 5 T + p T^{2} \) | 1.97.af |
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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
−9.398184286512178458263925867474, −8.961679325395712580723026277632, −7.59935786244490470822810350055, −6.71174548934044095208893948883, −5.98289624762003201649160243701, −5.65513065279316088246203234268, −4.30786919885967763645635100877, −3.22235141017577078764025505936, −2.26799229473408129186056872817, −1.13756958398551067927188738213,
1.13756958398551067927188738213, 2.26799229473408129186056872817, 3.22235141017577078764025505936, 4.30786919885967763645635100877, 5.65513065279316088246203234268, 5.98289624762003201649160243701, 6.71174548934044095208893948883, 7.59935786244490470822810350055, 8.961679325395712580723026277632, 9.398184286512178458263925867474
