| L(s) = 1 | − 2·5-s − 2·7-s + 2·13-s + 2·17-s + 2·19-s + 23-s − 25-s + 4·29-s + 4·35-s + 2·37-s + 2·43-s − 12·47-s − 3·49-s + 2·53-s − 12·59-s + 14·61-s − 4·65-s + 2·67-s − 6·73-s − 6·79-s + 4·83-s − 4·85-s + 18·89-s − 4·91-s − 4·95-s + 6·97-s + 101-s + ⋯ |
| L(s) = 1 | − 0.894·5-s − 0.755·7-s + 0.554·13-s + 0.485·17-s + 0.458·19-s + 0.208·23-s − 1/5·25-s + 0.742·29-s + 0.676·35-s + 0.328·37-s + 0.304·43-s − 1.75·47-s − 3/7·49-s + 0.274·53-s − 1.56·59-s + 1.79·61-s − 0.496·65-s + 0.244·67-s − 0.702·73-s − 0.675·79-s + 0.439·83-s − 0.433·85-s + 1.90·89-s − 0.419·91-s − 0.410·95-s + 0.609·97-s + 0.0995·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 100188 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 100188 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.427504597\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.427504597\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 11 | \( 1 \) | |
| 23 | \( 1 - T \) | |
| good | 5 | \( 1 + 2 T + p T^{2} \) | 1.5.c |
| 7 | \( 1 + 2 T + p T^{2} \) | 1.7.c |
| 13 | \( 1 - 2 T + p T^{2} \) | 1.13.ac |
| 17 | \( 1 - 2 T + p T^{2} \) | 1.17.ac |
| 19 | \( 1 - 2 T + p T^{2} \) | 1.19.ac |
| 29 | \( 1 - 4 T + p T^{2} \) | 1.29.ae |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 41 | \( 1 + p T^{2} \) | 1.41.a |
| 43 | \( 1 - 2 T + p T^{2} \) | 1.43.ac |
| 47 | \( 1 + 12 T + p T^{2} \) | 1.47.m |
| 53 | \( 1 - 2 T + p T^{2} \) | 1.53.ac |
| 59 | \( 1 + 12 T + p T^{2} \) | 1.59.m |
| 61 | \( 1 - 14 T + p T^{2} \) | 1.61.ao |
| 67 | \( 1 - 2 T + p T^{2} \) | 1.67.ac |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 + 6 T + p T^{2} \) | 1.73.g |
| 79 | \( 1 + 6 T + p T^{2} \) | 1.79.g |
| 83 | \( 1 - 4 T + p T^{2} \) | 1.83.ae |
| 89 | \( 1 - 18 T + p T^{2} \) | 1.89.as |
| 97 | \( 1 - 6 T + p T^{2} \) | 1.97.ag |
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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
−13.64764856502184, −13.24295850133248, −12.78511443377657, −12.24945503862102, −11.78124390393077, −11.40934915682070, −10.86264109854120, −10.30062346924461, −9.742946899832575, −9.416520324024332, −8.688953142785737, −8.225018280419849, −7.794280707484463, −7.234775836001162, −6.687455518813049, −6.179309768491683, −5.666910309697602, −4.918529120626628, −4.443517870880211, −3.701446617551816, −3.346075683500021, −2.851857167393124, −1.948421840752136, −1.118787608611294, −0.4209570795760576,
0.4209570795760576, 1.118787608611294, 1.948421840752136, 2.851857167393124, 3.346075683500021, 3.701446617551816, 4.443517870880211, 4.918529120626628, 5.666910309697602, 6.179309768491683, 6.687455518813049, 7.234775836001162, 7.794280707484463, 8.225018280419849, 8.688953142785737, 9.416520324024332, 9.742946899832575, 10.30062346924461, 10.86264109854120, 11.40934915682070, 11.78124390393077, 12.24945503862102, 12.78511443377657, 13.24295850133248, 13.64764856502184
