| L(s) = 1 | − 4·7-s − 3·9-s + 4·11-s − 2·13-s − 2·17-s + 4·19-s + 4·23-s + 2·29-s + 8·31-s + 6·37-s − 6·41-s + 8·43-s + 4·47-s + 9·49-s + 6·53-s − 4·59-s + 2·61-s + 12·63-s − 8·67-s + 6·73-s − 16·77-s + 9·81-s + 16·83-s − 6·89-s + 8·91-s + 14·97-s − 12·99-s + ⋯ |
| L(s) = 1 | − 1.51·7-s − 9-s + 1.20·11-s − 0.554·13-s − 0.485·17-s + 0.917·19-s + 0.834·23-s + 0.371·29-s + 1.43·31-s + 0.986·37-s − 0.937·41-s + 1.21·43-s + 0.583·47-s + 9/7·49-s + 0.824·53-s − 0.520·59-s + 0.256·61-s + 1.51·63-s − 0.977·67-s + 0.702·73-s − 1.82·77-s + 81-s + 1.75·83-s − 0.635·89-s + 0.838·91-s + 1.42·97-s − 1.20·99-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.276868154\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.276868154\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| good | 3 | \( 1 + p T^{2} \) | 1.3.a |
| 7 | \( 1 + 4 T + p T^{2} \) | 1.7.e |
| 11 | \( 1 - 4 T + p T^{2} \) | 1.11.ae |
| 13 | \( 1 + 2 T + p T^{2} \) | 1.13.c |
| 17 | \( 1 + 2 T + p T^{2} \) | 1.17.c |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 23 | \( 1 - 4 T + p T^{2} \) | 1.23.ae |
| 29 | \( 1 - 2 T + p T^{2} \) | 1.29.ac |
| 31 | \( 1 - 8 T + p T^{2} \) | 1.31.ai |
| 37 | \( 1 - 6 T + p T^{2} \) | 1.37.ag |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 - 8 T + p T^{2} \) | 1.43.ai |
| 47 | \( 1 - 4 T + p T^{2} \) | 1.47.ae |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 + 4 T + p T^{2} \) | 1.59.e |
| 61 | \( 1 - 2 T + p T^{2} \) | 1.61.ac |
| 67 | \( 1 + 8 T + p T^{2} \) | 1.67.i |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 - 6 T + p T^{2} \) | 1.73.ag |
| 79 | \( 1 + p T^{2} \) | 1.79.a |
| 83 | \( 1 - 16 T + p T^{2} \) | 1.83.aq |
| 89 | \( 1 + 6 T + p T^{2} \) | 1.89.g |
| 97 | \( 1 - 14 T + p T^{2} \) | 1.97.ao |
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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
−9.291763249756947680273677410722, −8.916494500187803195551794457755, −7.78749485529796929229571735762, −6.76167065115892628778425003984, −6.34551581674059305006577784340, −5.42444342291268605029621774940, −4.28670681048457803999561096522, −3.26209478229515851009460820825, −2.60068110583024471076066537498, −0.77815604840316859728659905693,
0.77815604840316859728659905693, 2.60068110583024471076066537498, 3.26209478229515851009460820825, 4.28670681048457803999561096522, 5.42444342291268605029621774940, 6.34551581674059305006577784340, 6.76167065115892628778425003984, 7.78749485529796929229571735762, 8.916494500187803195551794457755, 9.291763249756947680273677410722
