| L(s) = 1 | + 5-s + 7-s − 4·11-s − 13-s + 2·17-s − 8·19-s − 4·23-s + 25-s + 2·29-s − 4·31-s + 35-s + 2·37-s − 6·41-s + 8·43-s + 8·47-s + 49-s − 6·53-s − 4·55-s + 6·61-s − 65-s − 4·67-s + 8·71-s + 6·73-s − 4·77-s + 8·79-s + 12·83-s + 2·85-s + ⋯ |
| L(s) = 1 | + 0.447·5-s + 0.377·7-s − 1.20·11-s − 0.277·13-s + 0.485·17-s − 1.83·19-s − 0.834·23-s + 1/5·25-s + 0.371·29-s − 0.718·31-s + 0.169·35-s + 0.328·37-s − 0.937·41-s + 1.21·43-s + 1.16·47-s + 1/7·49-s − 0.824·53-s − 0.539·55-s + 0.768·61-s − 0.124·65-s − 0.488·67-s + 0.949·71-s + 0.702·73-s − 0.455·77-s + 0.900·79-s + 1.31·83-s + 0.216·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 65520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 65520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 5 | \( 1 - T \) | |
| 7 | \( 1 - T \) | |
| 13 | \( 1 + T \) | |
| good | 11 | \( 1 + 4 T + p T^{2} \) | 1.11.e |
| 17 | \( 1 - 2 T + p T^{2} \) | 1.17.ac |
| 19 | \( 1 + 8 T + p T^{2} \) | 1.19.i |
| 23 | \( 1 + 4 T + p T^{2} \) | 1.23.e |
| 29 | \( 1 - 2 T + p T^{2} \) | 1.29.ac |
| 31 | \( 1 + 4 T + p T^{2} \) | 1.31.e |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 - 8 T + p T^{2} \) | 1.43.ai |
| 47 | \( 1 - 8 T + p T^{2} \) | 1.47.ai |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 - 6 T + p T^{2} \) | 1.61.ag |
| 67 | \( 1 + 4 T + p T^{2} \) | 1.67.e |
| 71 | \( 1 - 8 T + p T^{2} \) | 1.71.ai |
| 73 | \( 1 - 6 T + p T^{2} \) | 1.73.ag |
| 79 | \( 1 - 8 T + p T^{2} \) | 1.79.ai |
| 83 | \( 1 - 12 T + p T^{2} \) | 1.83.am |
| 89 | \( 1 - 2 T + p T^{2} \) | 1.89.ac |
| 97 | \( 1 + 2 T + p T^{2} \) | 1.97.c |
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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
−14.34705871601059, −14.02671260796583, −13.49029858073432, −12.81748772860000, −12.60882886040104, −12.08025198851539, −11.35229324377280, −10.73645586912898, −10.52914845208528, −9.983822373409185, −9.402751170633621, −8.765062673921166, −8.290528914933780, −7.763890546525586, −7.353949469155369, −6.491516156143617, −6.143570781188710, −5.423561060476845, −5.041641850984570, −4.351758172070630, −3.793733845761847, −2.962249597947173, −2.161395729624188, −2.048020794813419, −0.8671895216213856, 0,
0.8671895216213856, 2.048020794813419, 2.161395729624188, 2.962249597947173, 3.793733845761847, 4.351758172070630, 5.041641850984570, 5.423561060476845, 6.143570781188710, 6.491516156143617, 7.353949469155369, 7.763890546525586, 8.290528914933780, 8.765062673921166, 9.402751170633621, 9.983822373409185, 10.52914845208528, 10.73645586912898, 11.35229324377280, 12.08025198851539, 12.60882886040104, 12.81748772860000, 13.49029858073432, 14.02671260796583, 14.34705871601059
