| L(s) = 1 | − 2·5-s + 7-s − 2·13-s − 6·17-s + 4·19-s − 4·23-s − 25-s − 6·29-s + 8·31-s − 2·35-s − 10·37-s + 10·41-s − 12·43-s − 8·47-s + 49-s − 6·53-s + 4·59-s − 10·61-s + 4·65-s − 12·67-s + 4·71-s + 2·73-s − 8·79-s + 4·83-s + 12·85-s − 6·89-s − 2·91-s + ⋯ |
| L(s) = 1 | − 0.894·5-s + 0.377·7-s − 0.554·13-s − 1.45·17-s + 0.917·19-s − 0.834·23-s − 1/5·25-s − 1.11·29-s + 1.43·31-s − 0.338·35-s − 1.64·37-s + 1.56·41-s − 1.82·43-s − 1.16·47-s + 1/7·49-s − 0.824·53-s + 0.520·59-s − 1.28·61-s + 0.496·65-s − 1.46·67-s + 0.474·71-s + 0.234·73-s − 0.900·79-s + 0.439·83-s + 1.30·85-s − 0.635·89-s − 0.209·91-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 7 | \( 1 - T \) | |
| good | 5 | \( 1 + 2 T + p T^{2} \) | 1.5.c |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 13 | \( 1 + 2 T + p T^{2} \) | 1.13.c |
| 17 | \( 1 + 6 T + p T^{2} \) | 1.17.g |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 23 | \( 1 + 4 T + p T^{2} \) | 1.23.e |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 - 8 T + p T^{2} \) | 1.31.ai |
| 37 | \( 1 + 10 T + p T^{2} \) | 1.37.k |
| 41 | \( 1 - 10 T + p T^{2} \) | 1.41.ak |
| 43 | \( 1 + 12 T + p T^{2} \) | 1.43.m |
| 47 | \( 1 + 8 T + p T^{2} \) | 1.47.i |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 - 4 T + p T^{2} \) | 1.59.ae |
| 61 | \( 1 + 10 T + p T^{2} \) | 1.61.k |
| 67 | \( 1 + 12 T + p T^{2} \) | 1.67.m |
| 71 | \( 1 - 4 T + p T^{2} \) | 1.71.ae |
| 73 | \( 1 - 2 T + p T^{2} \) | 1.73.ac |
| 79 | \( 1 + 8 T + p T^{2} \) | 1.79.i |
| 83 | \( 1 - 4 T + p T^{2} \) | 1.83.ae |
| 89 | \( 1 + 6 T + p T^{2} \) | 1.89.g |
| 97 | \( 1 - 10 T + p T^{2} \) | 1.97.ak |
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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
−9.542335673006963113033744072480, −8.614680372438593469438231251008, −7.86152616392232705724947299443, −7.17945773905277725475919160716, −6.19452637109873977406389525575, −4.99862586128493531522947252735, −4.26888101159701758252770076566, −3.21569388283931405752209731146, −1.87750590194886957605795356311, 0,
1.87750590194886957605795356311, 3.21569388283931405752209731146, 4.26888101159701758252770076566, 4.99862586128493531522947252735, 6.19452637109873977406389525575, 7.17945773905277725475919160716, 7.86152616392232705724947299443, 8.614680372438593469438231251008, 9.542335673006963113033744072480
