| L(s) = 1 | + 2-s − 4-s − 2·5-s + 4·7-s − 3·8-s − 2·10-s − 13-s + 4·14-s − 16-s + 2·17-s + 2·20-s − 25-s − 26-s − 4·28-s − 10·29-s + 4·31-s + 5·32-s + 2·34-s − 8·35-s − 2·37-s + 6·40-s + 6·41-s + 12·43-s + 9·49-s − 50-s + 52-s − 6·53-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1/2·4-s − 0.894·5-s + 1.51·7-s − 1.06·8-s − 0.632·10-s − 0.277·13-s + 1.06·14-s − 1/4·16-s + 0.485·17-s + 0.447·20-s − 1/5·25-s − 0.196·26-s − 0.755·28-s − 1.85·29-s + 0.718·31-s + 0.883·32-s + 0.342·34-s − 1.35·35-s − 0.328·37-s + 0.948·40-s + 0.937·41-s + 1.82·43-s + 9/7·49-s − 0.141·50-s + 0.138·52-s − 0.824·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 14157 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 14157 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) | |
| 11 | \( 1 \) | |
| 13 | \( 1 + T \) | |
| good | 2 | \( 1 - T + p T^{2} \) | 1.2.ab |
| 5 | \( 1 + 2 T + p T^{2} \) | 1.5.c |
| 7 | \( 1 - 4 T + p T^{2} \) | 1.7.ae |
| 17 | \( 1 - 2 T + p T^{2} \) | 1.17.ac |
| 19 | \( 1 + p T^{2} \) | 1.19.a |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 + 10 T + p T^{2} \) | 1.29.k |
| 31 | \( 1 - 4 T + p T^{2} \) | 1.31.ae |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 - 12 T + p T^{2} \) | 1.43.am |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 + 12 T + p T^{2} \) | 1.59.m |
| 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 | \( 1 + 2 T + p T^{2} \) | 1.73.c |
| 79 | \( 1 + 8 T + p T^{2} \) | 1.79.i |
| 83 | \( 1 - 4 T + p T^{2} \) | 1.83.ae |
| 89 | \( 1 - 2 T + p T^{2} \) | 1.89.ac |
| 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
−16.29233881215204, −15.64753596533844, −15.07242070959622, −14.76102497692392, −14.16627787772756, −13.81929457580130, −13.03548125702984, −12.37646953249578, −12.07633410331887, −11.26320155311943, −11.13923012328009, −10.21146835524115, −9.387585270190951, −8.914175585737838, −8.177840161459370, −7.663609664067637, −7.342291951502546, −6.079703866204255, −5.634325308991622, −4.862445839740569, −4.411942998125986, −3.877658638451044, −3.119682267210651, −2.151376949633022, −1.133250595165021, 0,
1.133250595165021, 2.151376949633022, 3.119682267210651, 3.877658638451044, 4.411942998125986, 4.862445839740569, 5.634325308991622, 6.079703866204255, 7.342291951502546, 7.663609664067637, 8.177840161459370, 8.914175585737838, 9.387585270190951, 10.21146835524115, 11.13923012328009, 11.26320155311943, 12.07633410331887, 12.37646953249578, 13.03548125702984, 13.81929457580130, 14.16627787772756, 14.76102497692392, 15.07242070959622, 15.64753596533844, 16.29233881215204
