| L(s) = 1 | + 5-s − 2·7-s + 11-s − 2·13-s + 2·17-s + 2·23-s + 25-s + 2·29-s − 4·31-s − 2·35-s − 6·37-s − 6·41-s + 6·43-s + 6·47-s − 3·49-s + 6·53-s + 55-s − 6·61-s − 2·65-s − 2·67-s + 4·71-s − 10·73-s − 2·77-s + 8·79-s − 14·83-s + 2·85-s − 10·89-s + ⋯ |
| L(s) = 1 | + 0.447·5-s − 0.755·7-s + 0.301·11-s − 0.554·13-s + 0.485·17-s + 0.417·23-s + 1/5·25-s + 0.371·29-s − 0.718·31-s − 0.338·35-s − 0.986·37-s − 0.937·41-s + 0.914·43-s + 0.875·47-s − 3/7·49-s + 0.824·53-s + 0.134·55-s − 0.768·61-s − 0.248·65-s − 0.244·67-s + 0.474·71-s − 1.17·73-s − 0.227·77-s + 0.900·79-s − 1.53·83-s + 0.216·85-s − 1.05·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 15840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 15840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) | |
| 11 | \( 1 - T \) | |
| good | 7 | \( 1 + 2 T + p T^{2} \) | 1.7.c |
| 13 | \( 1 + 2 T + p T^{2} \) | 1.13.c |
| 17 | \( 1 - 2 T + p T^{2} \) | 1.17.ac |
| 19 | \( 1 + p T^{2} \) | 1.19.a |
| 23 | \( 1 - 2 T + p T^{2} \) | 1.23.ac |
| 29 | \( 1 - 2 T + p T^{2} \) | 1.29.ac |
| 31 | \( 1 + 4 T + p T^{2} \) | 1.31.e |
| 37 | \( 1 + 6 T + p T^{2} \) | 1.37.g |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 - 6 T + p T^{2} \) | 1.43.ag |
| 47 | \( 1 - 6 T + p T^{2} \) | 1.47.ag |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 + 6 T + p T^{2} \) | 1.61.g |
| 67 | \( 1 + 2 T + p T^{2} \) | 1.67.c |
| 71 | \( 1 - 4 T + p T^{2} \) | 1.71.ae |
| 73 | \( 1 + 10 T + p T^{2} \) | 1.73.k |
| 79 | \( 1 - 8 T + p T^{2} \) | 1.79.ai |
| 83 | \( 1 + 14 T + p T^{2} \) | 1.83.o |
| 89 | \( 1 + 10 T + p T^{2} \) | 1.89.k |
| 97 | \( 1 + 14 T + p T^{2} \) | 1.97.o |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−16.36225522824477, −15.64438828502435, −15.23189639747727, −14.47537237966932, −14.10486155233238, −13.46564337049670, −12.92737738019408, −12.32037396970176, −11.98814222029171, −11.12643263709944, −10.54827839894908, −9.952136557909173, −9.534452279127191, −8.877922522823785, −8.375010308380520, −7.326928960425299, −7.103516536241197, −6.285701658930620, −5.717590236700582, −5.093838202894194, −4.310455201457120, −3.487681391835172, −2.897856440097014, −2.060677634070230, −1.144577944160745, 0,
1.144577944160745, 2.060677634070230, 2.897856440097014, 3.487681391835172, 4.310455201457120, 5.093838202894194, 5.717590236700582, 6.285701658930620, 7.103516536241197, 7.326928960425299, 8.375010308380520, 8.877922522823785, 9.534452279127191, 9.952136557909173, 10.54827839894908, 11.12643263709944, 11.98814222029171, 12.32037396970176, 12.92737738019408, 13.46564337049670, 14.10486155233238, 14.47537237966932, 15.23189639747727, 15.64438828502435, 16.36225522824477
