| L(s) = 1 | − 2-s − 3-s + 4-s + 3·5-s + 6-s + 7-s − 8-s + 9-s − 3·10-s − 12-s − 4·13-s − 14-s − 3·15-s + 16-s − 18-s + 5·19-s + 3·20-s − 21-s + 24-s + 4·25-s + 4·26-s − 27-s + 28-s − 6·29-s + 3·30-s + 7·31-s − 32-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s + 1.34·5-s + 0.408·6-s + 0.377·7-s − 0.353·8-s + 1/3·9-s − 0.948·10-s − 0.288·12-s − 1.10·13-s − 0.267·14-s − 0.774·15-s + 1/4·16-s − 0.235·18-s + 1.14·19-s + 0.670·20-s − 0.218·21-s + 0.204·24-s + 4/5·25-s + 0.784·26-s − 0.192·27-s + 0.188·28-s − 1.11·29-s + 0.547·30-s + 1.25·31-s − 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 209814 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 209814 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.020373816\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.020373816\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) | |
| 3 | \( 1 + T \) | |
| 11 | \( 1 \) | |
| 17 | \( 1 \) | |
| good | 5 | \( 1 - 3 T + p T^{2} \) | 1.5.ad |
| 7 | \( 1 - T + p T^{2} \) | 1.7.ab |
| 13 | \( 1 + 4 T + p T^{2} \) | 1.13.e |
| 19 | \( 1 - 5 T + p T^{2} \) | 1.19.af |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 - 7 T + p T^{2} \) | 1.31.ah |
| 37 | \( 1 - T + p T^{2} \) | 1.37.ab |
| 41 | \( 1 + p T^{2} \) | 1.41.a |
| 43 | \( 1 - 11 T + p T^{2} \) | 1.43.al |
| 47 | \( 1 + 6 T + p T^{2} \) | 1.47.g |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 - 15 T + p T^{2} \) | 1.59.ap |
| 61 | \( 1 - T + p T^{2} \) | 1.61.ab |
| 67 | \( 1 + 4 T + p T^{2} \) | 1.67.e |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 - 4 T + p T^{2} \) | 1.73.ae |
| 79 | \( 1 - 13 T + p T^{2} \) | 1.79.an |
| 83 | \( 1 - 9 T + p T^{2} \) | 1.83.aj |
| 89 | \( 1 + 6 T + p T^{2} \) | 1.89.g |
| 97 | \( 1 + 2 T + p T^{2} \) | 1.97.c |
| 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
−13.03130577185515, −12.42295923743056, −11.97613536767964, −11.63690547277108, −11.00453478406844, −10.67219709335076, −10.00037327404840, −9.814913542474258, −9.332332729380173, −9.077678430667759, −8.032071891352328, −7.985704913466391, −7.213963898326828, −6.795590105817868, −6.365994036895527, −5.610337805277466, −5.448860049670401, −4.957623061588450, −4.260857210702836, −3.558065630649571, −2.612331338157299, −2.448082584184442, −1.680869888547026, −1.165192105999938, −0.4899895590341042,
0.4899895590341042, 1.165192105999938, 1.680869888547026, 2.448082584184442, 2.612331338157299, 3.558065630649571, 4.260857210702836, 4.957623061588450, 5.448860049670401, 5.610337805277466, 6.365994036895527, 6.795590105817868, 7.213963898326828, 7.985704913466391, 8.032071891352328, 9.077678430667759, 9.332332729380173, 9.814913542474258, 10.00037327404840, 10.67219709335076, 11.00453478406844, 11.63690547277108, 11.97613536767964, 12.42295923743056, 13.03130577185515
