| L(s) = 1 | − 2-s − 3-s + 4-s − 2·5-s + 6-s − 7-s − 8-s + 9-s + 2·10-s + 6·11-s − 12-s + 4·13-s + 14-s + 2·15-s + 16-s − 6·17-s − 18-s + 4·19-s − 2·20-s + 21-s − 6·22-s − 4·23-s + 24-s − 25-s − 4·26-s − 27-s − 28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.894·5-s + 0.408·6-s − 0.377·7-s − 0.353·8-s + 1/3·9-s + 0.632·10-s + 1.80·11-s − 0.288·12-s + 1.10·13-s + 0.267·14-s + 0.516·15-s + 1/4·16-s − 1.45·17-s − 0.235·18-s + 0.917·19-s − 0.447·20-s + 0.218·21-s − 1.27·22-s − 0.834·23-s + 0.204·24-s − 1/5·25-s − 0.784·26-s − 0.192·27-s − 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 57498 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 57498 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) | |
| 3 | \( 1 + T \) | |
| 7 | \( 1 + T \) | |
| 37 | \( 1 \) | |
| good | 5 | \( 1 + 2 T + p T^{2} \) | 1.5.c |
| 11 | \( 1 - 6 T + p T^{2} \) | 1.11.ag |
| 13 | \( 1 - 4 T + p T^{2} \) | 1.13.ae |
| 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 | \( 1 + 2 T + p T^{2} \) | 1.29.c |
| 31 | \( 1 - 4 T + p T^{2} \) | 1.31.ae |
| 41 | \( 1 - 8 T + p T^{2} \) | 1.41.ai |
| 43 | \( 1 + 6 T + p T^{2} \) | 1.43.g |
| 47 | \( 1 + 4 T + p T^{2} \) | 1.47.e |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 - 4 T + p T^{2} \) | 1.59.ae |
| 61 | \( 1 + 12 T + p T^{2} \) | 1.61.m |
| 67 | \( 1 + p T^{2} \) | 1.67.a |
| 71 | \( 1 - 6 T + p T^{2} \) | 1.71.ag |
| 73 | \( 1 + 2 T + p T^{2} \) | 1.73.c |
| 79 | \( 1 - 14 T + p T^{2} \) | 1.79.ao |
| 83 | \( 1 + p T^{2} \) | 1.83.a |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 97 | \( 1 - 4 T + p T^{2} \) | 1.97.ae |
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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
−14.76162624584056, −14.09756263580270, −13.54658001703162, −13.11874636231435, −12.24090076539151, −11.88642630464215, −11.61588354691178, −11.05611156780443, −10.71463166636797, −9.904564046880830, −9.368694988749398, −9.053791589909793, −8.389678030994195, −7.926650893851486, −7.232879769302855, −6.710748085461033, −6.280819881115294, −5.913810890530252, −4.942048341724441, −4.142883323986510, −3.872877505395668, −3.276944508752092, −2.248461570272622, −1.455856105614000, −0.8462206920450151, 0,
0.8462206920450151, 1.455856105614000, 2.248461570272622, 3.276944508752092, 3.872877505395668, 4.142883323986510, 4.942048341724441, 5.913810890530252, 6.280819881115294, 6.710748085461033, 7.232879769302855, 7.926650893851486, 8.389678030994195, 9.053791589909793, 9.368694988749398, 9.904564046880830, 10.71463166636797, 11.05611156780443, 11.61588354691178, 11.88642630464215, 12.24090076539151, 13.11874636231435, 13.54658001703162, 14.09756263580270, 14.76162624584056
