| L(s) = 1 | + 2-s − 3-s + 4-s + 5-s − 6-s − 7-s + 8-s + 9-s + 10-s − 12-s + 2·13-s − 14-s − 15-s + 16-s + 18-s − 4·19-s + 20-s + 21-s − 24-s + 25-s + 2·26-s − 27-s − 28-s + 6·29-s − 30-s + 4·31-s + 32-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.447·5-s − 0.408·6-s − 0.377·7-s + 0.353·8-s + 1/3·9-s + 0.316·10-s − 0.288·12-s + 0.554·13-s − 0.267·14-s − 0.258·15-s + 1/4·16-s + 0.235·18-s − 0.917·19-s + 0.223·20-s + 0.218·21-s − 0.204·24-s + 1/5·25-s + 0.392·26-s − 0.192·27-s − 0.188·28-s + 1.11·29-s − 0.182·30-s + 0.718·31-s + 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 60690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 60690 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) | |
| 5 | \( 1 - T \) | |
| 7 | \( 1 + T \) | |
| 17 | \( 1 \) | |
| good | 11 | \( 1 + p T^{2} \) | 1.11.a |
| 13 | \( 1 - 2 T + p T^{2} \) | 1.13.ac |
| 19 | \( 1 + 4 T + p T^{2} \) | 1.19.e |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 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.g |
| 43 | \( 1 - 8 T + p T^{2} \) | 1.43.ai |
| 47 | \( 1 + 12 T + p T^{2} \) | 1.47.m |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 + 12 T + p T^{2} \) | 1.59.m |
| 61 | \( 1 + 2 T + p T^{2} \) | 1.61.c |
| 67 | \( 1 - 8 T + p T^{2} \) | 1.67.ai |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 + 14 T + p T^{2} \) | 1.73.o |
| 79 | \( 1 - 16 T + p T^{2} \) | 1.79.aq |
| 83 | \( 1 - 12 T + p T^{2} \) | 1.83.am |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 97 | \( 1 + 14 T + p T^{2} \) | 1.97.o |
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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
−14.70752005722261, −13.72276060474008, −13.63631490072341, −13.12155735209876, −12.43377693313195, −12.18183881928385, −11.65172487488005, −10.89183165240262, −10.64339554236297, −10.14746080134866, −9.497004971662801, −8.972461207736996, −8.211421849785026, −7.847574399692391, −6.862120849187877, −6.500618267655813, −6.273602921247745, −5.463102455331298, −5.076522308764008, −4.356648760331822, −3.909379970336924, −3.090596064394926, −2.559822878979962, −1.722814157597419, −1.063134808099558, 0,
1.063134808099558, 1.722814157597419, 2.559822878979962, 3.090596064394926, 3.909379970336924, 4.356648760331822, 5.076522308764008, 5.463102455331298, 6.273602921247745, 6.500618267655813, 6.862120849187877, 7.847574399692391, 8.211421849785026, 8.972461207736996, 9.497004971662801, 10.14746080134866, 10.64339554236297, 10.89183165240262, 11.65172487488005, 12.18183881928385, 12.43377693313195, 13.12155735209876, 13.63631490072341, 13.72276060474008, 14.70752005722261
