| L(s) = 1 | + 2-s + 3-s + 4-s − 2·5-s + 6-s + 7-s + 8-s + 9-s − 2·10-s + 12-s − 13-s + 14-s − 2·15-s + 16-s − 6·17-s + 18-s + 4·19-s − 2·20-s + 21-s + 4·23-s + 24-s − 25-s − 26-s + 27-s + 28-s + 10·29-s − 2·30-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 + 0.288·12-s − 0.277·13-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 + 0.834·23-s + 0.204·24-s − 1/5·25-s − 0.196·26-s + 0.192·27-s + 0.188·28-s + 1.85·29-s − 0.365·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 66066 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 66066 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) | |
| 11 | \( 1 \) | |
| 13 | \( 1 + T \) | |
| good | 5 | \( 1 + 2 T + p T^{2} \) | 1.5.c |
| 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.ae |
| 29 | \( 1 - 10 T + p T^{2} \) | 1.29.ak |
| 31 | \( 1 + 8 T + p T^{2} \) | 1.31.i |
| 37 | \( 1 - 6 T + p T^{2} \) | 1.37.ag |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 + 12 T + p T^{2} \) | 1.43.m |
| 47 | \( 1 - 12 T + p T^{2} \) | 1.47.am |
| 53 | \( 1 + 2 T + p T^{2} \) | 1.53.c |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 + 6 T + p T^{2} \) | 1.61.g |
| 67 | \( 1 + 4 T + p T^{2} \) | 1.67.e |
| 71 | \( 1 + 8 T + p T^{2} \) | 1.71.i |
| 73 | \( 1 + 10 T + p T^{2} \) | 1.73.k |
| 79 | \( 1 + 16 T + p T^{2} \) | 1.79.q |
| 83 | \( 1 + p T^{2} \) | 1.83.a |
| 89 | \( 1 + 14 T + p T^{2} \) | 1.89.o |
| 97 | \( 1 - 2 T + p T^{2} \) | 1.97.ac |
| 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
−14.36615197838279, −14.08534267890363, −13.31885619116779, −13.17350969367291, −12.44889553134295, −11.93136345320367, −11.57687045296905, −10.99450255038499, −10.62796542123639, −9.880756065356996, −9.290193672809863, −8.684443876718439, −8.325753131875882, −7.578491855250366, −7.248006820745175, −6.813470804606287, −6.013760520633535, −5.425579298034788, −4.568117183769385, −4.456995463339706, −3.808872150994388, −2.970840163376220, −2.734134703045550, −1.823886668225542, −1.087235756775336, 0,
1.087235756775336, 1.823886668225542, 2.734134703045550, 2.970840163376220, 3.808872150994388, 4.456995463339706, 4.568117183769385, 5.425579298034788, 6.013760520633535, 6.813470804606287, 7.248006820745175, 7.578491855250366, 8.325753131875882, 8.684443876718439, 9.290193672809863, 9.880756065356996, 10.62796542123639, 10.99450255038499, 11.57687045296905, 11.93136345320367, 12.44889553134295, 13.17350969367291, 13.31885619116779, 14.08534267890363, 14.36615197838279
