| L(s) = 1 | + 2-s − 3-s + 4-s + 5-s − 6-s − 3·7-s + 8-s + 9-s + 10-s + 11-s − 12-s + 2·13-s − 3·14-s − 15-s + 16-s + 3·17-s + 18-s + 20-s + 3·21-s + 22-s − 23-s − 24-s + 25-s + 2·26-s − 27-s − 3·28-s − 6·29-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.447·5-s − 0.408·6-s − 1.13·7-s + 0.353·8-s + 1/3·9-s + 0.316·10-s + 0.301·11-s − 0.288·12-s + 0.554·13-s − 0.801·14-s − 0.258·15-s + 1/4·16-s + 0.727·17-s + 0.235·18-s + 0.223·20-s + 0.654·21-s + 0.213·22-s − 0.208·23-s − 0.204·24-s + 1/5·25-s + 0.392·26-s − 0.192·27-s − 0.566·28-s − 1.11·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 317130 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 317130 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) | |
| 11 | \( 1 - T \) | |
| 31 | \( 1 \) | |
| good | 7 | \( 1 + 3 T + p T^{2} \) | 1.7.d |
| 13 | \( 1 - 2 T + p T^{2} \) | 1.13.ac |
| 17 | \( 1 - 3 T + p T^{2} \) | 1.17.ad |
| 19 | \( 1 + p T^{2} \) | 1.19.a |
| 23 | \( 1 + T + p T^{2} \) | 1.23.b |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 37 | \( 1 - 7 T + p T^{2} \) | 1.37.ah |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 - 9 T + p T^{2} \) | 1.43.aj |
| 47 | \( 1 + 10 T + p T^{2} \) | 1.47.k |
| 53 | \( 1 - 10 T + p T^{2} \) | 1.53.ak |
| 59 | \( 1 + 9 T + p T^{2} \) | 1.59.j |
| 61 | \( 1 + p T^{2} \) | 1.61.a |
| 67 | \( 1 + 11 T + p T^{2} \) | 1.67.l |
| 71 | \( 1 - 10 T + p T^{2} \) | 1.71.ak |
| 73 | \( 1 + 14 T + p T^{2} \) | 1.73.o |
| 79 | \( 1 - 4 T + p T^{2} \) | 1.79.ae |
| 83 | \( 1 - 7 T + p T^{2} \) | 1.83.ah |
| 89 | \( 1 + 8 T + p T^{2} \) | 1.89.i |
| 97 | \( 1 + 17 T + p T^{2} \) | 1.97.r |
| 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
−12.83268685203111, −12.49689484996071, −12.07252312866656, −11.55414517366186, −11.03449986351504, −10.75791091656365, −10.11624554938207, −9.738779701095615, −9.325852500479132, −8.917885133327993, −8.113754970405870, −7.615177383468151, −7.178443166218348, −6.596126562489479, −6.163863620349530, −5.859733959206633, −5.513611854566253, −4.840653381887238, −4.164985505400771, −3.898443694124467, −3.193021277508631, −2.823555284004483, −2.106905606123931, −1.416111776489565, −0.8545087613350686, 0,
0.8545087613350686, 1.416111776489565, 2.106905606123931, 2.823555284004483, 3.193021277508631, 3.898443694124467, 4.164985505400771, 4.840653381887238, 5.513611854566253, 5.859733959206633, 6.163863620349530, 6.596126562489479, 7.178443166218348, 7.615177383468151, 8.113754970405870, 8.917885133327993, 9.325852500479132, 9.738779701095615, 10.11624554938207, 10.75791091656365, 11.03449986351504, 11.55414517366186, 12.07252312866656, 12.49689484996071, 12.83268685203111
