| L(s) = 1 | − 2-s − 3-s + 4-s + 2·5-s + 6-s + 7-s − 8-s + 9-s − 2·10-s + 11-s − 12-s + 13-s − 14-s − 2·15-s + 16-s + 2·17-s − 18-s − 19-s + 2·20-s − 21-s − 22-s + 9·23-s + 24-s − 25-s − 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 + 0.301·11-s − 0.288·12-s + 0.277·13-s − 0.267·14-s − 0.516·15-s + 1/4·16-s + 0.485·17-s − 0.235·18-s − 0.229·19-s + 0.447·20-s − 0.218·21-s − 0.213·22-s + 1.87·23-s + 0.204·24-s − 1/5·25-s − 0.196·26-s − 0.192·27-s + 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1218 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1218 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.280903674\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.280903674\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) | |
| 29 | \( 1 + T \) | |
| good | 5 | \( 1 - 2 T + p T^{2} \) | 1.5.ac |
| 11 | \( 1 - T + p T^{2} \) | 1.11.ab |
| 13 | \( 1 - T + p T^{2} \) | 1.13.ab |
| 17 | \( 1 - 2 T + p T^{2} \) | 1.17.ac |
| 19 | \( 1 + T + p T^{2} \) | 1.19.b |
| 23 | \( 1 - 9 T + p T^{2} \) | 1.23.aj |
| 31 | \( 1 - 4 T + p T^{2} \) | 1.31.ae |
| 37 | \( 1 + 7 T + p T^{2} \) | 1.37.h |
| 41 | \( 1 + 4 T + p T^{2} \) | 1.41.e |
| 43 | \( 1 + 2 T + p T^{2} \) | 1.43.c |
| 47 | \( 1 - 3 T + p T^{2} \) | 1.47.ad |
| 53 | \( 1 + 3 T + p T^{2} \) | 1.53.d |
| 59 | \( 1 - 11 T + p T^{2} \) | 1.59.al |
| 61 | \( 1 + 8 T + p T^{2} \) | 1.61.i |
| 67 | \( 1 - 5 T + p T^{2} \) | 1.67.af |
| 71 | \( 1 - 8 T + p T^{2} \) | 1.71.ai |
| 73 | \( 1 - 9 T + p T^{2} \) | 1.73.aj |
| 79 | \( 1 + p T^{2} \) | 1.79.a |
| 83 | \( 1 - 3 T + p T^{2} \) | 1.83.ad |
| 89 | \( 1 - 4 T + p T^{2} \) | 1.89.ae |
| 97 | \( 1 + 5 T + p T^{2} \) | 1.97.f |
| 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
−9.750636650457613999857279771160, −9.019708957466045453666700520677, −8.230113202598052536406097684887, −7.15953407684935845381986827551, −6.49791822680995107940862041475, −5.60846649410397773373090401205, −4.85575760457774465109912609488, −3.41591857485656015479078198820, −2.06921955291521009176499334288, −1.02660567976177782004478306854,
1.02660567976177782004478306854, 2.06921955291521009176499334288, 3.41591857485656015479078198820, 4.85575760457774465109912609488, 5.60846649410397773373090401205, 6.49791822680995107940862041475, 7.15953407684935845381986827551, 8.230113202598052536406097684887, 9.019708957466045453666700520677, 9.750636650457613999857279771160
