| L(s) = 1 | + 2-s + 4-s − 5-s − 2·7-s + 8-s − 10-s + 2·11-s − 2·14-s + 16-s − 17-s + 4·19-s − 20-s + 2·22-s + 8·23-s + 25-s − 2·28-s + 6·29-s + 4·31-s + 32-s − 34-s + 2·35-s + 4·38-s − 40-s + 4·41-s − 4·43-s + 2·44-s + 8·46-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s − 0.447·5-s − 0.755·7-s + 0.353·8-s − 0.316·10-s + 0.603·11-s − 0.534·14-s + 1/4·16-s − 0.242·17-s + 0.917·19-s − 0.223·20-s + 0.426·22-s + 1.66·23-s + 1/5·25-s − 0.377·28-s + 1.11·29-s + 0.718·31-s + 0.176·32-s − 0.171·34-s + 0.338·35-s + 0.648·38-s − 0.158·40-s + 0.624·41-s − 0.609·43-s + 0.301·44-s + 1.17·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1530 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1530 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.435012470\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.435012470\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 5 | \( 1 + T \) | |
| 17 | \( 1 + T \) | |
| good | 7 | \( 1 + 2 T + p T^{2} \) | 1.7.c |
| 11 | \( 1 - 2 T + p T^{2} \) | 1.11.ac |
| 13 | \( 1 + p T^{2} \) | 1.13.a |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 23 | \( 1 - 8 T + p T^{2} \) | 1.23.ai |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 - 4 T + p T^{2} \) | 1.31.ae |
| 37 | \( 1 + p T^{2} \) | 1.37.a |
| 41 | \( 1 - 4 T + p T^{2} \) | 1.41.ae |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 - 8 T + p T^{2} \) | 1.47.ai |
| 53 | \( 1 + 2 T + p T^{2} \) | 1.53.c |
| 59 | \( 1 - 6 T + p T^{2} \) | 1.59.ag |
| 61 | \( 1 - 2 T + p T^{2} \) | 1.61.ac |
| 67 | \( 1 + 16 T + p T^{2} \) | 1.67.q |
| 71 | \( 1 + 8 T + p T^{2} \) | 1.71.i |
| 73 | \( 1 - 10 T + p T^{2} \) | 1.73.ak |
| 79 | \( 1 - 4 T + p T^{2} \) | 1.79.ae |
| 83 | \( 1 - 4 T + p T^{2} \) | 1.83.ae |
| 89 | \( 1 - 12 T + p T^{2} \) | 1.89.am |
| 97 | \( 1 + 2 T + p T^{2} \) | 1.97.c |
| 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
−9.436642361793284157931153389827, −8.722792546783662923335458147967, −7.65685461062244418021064264797, −6.87195763147489336075779868647, −6.29099144055718447981749957447, −5.21611144189162974556770560246, −4.40220651056535076491762714778, −3.42089241644053502284469489537, −2.72502778978520499524892639728, −1.04569548611098429720594531845,
1.04569548611098429720594531845, 2.72502778978520499524892639728, 3.42089241644053502284469489537, 4.40220651056535076491762714778, 5.21611144189162974556770560246, 6.29099144055718447981749957447, 6.87195763147489336075779868647, 7.65685461062244418021064264797, 8.722792546783662923335458147967, 9.436642361793284157931153389827
