| L(s) = 1 | + 2-s + 2·3-s + 4-s + 2·6-s − 2·7-s + 8-s + 9-s + 2·12-s + 2·13-s − 2·14-s + 16-s + 6·17-s + 18-s − 2·19-s − 4·21-s + 2·24-s + 2·26-s − 4·27-s − 2·28-s − 6·29-s + 10·31-s + 32-s + 6·34-s + 36-s − 2·38-s + 4·39-s − 6·41-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.15·3-s + 1/2·4-s + 0.816·6-s − 0.755·7-s + 0.353·8-s + 1/3·9-s + 0.577·12-s + 0.554·13-s − 0.534·14-s + 1/4·16-s + 1.45·17-s + 0.235·18-s − 0.458·19-s − 0.872·21-s + 0.408·24-s + 0.392·26-s − 0.769·27-s − 0.377·28-s − 1.11·29-s + 1.79·31-s + 0.176·32-s + 1.02·34-s + 1/6·36-s − 0.324·38-s + 0.640·39-s − 0.937·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 68450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 68450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.930893755\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.930893755\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) | |
| 5 | \( 1 \) | |
| 37 | \( 1 \) | |
| good | 3 | \( 1 - 2 T + p T^{2} \) | 1.3.ac |
| 7 | \( 1 + 2 T + p T^{2} \) | 1.7.c |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 13 | \( 1 - 2 T + p T^{2} \) | 1.13.ac |
| 17 | \( 1 - 6 T + p T^{2} \) | 1.17.ag |
| 19 | \( 1 + 2 T + p T^{2} \) | 1.19.c |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 - 10 T + p T^{2} \) | 1.31.ak |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 - 6 T + p T^{2} \) | 1.47.ag |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 - 6 T + p T^{2} \) | 1.59.ag |
| 61 | \( 1 - 10 T + p T^{2} \) | 1.61.ak |
| 67 | \( 1 + 2 T + p T^{2} \) | 1.67.c |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 + 2 T + p T^{2} \) | 1.73.c |
| 79 | \( 1 - 10 T + p T^{2} \) | 1.79.ak |
| 83 | \( 1 - 6 T + p T^{2} \) | 1.83.ag |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 97 | \( 1 - 2 T + p T^{2} \) | 1.97.ac |
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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
−14.21205625752531, −13.53742755593726, −13.36448418464851, −12.87606140298663, −12.22468622060044, −11.82199359601293, −11.27109327346146, −10.52980251285405, −10.06118220514135, −9.615722435441183, −9.066425220770185, −8.422135558375734, −8.010561411265991, −7.570490528757836, −6.775685817431505, −6.396787124930588, −5.724494965779151, −5.254048819025937, −4.447215765994622, −3.744891688462741, −3.401313985527390, −2.945180086774361, −2.262505901085463, −1.590563729683848, −0.6542036098171906,
0.6542036098171906, 1.590563729683848, 2.262505901085463, 2.945180086774361, 3.401313985527390, 3.744891688462741, 4.447215765994622, 5.254048819025937, 5.724494965779151, 6.396787124930588, 6.775685817431505, 7.570490528757836, 8.010561411265991, 8.422135558375734, 9.066425220770185, 9.615722435441183, 10.06118220514135, 10.52980251285405, 11.27109327346146, 11.82199359601293, 12.22468622060044, 12.87606140298663, 13.36448418464851, 13.53742755593726, 14.21205625752531
