| L(s) = 1 | − 2-s − 3-s + 4-s + 5-s + 6-s − 8-s + 9-s − 10-s + 2·11-s − 12-s − 15-s + 16-s − 2·17-s − 18-s − 19-s + 20-s − 2·22-s − 5·23-s + 24-s + 25-s − 27-s + 6·29-s + 30-s + 31-s − 32-s − 2·33-s + 2·34-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.447·5-s + 0.408·6-s − 0.353·8-s + 1/3·9-s − 0.316·10-s + 0.603·11-s − 0.288·12-s − 0.258·15-s + 1/4·16-s − 0.485·17-s − 0.235·18-s − 0.229·19-s + 0.223·20-s − 0.426·22-s − 1.04·23-s + 0.204·24-s + 1/5·25-s − 0.192·27-s + 1.11·29-s + 0.182·30-s + 0.179·31-s − 0.176·32-s − 0.348·33-s + 0.342·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 96330 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 96330 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7317094449\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7317094449\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) | |
| 13 | \( 1 \) | |
| 19 | \( 1 + T \) | |
| good | 7 | \( 1 + p T^{2} \) | 1.7.a |
| 11 | \( 1 - 2 T + p T^{2} \) | 1.11.ac |
| 17 | \( 1 + 2 T + p T^{2} \) | 1.17.c |
| 23 | \( 1 + 5 T + p T^{2} \) | 1.23.f |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 - T + p T^{2} \) | 1.31.ab |
| 37 | \( 1 + 9 T + p T^{2} \) | 1.37.j |
| 41 | \( 1 + 5 T + p T^{2} \) | 1.41.f |
| 43 | \( 1 - T + p T^{2} \) | 1.43.ab |
| 47 | \( 1 - T + p T^{2} \) | 1.47.ab |
| 53 | \( 1 + p T^{2} \) | 1.53.a |
| 59 | \( 1 + 9 T + p T^{2} \) | 1.59.j |
| 61 | \( 1 + 6 T + p T^{2} \) | 1.61.g |
| 67 | \( 1 + 8 T + p T^{2} \) | 1.67.i |
| 71 | \( 1 - 6 T + p T^{2} \) | 1.71.ag |
| 73 | \( 1 - 3 T + p T^{2} \) | 1.73.ad |
| 79 | \( 1 + 4 T + p T^{2} \) | 1.79.e |
| 83 | \( 1 - 12 T + p T^{2} \) | 1.83.am |
| 89 | \( 1 + 15 T + p T^{2} \) | 1.89.p |
| 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
−13.81287581735782, −13.38281838748589, −12.59577979441889, −12.14151962370166, −11.94974694491033, −11.19555621896940, −10.82517810600810, −10.21916121977146, −10.00777459003625, −9.243504984621052, −8.985901301116366, −8.240168901804676, −7.936863824914322, −7.094948958633944, −6.655302138325609, −6.320104762862926, −5.755255037024827, −5.096825417478648, −4.540579105453472, −3.900176044454537, −3.182976403272842, −2.479777329263240, −1.703851819315716, −1.347739485476306, −0.3166197341552121,
0.3166197341552121, 1.347739485476306, 1.703851819315716, 2.479777329263240, 3.182976403272842, 3.900176044454537, 4.540579105453472, 5.096825417478648, 5.755255037024827, 6.320104762862926, 6.655302138325609, 7.094948958633944, 7.936863824914322, 8.240168901804676, 8.985901301116366, 9.243504984621052, 10.00777459003625, 10.21916121977146, 10.82517810600810, 11.19555621896940, 11.94974694491033, 12.14151962370166, 12.59577979441889, 13.38281838748589, 13.81287581735782
