| L(s) = 1 | + 2-s − 3-s + 4-s + 5-s − 6-s + 4·7-s + 8-s + 9-s + 10-s − 12-s + 13-s + 4·14-s − 15-s + 16-s + 18-s − 4·19-s + 20-s − 4·21-s − 24-s + 25-s + 26-s − 27-s + 4·28-s − 6·29-s − 30-s + 4·31-s + 32-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.447·5-s − 0.408·6-s + 1.51·7-s + 0.353·8-s + 1/3·9-s + 0.316·10-s − 0.288·12-s + 0.277·13-s + 1.06·14-s − 0.258·15-s + 1/4·16-s + 0.235·18-s − 0.917·19-s + 0.223·20-s − 0.872·21-s − 0.204·24-s + 1/5·25-s + 0.196·26-s − 0.192·27-s + 0.755·28-s − 1.11·29-s − 0.182·30-s + 0.718·31-s + 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 112710 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 112710 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.681550851\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.681550851\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) | |
| 17 | \( 1 \) | |
| good | 7 | \( 1 - 4 T + p T^{2} \) | 1.7.ae |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 19 | \( 1 + 4 T + p T^{2} \) | 1.19.e |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 - 4 T + p T^{2} \) | 1.31.ae |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 + 12 T + p T^{2} \) | 1.59.m |
| 61 | \( 1 - 10 T + p T^{2} \) | 1.61.ak |
| 67 | \( 1 + 4 T + p T^{2} \) | 1.67.e |
| 71 | \( 1 - 12 T + p T^{2} \) | 1.71.am |
| 73 | \( 1 - 10 T + p T^{2} \) | 1.73.ak |
| 79 | \( 1 - 16 T + p T^{2} \) | 1.79.aq |
| 83 | \( 1 + 12 T + p T^{2} \) | 1.83.m |
| 89 | \( 1 + 6 T + p T^{2} \) | 1.89.g |
| 97 | \( 1 - 10 T + p T^{2} \) | 1.97.ak |
| 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
−13.77438902100766, −13.12646045534771, −12.69231943191801, −12.14746140808710, −11.66043771423855, −11.29300729869097, −10.78270437955318, −10.50169496649305, −9.844348253951393, −9.249722154826222, −8.499266945905637, −8.213795104277837, −7.614476101194170, −6.970858859025364, −6.534862440806541, −5.951154951168351, −5.413503353621660, −5.001659782900224, −4.562395965392309, −3.956853653480111, −3.390617374150948, −2.436416552300836, −1.911558870027347, −1.475314601268334, −0.6027610120711838,
0.6027610120711838, 1.475314601268334, 1.911558870027347, 2.436416552300836, 3.390617374150948, 3.956853653480111, 4.562395965392309, 5.001659782900224, 5.413503353621660, 5.951154951168351, 6.534862440806541, 6.970858859025364, 7.614476101194170, 8.213795104277837, 8.499266945905637, 9.249722154826222, 9.844348253951393, 10.50169496649305, 10.78270437955318, 11.29300729869097, 11.66043771423855, 12.14746140808710, 12.69231943191801, 13.12646045534771, 13.77438902100766
