| 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 − 18-s − 8·19-s − 20-s − 2·22-s − 8·23-s + 24-s + 25-s − 27-s + 8·29-s − 30-s − 2·31-s − 32-s − 2·33-s + 36-s − 8·37-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 − 0.235·18-s − 1.83·19-s − 0.223·20-s − 0.426·22-s − 1.66·23-s + 0.204·24-s + 1/5·25-s − 0.192·27-s + 1.48·29-s − 0.182·30-s − 0.359·31-s − 0.176·32-s − 0.348·33-s + 1/6·36-s − 1.31·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 248430 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 248430 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6430090601\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6430090601\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) | |
| 7 | \( 1 \) | |
| 13 | \( 1 \) | |
| good | 11 | \( 1 - 2 T + p T^{2} \) | 1.11.ac |
| 17 | \( 1 + p T^{2} \) | 1.17.a |
| 19 | \( 1 + 8 T + p T^{2} \) | 1.19.i |
| 23 | \( 1 + 8 T + p T^{2} \) | 1.23.i |
| 29 | \( 1 - 8 T + p T^{2} \) | 1.29.ai |
| 31 | \( 1 + 2 T + p T^{2} \) | 1.31.c |
| 37 | \( 1 + 8 T + p T^{2} \) | 1.37.i |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 - 10 T + p T^{2} \) | 1.43.ak |
| 47 | \( 1 + 2 T + p T^{2} \) | 1.47.c |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 - 4 T + p T^{2} \) | 1.59.ae |
| 61 | \( 1 + 2 T + p T^{2} \) | 1.61.c |
| 67 | \( 1 - 14 T + p T^{2} \) | 1.67.ao |
| 71 | \( 1 + 12 T + p T^{2} \) | 1.71.m |
| 73 | \( 1 + 2 T + p T^{2} \) | 1.73.c |
| 79 | \( 1 + 8 T + p T^{2} \) | 1.79.i |
| 83 | \( 1 + 16 T + p T^{2} \) | 1.83.q |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 97 | \( 1 - 10 T + p T^{2} \) | 1.97.ak |
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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
−12.68340094660966, −12.35971548591294, −11.80340084467184, −11.50903157091548, −10.94397974258146, −10.52185380607875, −10.14283219064855, −9.753604785994265, −8.926315139956543, −8.749085800092480, −8.232074426416181, −7.695495841681035, −7.242296900759234, −6.643224751631198, −6.254450045670640, −5.936829441702688, −5.237124219815818, −4.397501960819662, −4.273312390698559, −3.634918921789434, −2.903825007072042, −2.172752466203367, −1.769106398459862, −0.9638648390678255, −0.2986839396805136,
0.2986839396805136, 0.9638648390678255, 1.769106398459862, 2.172752466203367, 2.903825007072042, 3.634918921789434, 4.273312390698559, 4.397501960819662, 5.237124219815818, 5.936829441702688, 6.254450045670640, 6.643224751631198, 7.242296900759234, 7.695495841681035, 8.232074426416181, 8.749085800092480, 8.926315139956543, 9.753604785994265, 10.14283219064855, 10.52185380607875, 10.94397974258146, 11.50903157091548, 11.80340084467184, 12.35971548591294, 12.68340094660966
