| L(s) = 1 | + 2-s + 3-s + 4-s − 2·5-s + 6-s + 7-s + 8-s + 9-s − 2·10-s − 4·11-s + 12-s − 4·13-s + 14-s − 2·15-s + 16-s − 17-s + 18-s − 3·19-s − 2·20-s + 21-s − 4·22-s + 24-s − 25-s − 4·26-s + 27-s + 28-s − 6·29-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.894·5-s + 0.408·6-s + 0.377·7-s + 0.353·8-s + 1/3·9-s − 0.632·10-s − 1.20·11-s + 0.288·12-s − 1.10·13-s + 0.267·14-s − 0.516·15-s + 1/4·16-s − 0.242·17-s + 0.235·18-s − 0.688·19-s − 0.447·20-s + 0.218·21-s − 0.852·22-s + 0.204·24-s − 1/5·25-s − 0.784·26-s + 0.192·27-s + 0.188·28-s − 1.11·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 53958 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 53958 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.653498431\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.653498431\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) | |
| 17 | \( 1 + T \) | |
| 23 | \( 1 \) | |
| good | 5 | \( 1 + 2 T + p T^{2} \) | 1.5.c |
| 7 | \( 1 - T + p T^{2} \) | 1.7.ab |
| 11 | \( 1 + 4 T + p T^{2} \) | 1.11.e |
| 13 | \( 1 + 4 T + p T^{2} \) | 1.13.e |
| 19 | \( 1 + 3 T + p T^{2} \) | 1.19.d |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 + T + p T^{2} \) | 1.31.b |
| 37 | \( 1 - 9 T + p T^{2} \) | 1.37.aj |
| 41 | \( 1 + 7 T + p T^{2} \) | 1.41.h |
| 43 | \( 1 - 7 T + p T^{2} \) | 1.43.ah |
| 47 | \( 1 + 5 T + p T^{2} \) | 1.47.f |
| 53 | \( 1 + 13 T + p T^{2} \) | 1.53.n |
| 59 | \( 1 + 10 T + p T^{2} \) | 1.59.k |
| 61 | \( 1 - 5 T + p T^{2} \) | 1.61.af |
| 67 | \( 1 + 4 T + p T^{2} \) | 1.67.e |
| 71 | \( 1 + 6 T + p T^{2} \) | 1.71.g |
| 73 | \( 1 - 4 T + p T^{2} \) | 1.73.ae |
| 79 | \( 1 - 4 T + p T^{2} \) | 1.79.ae |
| 83 | \( 1 - 10 T + p T^{2} \) | 1.83.ak |
| 89 | \( 1 + 16 T + p T^{2} \) | 1.89.q |
| 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
−14.59441256481312, −13.99037447341482, −13.32667042321618, −12.88763955275737, −12.60016215176250, −11.93895319630165, −11.40795985500082, −10.94416791644384, −10.49472843676304, −9.702782325956783, −9.389232908774729, −8.452608384702924, −8.012162505763057, −7.610271902973523, −7.280853440223941, −6.463859220504275, −5.844420109156968, −5.079549244143838, −4.638067636325898, −4.218520063420016, −3.413864107954750, −2.932730175738162, −2.224699116061241, −1.713446532667137, −0.3519215995162802,
0.3519215995162802, 1.713446532667137, 2.224699116061241, 2.932730175738162, 3.413864107954750, 4.218520063420016, 4.638067636325898, 5.079549244143838, 5.844420109156968, 6.463859220504275, 7.280853440223941, 7.610271902973523, 8.012162505763057, 8.452608384702924, 9.389232908774729, 9.702782325956783, 10.49472843676304, 10.94416791644384, 11.40795985500082, 11.93895319630165, 12.60016215176250, 12.88763955275737, 13.32667042321618, 13.99037447341482, 14.59441256481312
