| L(s) = 1 | + 2-s + 3-s + 4-s − 5-s + 6-s + 7-s + 8-s + 9-s − 10-s + 5·11-s + 12-s + 2·13-s + 14-s − 15-s + 16-s + 3·17-s + 18-s − 20-s + 21-s + 5·22-s − 23-s + 24-s − 4·25-s + 2·26-s + 27-s + 28-s + 10·29-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.447·5-s + 0.408·6-s + 0.377·7-s + 0.353·8-s + 1/3·9-s − 0.316·10-s + 1.50·11-s + 0.288·12-s + 0.554·13-s + 0.267·14-s − 0.258·15-s + 1/4·16-s + 0.727·17-s + 0.235·18-s − 0.223·20-s + 0.218·21-s + 1.06·22-s − 0.208·23-s + 0.204·24-s − 4/5·25-s + 0.392·26-s + 0.192·27-s + 0.188·28-s + 1.85·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 348726 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 348726 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(9.537678916\) |
| \(L(\frac12)\) |
\(\approx\) |
\(9.537678916\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) | |
| 7 | \( 1 - T \) | |
| 19 | \( 1 \) | |
| 23 | \( 1 + T \) | |
| good | 5 | \( 1 + T + p T^{2} \) | 1.5.b |
| 11 | \( 1 - 5 T + p T^{2} \) | 1.11.af |
| 13 | \( 1 - 2 T + p T^{2} \) | 1.13.ac |
| 17 | \( 1 - 3 T + p T^{2} \) | 1.17.ad |
| 29 | \( 1 - 10 T + p T^{2} \) | 1.29.ak |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 - 7 T + p T^{2} \) | 1.37.ah |
| 41 | \( 1 + 5 T + p T^{2} \) | 1.41.f |
| 43 | \( 1 + p T^{2} \) | 1.43.a |
| 47 | \( 1 - 6 T + p T^{2} \) | 1.47.ag |
| 53 | \( 1 - 2 T + p T^{2} \) | 1.53.ac |
| 59 | \( 1 + 12 T + p T^{2} \) | 1.59.m |
| 61 | \( 1 - 8 T + p T^{2} \) | 1.61.ai |
| 67 | \( 1 - 14 T + p T^{2} \) | 1.67.ao |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 + 4 T + p T^{2} \) | 1.73.e |
| 79 | \( 1 - 6 T + p T^{2} \) | 1.79.ag |
| 83 | \( 1 - 14 T + p T^{2} \) | 1.83.ao |
| 89 | \( 1 - 9 T + p T^{2} \) | 1.89.aj |
| 97 | \( 1 - 8 T + p T^{2} \) | 1.97.ai |
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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
−12.29069994596148, −12.25439515867458, −11.70227316122387, −11.42893919118479, −10.79702606837647, −10.34878126423259, −9.818040705949236, −9.334743146163211, −8.895569280970217, −8.236439920461960, −8.055564501583467, −7.533174770702552, −6.827908266138062, −6.617936460370538, −5.964304834658267, −5.638051650554867, −4.666229739208848, −4.581751288148404, −3.904641810369464, −3.501854040741414, −3.164783595089622, −2.298584283248947, −1.902848450755926, −1.070254694917005, −0.7952959456285460,
0.7952959456285460, 1.070254694917005, 1.902848450755926, 2.298584283248947, 3.164783595089622, 3.501854040741414, 3.904641810369464, 4.581751288148404, 4.666229739208848, 5.638051650554867, 5.964304834658267, 6.617936460370538, 6.827908266138062, 7.533174770702552, 8.055564501583467, 8.236439920461960, 8.895569280970217, 9.334743146163211, 9.818040705949236, 10.34878126423259, 10.79702606837647, 11.42893919118479, 11.70227316122387, 12.25439515867458, 12.29069994596148
