| L(s) = 1 | − 2-s − 3-s + 4-s + 5-s + 6-s − 4·7-s − 8-s + 9-s − 10-s − 5·11-s − 12-s + 4·14-s − 15-s + 16-s − 18-s + 20-s + 4·21-s + 5·22-s + 2·23-s + 24-s − 4·25-s − 27-s − 4·28-s − 3·29-s + 30-s + 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 − 1.50·11-s − 0.288·12-s + 1.06·14-s − 0.258·15-s + 1/4·16-s − 0.235·18-s + 0.223·20-s + 0.872·21-s + 1.06·22-s + 0.417·23-s + 0.204·24-s − 4/5·25-s − 0.192·27-s − 0.755·28-s − 0.557·29-s + 0.182·30-s + 0.179·31-s − 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 293046 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 293046 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) | |
| 13 | \( 1 \) | |
| 17 | \( 1 \) | |
| good | 5 | \( 1 - T + p T^{2} \) | 1.5.ab |
| 7 | \( 1 + 4 T + p T^{2} \) | 1.7.e |
| 11 | \( 1 + 5 T + p T^{2} \) | 1.11.f |
| 19 | \( 1 + p T^{2} \) | 1.19.a |
| 23 | \( 1 - 2 T + p T^{2} \) | 1.23.ac |
| 29 | \( 1 + 3 T + p T^{2} \) | 1.29.d |
| 31 | \( 1 - T + p T^{2} \) | 1.31.ab |
| 37 | \( 1 - 4 T + p T^{2} \) | 1.37.ae |
| 41 | \( 1 + 2 T + p T^{2} \) | 1.41.c |
| 43 | \( 1 - 8 T + p T^{2} \) | 1.43.ai |
| 47 | \( 1 + 6 T + p T^{2} \) | 1.47.g |
| 53 | \( 1 + T + p T^{2} \) | 1.53.b |
| 59 | \( 1 - T + p T^{2} \) | 1.59.ab |
| 61 | \( 1 - 10 T + p T^{2} \) | 1.61.ak |
| 67 | \( 1 - 2 T + p T^{2} \) | 1.67.ac |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 - 13 T + p T^{2} \) | 1.73.an |
| 79 | \( 1 + 3 T + p T^{2} \) | 1.79.d |
| 83 | \( 1 + 4 T + p T^{2} \) | 1.83.e |
| 89 | \( 1 + 8 T + p T^{2} \) | 1.89.i |
| 97 | \( 1 + 17 T + p T^{2} \) | 1.97.r |
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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
−12.93054441717456, −12.61766915196098, −12.03838493520686, −11.31144135913463, −11.14159752279171, −10.49782164025954, −10.09806970321385, −9.791975245979830, −9.438307385663508, −8.930732121481060, −8.240465666474875, −7.835325970255049, −7.348698025015112, −6.751493829182815, −6.496123051419343, −5.822265948133273, −5.550837488920721, −5.084469986989174, −4.265170288520905, −3.695578571502725, −3.060547041498948, −2.565234700697821, −2.143103646582035, −1.281439099059823, −0.5232811529177144, 0,
0.5232811529177144, 1.281439099059823, 2.143103646582035, 2.565234700697821, 3.060547041498948, 3.695578571502725, 4.265170288520905, 5.084469986989174, 5.550837488920721, 5.822265948133273, 6.496123051419343, 6.751493829182815, 7.348698025015112, 7.835325970255049, 8.240465666474875, 8.930732121481060, 9.438307385663508, 9.791975245979830, 10.09806970321385, 10.49782164025954, 11.14159752279171, 11.31144135913463, 12.03838493520686, 12.61766915196098, 12.93054441717456
