| L(s) = 1 | + 2-s − 3-s + 4-s − 5-s − 6-s + 7-s + 8-s + 9-s − 10-s − 12-s − 13-s + 14-s + 15-s + 16-s − 4·17-s + 18-s − 20-s − 21-s + 23-s − 24-s − 4·25-s − 26-s − 27-s + 28-s − 29-s + 30-s − 2·31-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 − 0.288·12-s − 0.277·13-s + 0.267·14-s + 0.258·15-s + 1/4·16-s − 0.970·17-s + 0.235·18-s − 0.223·20-s − 0.218·21-s + 0.208·23-s − 0.204·24-s − 4/5·25-s − 0.196·26-s − 0.192·27-s + 0.188·28-s − 0.185·29-s + 0.182·30-s − 0.359·31-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
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) | |
| 7 | \( 1 - T \) | |
| 19 | \( 1 \) | |
| 23 | \( 1 - T \) | |
| good | 5 | \( 1 + T + p T^{2} \) | 1.5.b |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 13 | \( 1 + T + p T^{2} \) | 1.13.b |
| 17 | \( 1 + 4 T + p T^{2} \) | 1.17.e |
| 29 | \( 1 + T + p T^{2} \) | 1.29.b |
| 31 | \( 1 + 2 T + p T^{2} \) | 1.31.c |
| 37 | \( 1 - 7 T + p T^{2} \) | 1.37.ah |
| 41 | \( 1 + 3 T + p T^{2} \) | 1.41.d |
| 43 | \( 1 + 5 T + p T^{2} \) | 1.43.f |
| 47 | \( 1 - 11 T + p T^{2} \) | 1.47.al |
| 53 | \( 1 + 8 T + p T^{2} \) | 1.53.i |
| 59 | \( 1 - 14 T + p T^{2} \) | 1.59.ao |
| 61 | \( 1 - 2 T + p T^{2} \) | 1.61.ac |
| 67 | \( 1 - 4 T + p T^{2} \) | 1.67.ae |
| 71 | \( 1 + 14 T + p T^{2} \) | 1.71.o |
| 73 | \( 1 + 4 T + p T^{2} \) | 1.73.e |
| 79 | \( 1 - 16 T + p T^{2} \) | 1.79.aq |
| 83 | \( 1 + 4 T + p T^{2} \) | 1.83.e |
| 89 | \( 1 - 18 T + p T^{2} \) | 1.89.as |
| 97 | \( 1 - T + p T^{2} \) | 1.97.ab |
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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
−12.83677334169475, −12.25192849597831, −11.75449958059924, −11.59155312445156, −11.13229975435680, −10.59896600496424, −10.31237400269913, −9.613372911848427, −9.190969437727041, −8.623769428352827, −8.051108602230691, −7.593442552471810, −7.248952552165505, −6.541482301459494, −6.386075765840649, −5.596017559980873, −5.324509272920762, −4.722598041701246, −4.295100402266782, −3.880264630078239, −3.343707195405244, −2.539698960312160, −2.156949923970506, −1.483746358430867, −0.7309070602932734, 0,
0.7309070602932734, 1.483746358430867, 2.156949923970506, 2.539698960312160, 3.343707195405244, 3.880264630078239, 4.295100402266782, 4.722598041701246, 5.324509272920762, 5.596017559980873, 6.386075765840649, 6.541482301459494, 7.248952552165505, 7.593442552471810, 8.051108602230691, 8.623769428352827, 9.190969437727041, 9.613372911848427, 10.31237400269913, 10.59896600496424, 11.13229975435680, 11.59155312445156, 11.75449958059924, 12.25192849597831, 12.83677334169475
