| L(s) = 1 | + 2-s − 3-s + 4-s + 5-s − 6-s + 2·7-s + 8-s − 2·9-s + 10-s − 12-s + 13-s + 2·14-s − 15-s + 16-s + 17-s − 2·18-s + 6·19-s + 20-s − 2·21-s − 9·23-s − 24-s + 25-s + 26-s + 5·27-s + 2·28-s − 5·29-s − 30-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.447·5-s − 0.408·6-s + 0.755·7-s + 0.353·8-s − 2/3·9-s + 0.316·10-s − 0.288·12-s + 0.277·13-s + 0.534·14-s − 0.258·15-s + 1/4·16-s + 0.242·17-s − 0.471·18-s + 1.37·19-s + 0.223·20-s − 0.436·21-s − 1.87·23-s − 0.204·24-s + 1/5·25-s + 0.196·26-s + 0.962·27-s + 0.377·28-s − 0.928·29-s − 0.182·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 15730 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 15730 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) | |
| 5 | \( 1 - T \) | |
| 11 | \( 1 \) | |
| 13 | \( 1 - T \) | |
| good | 3 | \( 1 + T + p T^{2} \) | 1.3.b |
| 7 | \( 1 - 2 T + p T^{2} \) | 1.7.ac |
| 17 | \( 1 - T + p T^{2} \) | 1.17.ab |
| 19 | \( 1 - 6 T + p T^{2} \) | 1.19.ag |
| 23 | \( 1 + 9 T + p T^{2} \) | 1.23.j |
| 29 | \( 1 + 5 T + p T^{2} \) | 1.29.f |
| 31 | \( 1 - 2 T + p T^{2} \) | 1.31.ac |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 + 13 T + p T^{2} \) | 1.43.n |
| 47 | \( 1 + 6 T + p T^{2} \) | 1.47.g |
| 53 | \( 1 - 7 T + p T^{2} \) | 1.53.ah |
| 59 | \( 1 + 8 T + p T^{2} \) | 1.59.i |
| 61 | \( 1 + 7 T + p T^{2} \) | 1.61.h |
| 67 | \( 1 - 2 T + p T^{2} \) | 1.67.ac |
| 71 | \( 1 - 10 T + p T^{2} \) | 1.71.ak |
| 73 | \( 1 + 4 T + p T^{2} \) | 1.73.e |
| 79 | \( 1 + 11 T + p T^{2} \) | 1.79.l |
| 83 | \( 1 + 16 T + p T^{2} \) | 1.83.q |
| 89 | \( 1 - 14 T + p T^{2} \) | 1.89.ao |
| 97 | \( 1 + 10 T + p T^{2} \) | 1.97.k |
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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
−16.27251359438103, −15.75318716233322, −15.00589137235881, −14.55285742650321, −13.86439531736435, −13.75093128973572, −13.00031328089654, −12.11644998647781, −11.85418358043084, −11.39550649991884, −10.83089456451405, −10.04926742024252, −9.720930593784566, −8.686162552247920, −8.167291197982559, −7.581812224653428, −6.749582456442370, −6.155560954347093, −5.540018914342789, −5.189446767927238, −4.499398994210378, −3.573995638716090, −3.004757908663798, −1.953162480185760, −1.378827286293878, 0,
1.378827286293878, 1.953162480185760, 3.004757908663798, 3.573995638716090, 4.499398994210378, 5.189446767927238, 5.540018914342789, 6.155560954347093, 6.749582456442370, 7.581812224653428, 8.167291197982559, 8.686162552247920, 9.720930593784566, 10.04926742024252, 10.83089456451405, 11.39550649991884, 11.85418358043084, 12.11644998647781, 13.00031328089654, 13.75093128973572, 13.86439531736435, 14.55285742650321, 15.00589137235881, 15.75318716233322, 16.27251359438103
