| L(s) = 1 | + 2-s − 3-s + 4-s − 5-s − 6-s − 2·7-s + 8-s + 9-s − 10-s − 12-s + 6·13-s − 2·14-s + 15-s + 16-s − 2·17-s + 18-s + 4·19-s − 20-s + 2·21-s − 6·23-s − 24-s + 25-s + 6·26-s − 27-s − 2·28-s + 30-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 − 0.755·7-s + 0.353·8-s + 1/3·9-s − 0.316·10-s − 0.288·12-s + 1.66·13-s − 0.534·14-s + 0.258·15-s + 1/4·16-s − 0.485·17-s + 0.235·18-s + 0.917·19-s − 0.223·20-s + 0.436·21-s − 1.25·23-s − 0.204·24-s + 1/5·25-s + 1.17·26-s − 0.192·27-s − 0.377·28-s + 0.182·30-s + 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 25230 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25230 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) | |
| 5 | \( 1 + T \) | |
| 29 | \( 1 \) | |
| good | 7 | \( 1 + 2 T + p T^{2} \) | 1.7.c |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 13 | \( 1 - 6 T + p T^{2} \) | 1.13.ag |
| 17 | \( 1 + 2 T + p T^{2} \) | 1.17.c |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 23 | \( 1 + 6 T + p T^{2} \) | 1.23.g |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 41 | \( 1 + p T^{2} \) | 1.41.a |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 + 8 T + p T^{2} \) | 1.47.i |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 + p T^{2} \) | 1.61.a |
| 67 | \( 1 - 2 T + p T^{2} \) | 1.67.ac |
| 71 | \( 1 - 8 T + p T^{2} \) | 1.71.ai |
| 73 | \( 1 - 14 T + p T^{2} \) | 1.73.ao |
| 79 | \( 1 - 4 T + p T^{2} \) | 1.79.ae |
| 83 | \( 1 + 6 T + p T^{2} \) | 1.83.g |
| 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
−15.74561269455847, −15.30744995962117, −14.46082724723667, −13.88003592471706, −13.49821689793990, −12.84115295121463, −12.56405548311904, −11.73719565366798, −11.43506414462200, −10.95932403306182, −10.30263532009209, −9.701781325198008, −9.140414940802291, −8.157121033455616, −7.981591951588943, −6.944558149781668, −6.527117317919859, −6.081553717875700, −5.450450726778701, −4.773224122309005, −3.985492284578712, −3.587128712829573, −2.960191306031534, −1.894285670217101, −1.075358339364070, 0,
1.075358339364070, 1.894285670217101, 2.960191306031534, 3.587128712829573, 3.985492284578712, 4.773224122309005, 5.450450726778701, 6.081553717875700, 6.527117317919859, 6.944558149781668, 7.981591951588943, 8.157121033455616, 9.140414940802291, 9.701781325198008, 10.30263532009209, 10.95932403306182, 11.43506414462200, 11.73719565366798, 12.56405548311904, 12.84115295121463, 13.49821689793990, 13.88003592471706, 14.46082724723667, 15.30744995962117, 15.74561269455847
