| L(s) = 1 | − 2-s − 3-s + 4-s − 5-s + 6-s − 4·7-s − 8-s − 2·9-s + 10-s − 12-s − 13-s + 4·14-s + 15-s + 16-s + 3·17-s + 2·18-s − 6·19-s − 20-s + 4·21-s + 5·23-s + 24-s + 25-s + 26-s + 5·27-s − 4·28-s − 7·29-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 − 1.51·7-s − 0.353·8-s − 2/3·9-s + 0.316·10-s − 0.288·12-s − 0.277·13-s + 1.06·14-s + 0.258·15-s + 1/4·16-s + 0.727·17-s + 0.471·18-s − 1.37·19-s − 0.223·20-s + 0.872·21-s + 1.04·23-s + 0.204·24-s + 1/5·25-s + 0.196·26-s + 0.962·27-s − 0.755·28-s − 1.29·29-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 | \( 1 + 4 T + p T^{2} \) | 1.7.e |
| 17 | \( 1 - 3 T + p T^{2} \) | 1.17.ad |
| 19 | \( 1 + 6 T + p T^{2} \) | 1.19.g |
| 23 | \( 1 - 5 T + p T^{2} \) | 1.23.af |
| 29 | \( 1 + 7 T + p T^{2} \) | 1.29.h |
| 31 | \( 1 - 2 T + p T^{2} \) | 1.31.ac |
| 37 | \( 1 + 4 T + p T^{2} \) | 1.37.e |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 + T + p T^{2} \) | 1.43.b |
| 47 | \( 1 + 8 T + p T^{2} \) | 1.47.i |
| 53 | \( 1 + 5 T + p T^{2} \) | 1.53.f |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 + 5 T + p T^{2} \) | 1.61.f |
| 67 | \( 1 + 8 T + p T^{2} \) | 1.67.i |
| 71 | \( 1 - 6 T + p T^{2} \) | 1.71.ag |
| 73 | \( 1 + 2 T + p T^{2} \) | 1.73.c |
| 79 | \( 1 - 5 T + p T^{2} \) | 1.79.af |
| 83 | \( 1 + 10 T + p T^{2} \) | 1.83.k |
| 89 | \( 1 + 8 T + p T^{2} \) | 1.89.i |
| 97 | \( 1 + p T^{2} \) | 1.97.a |
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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
−16.57158796998277, −16.26759265103839, −15.52790154736908, −15.01585588601876, −14.57495412186215, −13.66562462351346, −12.99489772892293, −12.54631476256060, −12.06612535882343, −11.39224280101829, −10.89382580138450, −10.32067101647553, −9.775425167757906, −9.138594267873080, −8.634005753911719, −7.977706813877474, −7.208981968880639, −6.628603460139287, −6.226717567225416, −5.501937764126590, −4.800605312657583, −3.724107293071552, −3.186133828969894, −2.519003034661403, −1.349552623584849, 0, 0,
1.349552623584849, 2.519003034661403, 3.186133828969894, 3.724107293071552, 4.800605312657583, 5.501937764126590, 6.226717567225416, 6.628603460139287, 7.208981968880639, 7.977706813877474, 8.634005753911719, 9.138594267873080, 9.775425167757906, 10.32067101647553, 10.89382580138450, 11.39224280101829, 12.06612535882343, 12.54631476256060, 12.99489772892293, 13.66562462351346, 14.57495412186215, 15.01585588601876, 15.52790154736908, 16.26759265103839, 16.57158796998277
