| L(s) = 1 | − 2-s − 3-s + 4-s − 5-s + 6-s − 8-s − 2·9-s + 10-s + 3·11-s − 12-s + 5·13-s + 15-s + 16-s + 2·18-s + 2·19-s − 20-s − 3·22-s + 24-s + 25-s − 5·26-s + 5·27-s − 6·29-s − 30-s + 4·31-s − 32-s − 3·33-s − 2·36-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.447·5-s + 0.408·6-s − 0.353·8-s − 2/3·9-s + 0.316·10-s + 0.904·11-s − 0.288·12-s + 1.38·13-s + 0.258·15-s + 1/4·16-s + 0.471·18-s + 0.458·19-s − 0.223·20-s − 0.639·22-s + 0.204·24-s + 1/5·25-s − 0.980·26-s + 0.962·27-s − 1.11·29-s − 0.182·30-s + 0.718·31-s − 0.176·32-s − 0.522·33-s − 1/3·36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 141610 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 141610 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) | |
| 7 | \( 1 \) | |
| 17 | \( 1 \) | |
| good | 3 | \( 1 + T + p T^{2} \) | 1.3.b |
| 11 | \( 1 - 3 T + p T^{2} \) | 1.11.ad |
| 13 | \( 1 - 5 T + p T^{2} \) | 1.13.af |
| 19 | \( 1 - 2 T + p T^{2} \) | 1.19.ac |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 - 4 T + p T^{2} \) | 1.31.ae |
| 37 | \( 1 - 4 T + p T^{2} \) | 1.37.ae |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 - 8 T + p T^{2} \) | 1.43.ai |
| 47 | \( 1 + 6 T + p T^{2} \) | 1.47.g |
| 53 | \( 1 - 9 T + p T^{2} \) | 1.53.aj |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 - 4 T + p T^{2} \) | 1.61.ae |
| 67 | \( 1 - 2 T + p T^{2} \) | 1.67.ac |
| 71 | \( 1 + 9 T + p T^{2} \) | 1.71.j |
| 73 | \( 1 - 10 T + p T^{2} \) | 1.73.ak |
| 79 | \( 1 + 5 T + p T^{2} \) | 1.79.f |
| 83 | \( 1 + p T^{2} \) | 1.83.a |
| 89 | \( 1 - 9 T + p T^{2} \) | 1.89.aj |
| 97 | \( 1 - 4 T + p T^{2} \) | 1.97.ae |
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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
−13.59328856751813, −13.15648509543352, −12.51187469079731, −11.96027576819737, −11.60789188704165, −11.21846324381520, −10.96455108982131, −10.29021260940970, −9.809482269658397, −9.183447030214558, −8.714669437760914, −8.483799298467268, −7.786799066432873, −7.349923591012305, −6.651538256848932, −6.291247686434194, −5.843230531059215, −5.301189006813960, −4.613023844647634, −3.835775320496621, −3.550971683387179, −2.828249369108764, −2.097620545873325, −1.229863280959843, −0.8660233642647116, 0,
0.8660233642647116, 1.229863280959843, 2.097620545873325, 2.828249369108764, 3.550971683387179, 3.835775320496621, 4.613023844647634, 5.301189006813960, 5.843230531059215, 6.291247686434194, 6.651538256848932, 7.349923591012305, 7.786799066432873, 8.483799298467268, 8.714669437760914, 9.183447030214558, 9.809482269658397, 10.29021260940970, 10.96455108982131, 11.21846324381520, 11.60789188704165, 11.96027576819737, 12.51187469079731, 13.15648509543352, 13.59328856751813
