| L(s) = 1 | + 2-s − 3-s + 4-s − 6-s + 3·7-s + 8-s + 9-s − 11-s − 12-s − 2·13-s + 3·14-s + 16-s − 3·17-s + 18-s − 2·19-s − 3·21-s − 22-s + 7·23-s − 24-s − 5·25-s − 2·26-s − 27-s + 3·28-s + 9·31-s + 32-s + 33-s − 3·34-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.408·6-s + 1.13·7-s + 0.353·8-s + 1/3·9-s − 0.301·11-s − 0.288·12-s − 0.554·13-s + 0.801·14-s + 1/4·16-s − 0.727·17-s + 0.235·18-s − 0.458·19-s − 0.654·21-s − 0.213·22-s + 1.45·23-s − 0.204·24-s − 25-s − 0.392·26-s − 0.192·27-s + 0.566·28-s + 1.61·31-s + 0.176·32-s + 0.174·33-s − 0.514·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 55506 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 55506 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.683457190\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.683457190\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) | |
| 11 | \( 1 + T \) | |
| 29 | \( 1 \) | |
| good | 5 | \( 1 + p T^{2} \) | 1.5.a |
| 7 | \( 1 - 3 T + p T^{2} \) | 1.7.ad |
| 13 | \( 1 + 2 T + p T^{2} \) | 1.13.c |
| 17 | \( 1 + 3 T + p T^{2} \) | 1.17.d |
| 19 | \( 1 + 2 T + p T^{2} \) | 1.19.c |
| 23 | \( 1 - 7 T + p T^{2} \) | 1.23.ah |
| 31 | \( 1 - 9 T + p T^{2} \) | 1.31.aj |
| 37 | \( 1 - 8 T + p T^{2} \) | 1.37.ai |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 - 6 T + p T^{2} \) | 1.43.ag |
| 47 | \( 1 + T + p T^{2} \) | 1.47.b |
| 53 | \( 1 - 14 T + p T^{2} \) | 1.53.ao |
| 59 | \( 1 - 4 T + p T^{2} \) | 1.59.ae |
| 61 | \( 1 + 8 T + p T^{2} \) | 1.61.i |
| 67 | \( 1 - 12 T + p T^{2} \) | 1.67.am |
| 71 | \( 1 + 3 T + p T^{2} \) | 1.71.d |
| 73 | \( 1 - 11 T + p T^{2} \) | 1.73.al |
| 79 | \( 1 - 5 T + p T^{2} \) | 1.79.af |
| 83 | \( 1 + p T^{2} \) | 1.83.a |
| 89 | \( 1 - 13 T + p T^{2} \) | 1.89.an |
| 97 | \( 1 + 17 T + p T^{2} \) | 1.97.r |
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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
−14.53015234829960, −13.65604738517127, −13.52678570569508, −12.91714457185551, −12.25527940890326, −11.93039322684828, −11.25332853693808, −11.10133905683558, −10.48409848137499, −9.909784075805277, −9.302696490952916, −8.533960374934098, −8.053115845669896, −7.531024014591232, −6.888590107925235, −6.445747978298391, −5.761337672295273, −5.161818545989379, −4.769655910794908, −4.307790577686629, −3.659846019704098, −2.581169213996608, −2.322634165704389, −1.381747108882604, −0.6304704869910557,
0.6304704869910557, 1.381747108882604, 2.322634165704389, 2.581169213996608, 3.659846019704098, 4.307790577686629, 4.769655910794908, 5.161818545989379, 5.761337672295273, 6.445747978298391, 6.888590107925235, 7.531024014591232, 8.053115845669896, 8.533960374934098, 9.302696490952916, 9.909784075805277, 10.48409848137499, 11.10133905683558, 11.25332853693808, 11.93039322684828, 12.25527940890326, 12.91714457185551, 13.52678570569508, 13.65604738517127, 14.53015234829960
