| L(s) = 1 | + 2-s − 3-s + 4-s − 6-s + 5·7-s + 8-s − 2·9-s + 4·11-s − 12-s − 3·13-s + 5·14-s + 16-s − 17-s − 2·18-s − 2·19-s − 5·21-s + 4·22-s + 8·23-s − 24-s − 3·26-s + 5·27-s + 5·28-s − 5·31-s + 32-s − 4·33-s − 34-s − 2·36-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.408·6-s + 1.88·7-s + 0.353·8-s − 2/3·9-s + 1.20·11-s − 0.288·12-s − 0.832·13-s + 1.33·14-s + 1/4·16-s − 0.242·17-s − 0.471·18-s − 0.458·19-s − 1.09·21-s + 0.852·22-s + 1.66·23-s − 0.204·24-s − 0.588·26-s + 0.962·27-s + 0.944·28-s − 0.898·31-s + 0.176·32-s − 0.696·33-s − 0.171·34-s − 1/3·36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 850 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 850 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.387569131\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.387569131\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 17 | \( 1 + T \) | |
| good | 3 | \( 1 + T + p T^{2} \) | 1.3.b |
| 7 | \( 1 - 5 T + p T^{2} \) | 1.7.af |
| 11 | \( 1 - 4 T + p T^{2} \) | 1.11.ae |
| 13 | \( 1 + 3 T + p T^{2} \) | 1.13.d |
| 19 | \( 1 + 2 T + p T^{2} \) | 1.19.c |
| 23 | \( 1 - 8 T + p T^{2} \) | 1.23.ai |
| 29 | \( 1 + p T^{2} \) | 1.29.a |
| 31 | \( 1 + 5 T + p T^{2} \) | 1.31.f |
| 37 | \( 1 - 12 T + p T^{2} \) | 1.37.am |
| 41 | \( 1 + 10 T + p T^{2} \) | 1.41.k |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 - 2 T + p T^{2} \) | 1.47.ac |
| 53 | \( 1 - T + p T^{2} \) | 1.53.ab |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 - 2 T + p T^{2} \) | 1.61.ac |
| 67 | \( 1 + 8 T + p T^{2} \) | 1.67.i |
| 71 | \( 1 - 5 T + p T^{2} \) | 1.71.af |
| 73 | \( 1 + 4 T + p T^{2} \) | 1.73.e |
| 79 | \( 1 + 17 T + p T^{2} \) | 1.79.r |
| 83 | \( 1 - 16 T + p T^{2} \) | 1.83.aq |
| 89 | \( 1 + 6 T + p T^{2} \) | 1.89.g |
| 97 | \( 1 + 16 T + p T^{2} \) | 1.97.q |
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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
−10.64713753124356970889274641864, −9.237472952371987631199225800681, −8.453864351318274253193293581515, −7.47210562821160363943359837889, −6.62406300598745126805453340109, −5.54934722007244670213003825632, −4.89676996692636886203356262691, −4.14295887064175091644355723315, −2.60738907647638901911378266781, −1.33537369170169404828229047839,
1.33537369170169404828229047839, 2.60738907647638901911378266781, 4.14295887064175091644355723315, 4.89676996692636886203356262691, 5.54934722007244670213003825632, 6.62406300598745126805453340109, 7.47210562821160363943359837889, 8.453864351318274253193293581515, 9.237472952371987631199225800681, 10.64713753124356970889274641864
