| L(s) = 1 | + 2-s − 3·3-s + 4-s + 2·5-s − 3·6-s + 2·7-s + 8-s + 6·9-s + 2·10-s + 2·11-s − 3·12-s + 13-s + 2·14-s − 6·15-s + 16-s + 6·18-s + 2·20-s − 6·21-s + 2·22-s − 4·23-s − 3·24-s − 25-s + 26-s − 9·27-s + 2·28-s − 2·29-s − 6·30-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.73·3-s + 1/2·4-s + 0.894·5-s − 1.22·6-s + 0.755·7-s + 0.353·8-s + 2·9-s + 0.632·10-s + 0.603·11-s − 0.866·12-s + 0.277·13-s + 0.534·14-s − 1.54·15-s + 1/4·16-s + 1.41·18-s + 0.447·20-s − 1.30·21-s + 0.426·22-s − 0.834·23-s − 0.612·24-s − 1/5·25-s + 0.196·26-s − 1.73·27-s + 0.377·28-s − 0.371·29-s − 1.09·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43706 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43706 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) | |
| 13 | \( 1 - T \) | |
| 41 | \( 1 \) | |
| good | 3 | \( 1 + p T + p T^{2} \) | 1.3.d |
| 5 | \( 1 - 2 T + p T^{2} \) | 1.5.ac |
| 7 | \( 1 - 2 T + p T^{2} \) | 1.7.ac |
| 11 | \( 1 - 2 T + p T^{2} \) | 1.11.ac |
| 17 | \( 1 + p T^{2} \) | 1.17.a |
| 19 | \( 1 + p T^{2} \) | 1.19.a |
| 23 | \( 1 + 4 T + p T^{2} \) | 1.23.e |
| 29 | \( 1 + 2 T + p T^{2} \) | 1.29.c |
| 31 | \( 1 - T + p T^{2} \) | 1.31.ab |
| 37 | \( 1 + p T^{2} \) | 1.37.a |
| 43 | \( 1 + 2 T + p T^{2} \) | 1.43.c |
| 47 | \( 1 - 2 T + p T^{2} \) | 1.47.ac |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 + 7 T + p T^{2} \) | 1.59.h |
| 61 | \( 1 - 13 T + p T^{2} \) | 1.61.an |
| 67 | \( 1 - 2 T + p T^{2} \) | 1.67.ac |
| 71 | \( 1 + 4 T + p T^{2} \) | 1.71.e |
| 73 | \( 1 + 10 T + p T^{2} \) | 1.73.k |
| 79 | \( 1 - 4 T + p T^{2} \) | 1.79.ae |
| 83 | \( 1 + 15 T + p T^{2} \) | 1.83.p |
| 89 | \( 1 - 3 T + p T^{2} \) | 1.89.ad |
| 97 | \( 1 + 11 T + p T^{2} \) | 1.97.l |
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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
−14.89526112762287, −14.21132685023984, −14.03333542991932, −13.13443217146624, −12.99863891426469, −12.13289832838409, −11.91458440990256, −11.32276935578740, −11.01327132313725, −10.39344661075830, −9.896549729490430, −9.431983733768587, −8.531039131349398, −7.876935150183156, −7.148462216164532, −6.642487012332020, −6.079152054009087, −5.740694177998739, −5.271715630211767, −4.603153506185576, −4.207562496296773, −3.426903485792100, −2.322319318670954, −1.624684355084606, −1.180139975802423, 0,
1.180139975802423, 1.624684355084606, 2.322319318670954, 3.426903485792100, 4.207562496296773, 4.603153506185576, 5.271715630211767, 5.740694177998739, 6.079152054009087, 6.642487012332020, 7.148462216164532, 7.876935150183156, 8.531039131349398, 9.431983733768587, 9.896549729490430, 10.39344661075830, 11.01327132313725, 11.32276935578740, 11.91458440990256, 12.13289832838409, 12.99863891426469, 13.13443217146624, 14.03333542991932, 14.21132685023984, 14.89526112762287
