| L(s) = 1 | − 3-s + 3·7-s + 9-s + 3·11-s − 17-s − 7·19-s − 3·21-s − 4·23-s − 27-s + 29-s − 10·31-s − 3·33-s + 37-s − 3·41-s + 10·43-s − 3·47-s + 2·49-s + 51-s − 9·53-s + 7·57-s − 10·59-s − 12·61-s + 3·63-s − 10·67-s + 4·69-s − 13·73-s + 9·77-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1.13·7-s + 1/3·9-s + 0.904·11-s − 0.242·17-s − 1.60·19-s − 0.654·21-s − 0.834·23-s − 0.192·27-s + 0.185·29-s − 1.79·31-s − 0.522·33-s + 0.164·37-s − 0.468·41-s + 1.52·43-s − 0.437·47-s + 2/7·49-s + 0.140·51-s − 1.23·53-s + 0.927·57-s − 1.30·59-s − 1.53·61-s + 0.377·63-s − 1.22·67-s + 0.481·69-s − 1.52·73-s + 1.02·77-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 3 | \( 1 + T \) | |
| 5 | \( 1 \) | |
| 17 | \( 1 + T \) | |
| good | 7 | \( 1 - 3 T + p T^{2} \) | 1.7.ad |
| 11 | \( 1 - 3 T + p T^{2} \) | 1.11.ad |
| 13 | \( 1 + p T^{2} \) | 1.13.a |
| 19 | \( 1 + 7 T + p T^{2} \) | 1.19.h |
| 23 | \( 1 + 4 T + p T^{2} \) | 1.23.e |
| 29 | \( 1 - T + p T^{2} \) | 1.29.ab |
| 31 | \( 1 + 10 T + p T^{2} \) | 1.31.k |
| 37 | \( 1 - T + p T^{2} \) | 1.37.ab |
| 41 | \( 1 + 3 T + p T^{2} \) | 1.41.d |
| 43 | \( 1 - 10 T + p T^{2} \) | 1.43.ak |
| 47 | \( 1 + 3 T + p T^{2} \) | 1.47.d |
| 53 | \( 1 + 9 T + p T^{2} \) | 1.53.j |
| 59 | \( 1 + 10 T + p T^{2} \) | 1.59.k |
| 61 | \( 1 + 12 T + p T^{2} \) | 1.61.m |
| 67 | \( 1 + 10 T + p T^{2} \) | 1.67.k |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 + 13 T + p T^{2} \) | 1.73.n |
| 79 | \( 1 - 2 T + p T^{2} \) | 1.79.ac |
| 83 | \( 1 + 4 T + p T^{2} \) | 1.83.e |
| 89 | \( 1 + 10 T + p T^{2} \) | 1.89.k |
| 97 | \( 1 - 6 T + p T^{2} \) | 1.97.ag |
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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
−7.75131345520724576689416812048, −7.23485875315271199981423414561, −6.16793394019973628172404866358, −5.93958451969118565483249189289, −4.65902178664894086067273107877, −4.46161374092450945700867361980, −3.45677028797570232560198048618, −2.05597579960511588497965593428, −1.48684886570517259415575052923, 0,
1.48684886570517259415575052923, 2.05597579960511588497965593428, 3.45677028797570232560198048618, 4.46161374092450945700867361980, 4.65902178664894086067273107877, 5.93958451969118565483249189289, 6.16793394019973628172404866358, 7.23485875315271199981423414561, 7.75131345520724576689416812048
