| L(s) = 1 | + 7-s − 3·9-s − 11-s + 2·13-s − 4·17-s + 2·19-s − 5·23-s − 29-s − 2·31-s + 3·37-s + 12·41-s + 11·43-s − 2·47-s + 49-s + 6·53-s + 10·59-s − 4·61-s − 3·63-s + 67-s − 3·71-s − 77-s − 9·79-s + 9·81-s − 2·83-s − 6·89-s + 2·91-s − 14·97-s + ⋯ |
| L(s) = 1 | + 0.377·7-s − 9-s − 0.301·11-s + 0.554·13-s − 0.970·17-s + 0.458·19-s − 1.04·23-s − 0.185·29-s − 0.359·31-s + 0.493·37-s + 1.87·41-s + 1.67·43-s − 0.291·47-s + 1/7·49-s + 0.824·53-s + 1.30·59-s − 0.512·61-s − 0.377·63-s + 0.122·67-s − 0.356·71-s − 0.113·77-s − 1.01·79-s + 81-s − 0.219·83-s − 0.635·89-s + 0.209·91-s − 1.42·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 11200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 5 | \( 1 \) | |
| 7 | \( 1 - T \) | |
| good | 3 | \( 1 + p T^{2} \) | 1.3.a |
| 11 | \( 1 + T + p T^{2} \) | 1.11.b |
| 13 | \( 1 - 2 T + p T^{2} \) | 1.13.ac |
| 17 | \( 1 + 4 T + p T^{2} \) | 1.17.e |
| 19 | \( 1 - 2 T + p T^{2} \) | 1.19.ac |
| 23 | \( 1 + 5 T + p T^{2} \) | 1.23.f |
| 29 | \( 1 + T + p T^{2} \) | 1.29.b |
| 31 | \( 1 + 2 T + p T^{2} \) | 1.31.c |
| 37 | \( 1 - 3 T + p T^{2} \) | 1.37.ad |
| 41 | \( 1 - 12 T + p T^{2} \) | 1.41.am |
| 43 | \( 1 - 11 T + p T^{2} \) | 1.43.al |
| 47 | \( 1 + 2 T + p T^{2} \) | 1.47.c |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 - 10 T + p T^{2} \) | 1.59.ak |
| 61 | \( 1 + 4 T + p T^{2} \) | 1.61.e |
| 67 | \( 1 - T + p T^{2} \) | 1.67.ab |
| 71 | \( 1 + 3 T + p T^{2} \) | 1.71.d |
| 73 | \( 1 + p T^{2} \) | 1.73.a |
| 79 | \( 1 + 9 T + p T^{2} \) | 1.79.j |
| 83 | \( 1 + 2 T + p T^{2} \) | 1.83.c |
| 89 | \( 1 + 6 T + p T^{2} \) | 1.89.g |
| 97 | \( 1 + 14 T + p T^{2} \) | 1.97.o |
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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
−16.70705630925623, −16.06740615208795, −15.81801124530565, −14.91614573630964, −14.52796148492449, −13.87695415955428, −13.47957351671773, −12.74021628875876, −12.15064898249127, −11.40476836827563, −11.08341405319232, −10.54988671453806, −9.644841422452635, −9.109555719183774, −8.486753071590102, −7.946868806839628, −7.319218706959483, −6.479585974294943, −5.744889049711036, −5.440717142947947, −4.325523211970281, −3.916400539483368, −2.772771597588758, −2.311764166547204, −1.155418069093022, 0,
1.155418069093022, 2.311764166547204, 2.772771597588758, 3.916400539483368, 4.325523211970281, 5.440717142947947, 5.744889049711036, 6.479585974294943, 7.319218706959483, 7.946868806839628, 8.486753071590102, 9.109555719183774, 9.644841422452635, 10.54988671453806, 11.08341405319232, 11.40476836827563, 12.15064898249127, 12.74021628875876, 13.47957351671773, 13.87695415955428, 14.52796148492449, 14.91614573630964, 15.81801124530565, 16.06740615208795, 16.70705630925623
