| L(s) = 1 | − 5-s − 2·7-s + 4·11-s + 2·13-s + 5·17-s − 5·19-s − 23-s + 25-s + 2·29-s − 7·31-s + 2·35-s + 6·37-s + 4·43-s − 4·47-s − 3·49-s − 9·53-s − 4·55-s + 14·59-s + 11·61-s − 2·65-s + 14·67-s − 12·73-s − 8·77-s + 3·79-s − 83-s − 5·85-s − 4·91-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 0.755·7-s + 1.20·11-s + 0.554·13-s + 1.21·17-s − 1.14·19-s − 0.208·23-s + 1/5·25-s + 0.371·29-s − 1.25·31-s + 0.338·35-s + 0.986·37-s + 0.609·43-s − 0.583·47-s − 3/7·49-s − 1.23·53-s − 0.539·55-s + 1.82·59-s + 1.40·61-s − 0.248·65-s + 1.71·67-s − 1.40·73-s − 0.911·77-s + 0.337·79-s − 0.109·83-s − 0.542·85-s − 0.419·91-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8640 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8640 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.740398077\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.740398077\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 5 | \( 1 + T \) | |
| good | 7 | \( 1 + 2 T + p T^{2} \) | 1.7.c |
| 11 | \( 1 - 4 T + p T^{2} \) | 1.11.ae |
| 13 | \( 1 - 2 T + p T^{2} \) | 1.13.ac |
| 17 | \( 1 - 5 T + p T^{2} \) | 1.17.af |
| 19 | \( 1 + 5 T + p T^{2} \) | 1.19.f |
| 23 | \( 1 + T + p T^{2} \) | 1.23.b |
| 29 | \( 1 - 2 T + p T^{2} \) | 1.29.ac |
| 31 | \( 1 + 7 T + p T^{2} \) | 1.31.h |
| 37 | \( 1 - 6 T + p T^{2} \) | 1.37.ag |
| 41 | \( 1 + p T^{2} \) | 1.41.a |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 + 4 T + p T^{2} \) | 1.47.e |
| 53 | \( 1 + 9 T + p T^{2} \) | 1.53.j |
| 59 | \( 1 - 14 T + p T^{2} \) | 1.59.ao |
| 61 | \( 1 - 11 T + p T^{2} \) | 1.61.al |
| 67 | \( 1 - 14 T + p T^{2} \) | 1.67.ao |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 + 12 T + p T^{2} \) | 1.73.m |
| 79 | \( 1 - 3 T + p T^{2} \) | 1.79.ad |
| 83 | \( 1 + T + p T^{2} \) | 1.83.b |
| 89 | \( 1 + p T^{2} \) | 1.89.a |
| 97 | \( 1 - 16 T + p T^{2} \) | 1.97.aq |
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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
−7.80682575825068988107180784860, −6.97582489844166987098471330845, −6.42323440035188465542207462285, −5.87926903845192686452558018879, −4.97725659093239021802777561752, −3.86804861284948376825381462383, −3.77852652903357473425653255282, −2.76179113918930563985009936390, −1.65614461383301376595013047379, −0.66140698487738723827485617052,
0.66140698487738723827485617052, 1.65614461383301376595013047379, 2.76179113918930563985009936390, 3.77852652903357473425653255282, 3.86804861284948376825381462383, 4.97725659093239021802777561752, 5.87926903845192686452558018879, 6.42323440035188465542207462285, 6.97582489844166987098471330845, 7.80682575825068988107180784860
