| L(s) = 1 | + 3-s − 7-s − 2·9-s + 4·13-s + 4·19-s − 21-s − 5·27-s − 6·29-s + 10·31-s − 8·37-s + 4·39-s − 3·41-s − 43-s + 9·47-s − 6·49-s + 12·53-s + 4·57-s − 6·59-s + 11·61-s + 2·63-s − 67-s + 6·71-s − 8·73-s − 14·79-s + 81-s − 12·83-s − 6·87-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.377·7-s − 2/3·9-s + 1.10·13-s + 0.917·19-s − 0.218·21-s − 0.962·27-s − 1.11·29-s + 1.79·31-s − 1.31·37-s + 0.640·39-s − 0.468·41-s − 0.152·43-s + 1.31·47-s − 6/7·49-s + 1.64·53-s + 0.529·57-s − 0.781·59-s + 1.40·61-s + 0.251·63-s − 0.122·67-s + 0.712·71-s − 0.936·73-s − 1.57·79-s + 1/9·81-s − 1.31·83-s − 0.643·87-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 48400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 48400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 11 | \( 1 \) | |
| good | 3 | \( 1 - T + p T^{2} \) | 1.3.ab |
| 7 | \( 1 + T + p T^{2} \) | 1.7.b |
| 13 | \( 1 - 4 T + p T^{2} \) | 1.13.ae |
| 17 | \( 1 + p T^{2} \) | 1.17.a |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 - 10 T + p T^{2} \) | 1.31.ak |
| 37 | \( 1 + 8 T + p T^{2} \) | 1.37.i |
| 41 | \( 1 + 3 T + p T^{2} \) | 1.41.d |
| 43 | \( 1 + T + p T^{2} \) | 1.43.b |
| 47 | \( 1 - 9 T + p T^{2} \) | 1.47.aj |
| 53 | \( 1 - 12 T + p T^{2} \) | 1.53.am |
| 59 | \( 1 + 6 T + p T^{2} \) | 1.59.g |
| 61 | \( 1 - 11 T + p T^{2} \) | 1.61.al |
| 67 | \( 1 + T + p T^{2} \) | 1.67.b |
| 71 | \( 1 - 6 T + p T^{2} \) | 1.71.ag |
| 73 | \( 1 + 8 T + p T^{2} \) | 1.73.i |
| 79 | \( 1 + 14 T + p T^{2} \) | 1.79.o |
| 83 | \( 1 + 12 T + p T^{2} \) | 1.83.m |
| 89 | \( 1 + 15 T + p T^{2} \) | 1.89.p |
| 97 | \( 1 + 8 T + p T^{2} \) | 1.97.i |
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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
−14.85903505205384, −14.13057721633686, −13.79243514221465, −13.44875064101842, −12.92988326325228, −12.13303435520696, −11.79632223945668, −11.19328318480173, −10.75435011903306, −9.946465931004438, −9.693367960928483, −8.925913013547057, −8.481002269261906, −8.228314755450613, −7.341432809833588, −6.949273840270014, −6.210522911821084, −5.619505149877714, −5.284334294887753, −4.219966045508317, −3.789489466819388, −3.073310879421270, −2.714028990245800, −1.772066939838338, −1.036666467267647, 0,
1.036666467267647, 1.772066939838338, 2.714028990245800, 3.073310879421270, 3.789489466819388, 4.219966045508317, 5.284334294887753, 5.619505149877714, 6.210522911821084, 6.949273840270014, 7.341432809833588, 8.228314755450613, 8.481002269261906, 8.925913013547057, 9.693367960928483, 9.946465931004438, 10.75435011903306, 11.19328318480173, 11.79632223945668, 12.13303435520696, 12.92988326325228, 13.44875064101842, 13.79243514221465, 14.13057721633686, 14.85903505205384
