| L(s) = 1 | − 2·7-s − 2·11-s + 6·13-s + 17-s − 3·19-s − 7·23-s − 6·29-s − 31-s − 8·37-s + 10·41-s + 2·43-s − 8·47-s − 3·49-s − 9·53-s − 10·59-s − 5·61-s − 8·67-s + 6·71-s + 2·73-s + 4·77-s − 15·79-s + 13·83-s − 2·89-s − 12·91-s − 8·97-s + 101-s + 103-s + ⋯ |
| L(s) = 1 | − 0.755·7-s − 0.603·11-s + 1.66·13-s + 0.242·17-s − 0.688·19-s − 1.45·23-s − 1.11·29-s − 0.179·31-s − 1.31·37-s + 1.56·41-s + 0.304·43-s − 1.16·47-s − 3/7·49-s − 1.23·53-s − 1.30·59-s − 0.640·61-s − 0.977·67-s + 0.712·71-s + 0.234·73-s + 0.455·77-s − 1.68·79-s + 1.42·83-s − 0.211·89-s − 1.25·91-s − 0.812·97-s + 0.0995·101-s + 0.0985·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9958100024\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9958100024\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| good | 7 | \( 1 + 2 T + p T^{2} \) | 1.7.c |
| 11 | \( 1 + 2 T + p T^{2} \) | 1.11.c |
| 13 | \( 1 - 6 T + p T^{2} \) | 1.13.ag |
| 17 | \( 1 - T + p T^{2} \) | 1.17.ab |
| 19 | \( 1 + 3 T + p T^{2} \) | 1.19.d |
| 23 | \( 1 + 7 T + p T^{2} \) | 1.23.h |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 + T + p T^{2} \) | 1.31.b |
| 37 | \( 1 + 8 T + p T^{2} \) | 1.37.i |
| 41 | \( 1 - 10 T + p T^{2} \) | 1.41.ak |
| 43 | \( 1 - 2 T + p T^{2} \) | 1.43.ac |
| 47 | \( 1 + 8 T + p T^{2} \) | 1.47.i |
| 53 | \( 1 + 9 T + p T^{2} \) | 1.53.j |
| 59 | \( 1 + 10 T + p T^{2} \) | 1.59.k |
| 61 | \( 1 + 5 T + p T^{2} \) | 1.61.f |
| 67 | \( 1 + 8 T + p T^{2} \) | 1.67.i |
| 71 | \( 1 - 6 T + p T^{2} \) | 1.71.ag |
| 73 | \( 1 - 2 T + p T^{2} \) | 1.73.ac |
| 79 | \( 1 + 15 T + p T^{2} \) | 1.79.p |
| 83 | \( 1 - 13 T + p T^{2} \) | 1.83.an |
| 89 | \( 1 + 2 T + p T^{2} \) | 1.89.c |
| 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
−14.66287909844260, −14.08407006478117, −13.62931630893732, −13.16210194723940, −12.56658906511716, −12.37788310184668, −11.38660207308208, −11.08357656485475, −10.52902814844480, −10.02031854381001, −9.403895103173815, −8.925734018766596, −8.284549411961494, −7.837671391341640, −7.243213634709655, −6.390068684194042, −6.076561076420988, −5.667909280193447, −4.754404213633312, −4.109754760602964, −3.471872051734903, −3.079768054084759, −2.040881618723694, −1.520177567166843, −0.3501054927837006,
0.3501054927837006, 1.520177567166843, 2.040881618723694, 3.079768054084759, 3.471872051734903, 4.109754760602964, 4.754404213633312, 5.667909280193447, 6.076561076420988, 6.390068684194042, 7.243213634709655, 7.837671391341640, 8.284549411961494, 8.925734018766596, 9.403895103173815, 10.02031854381001, 10.52902814844480, 11.08357656485475, 11.38660207308208, 12.37788310184668, 12.56658906511716, 13.16210194723940, 13.62931630893732, 14.08407006478117, 14.66287909844260
