L(s) = 1 | + (−0.5 + 0.866i)2-s + (0.349 − 1.69i)3-s + (−0.499 − 0.866i)4-s − 3.69·5-s + (1.29 + 1.15i)6-s + 0.999·8-s + (−2.75 − 1.18i)9-s + (1.84 − 3.20i)10-s − 1.47·11-s + (−1.64 + 0.545i)12-s + (1.34 − 2.33i)13-s + (−1.29 + 6.27i)15-s + (−0.5 + 0.866i)16-s + (−3.28 + 5.69i)17-s + (2.40 − 1.79i)18-s + (0.444 + 0.769i)19-s + ⋯ |
L(s) = 1 | + (−0.353 + 0.612i)2-s + (0.201 − 0.979i)3-s + (−0.249 − 0.433i)4-s − 1.65·5-s + (0.528 + 0.469i)6-s + 0.353·8-s + (−0.918 − 0.395i)9-s + (0.584 − 1.01i)10-s − 0.445·11-s + (−0.474 + 0.157i)12-s + (0.374 − 0.648i)13-s + (−0.334 + 1.62i)15-s + (−0.125 + 0.216i)16-s + (−0.797 + 1.38i)17-s + (0.566 − 0.422i)18-s + (0.101 + 0.176i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.105 - 0.994i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.105 - 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.393353 + 0.353919i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.393353 + 0.353919i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.5 - 0.866i)T \) |
| 3 | \( 1 + (-0.349 + 1.69i)T \) |
| 7 | \( 1 \) |
good | 5 | \( 1 + 3.69T + 5T^{2} \) |
| 11 | \( 1 + 1.47T + 11T^{2} \) |
| 13 | \( 1 + (-1.34 + 2.33i)T + (-6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + (3.28 - 5.69i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-0.444 - 0.769i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 - 6.28T + 23T^{2} \) |
| 29 | \( 1 + (-1.25 - 2.17i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (-3.40 - 5.89i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (1.38 + 2.40i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-2.05 + 3.56i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (-0.00618 - 0.0107i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (3.49 - 6.05i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (1.60 - 2.78i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-3.45 - 5.98i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (2.86 - 4.96i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-4.73 - 8.19i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + 5.46T + 71T^{2} \) |
| 73 | \( 1 + (-6.03 + 10.4i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (5.72 - 9.91i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (2.23 + 3.87i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (-4.43 - 7.68i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-6.58 - 11.4i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−10.58119073783996658411962792470, −8.933936896224472751734331252135, −8.435860508768538897893589983370, −7.80923242654595105303337436982, −7.12172384444572712288304201687, −6.34169763381382009073736607423, −5.19285808532713905336542272583, −3.98230970520837817897675783341, −2.92332368580282154704752283112, −1.11203542182934584380599048891,
0.33602037307279294487986369816, 2.64339038750555552170518651663, 3.48792214310596802882947827575, 4.42659095115916962199122121819, 4.97799916140919826475581289185, 6.73932158769527601795334425372, 7.70152529228599840352253621003, 8.437065181079716618778883521024, 9.131850002937741128770107086744, 9.914022886927405518531009348649