L(s) = 1 | + (−0.5 − 0.866i)2-s + (0.5 − 0.866i)3-s + (−0.499 + 0.866i)4-s − 0.999·6-s + (−2.5 − 0.866i)7-s + 0.999·8-s + (−0.499 − 0.866i)9-s + (2.5 − 4.33i)11-s + (0.499 + 0.866i)12-s − 13-s + (0.500 + 2.59i)14-s + (−0.5 − 0.866i)16-s + (−3.5 + 6.06i)17-s + (−0.499 + 0.866i)18-s + (−3.5 − 6.06i)19-s + ⋯ |
L(s) = 1 | + (−0.353 − 0.612i)2-s + (0.288 − 0.499i)3-s + (−0.249 + 0.433i)4-s − 0.408·6-s + (−0.944 − 0.327i)7-s + 0.353·8-s + (−0.166 − 0.288i)9-s + (0.753 − 1.30i)11-s + (0.144 + 0.249i)12-s − 0.277·13-s + (0.133 + 0.694i)14-s + (−0.125 − 0.216i)16-s + (−0.848 + 1.47i)17-s + (−0.117 + 0.204i)18-s + (−0.802 − 1.39i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.991 + 0.126i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.991 + 0.126i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.0485470 - 0.765000i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0485470 - 0.765000i\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (-0.5 + 0.866i)T \) |
| 7 | \( 1 + (2.5 + 0.866i)T \) |
| 13 | \( 1 + T \) |
good | 5 | \( 1 + (-2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (-2.5 + 4.33i)T + (-5.5 - 9.52i)T^{2} \) |
| 17 | \( 1 + (3.5 - 6.06i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (3.5 + 6.06i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (1 + 1.73i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + 9T + 29T^{2} \) |
| 31 | \( 1 + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (2 + 3.46i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 - 4T + 41T^{2} \) |
| 43 | \( 1 - 2T + 43T^{2} \) |
| 47 | \( 1 + (-1.5 - 2.59i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (0.5 - 0.866i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (3.5 - 6.06i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (6.5 + 11.2i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (1.5 - 2.59i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 - 9T + 71T^{2} \) |
| 73 | \( 1 + (-5 + 8.66i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (7 + 12.1i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 - 16T + 83T^{2} \) |
| 89 | \( 1 + (-6 - 10.3i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 - 6T + 97T^{2} \) |
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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
−10.65989570045205121531432061604, −9.203568794378694818929019739795, −8.920030904475143983413660957657, −7.85979638002388090504422929786, −6.67734413253206016608384020228, −6.09817583210124852133315326507, −4.26515255943360664778161075487, −3.36354974527562499121881901580, −2.16862647203008153376810629194, −0.46035840243235644615126640601,
2.10050992828243298710176986334, 3.64135852906212428409458204719, 4.69201831469839573363367913387, 5.79347851888772012721116922892, 6.85859028368445848078516211790, 7.50567041258004675739322081623, 8.814066218781846911788144475148, 9.483528730192644470497644354109, 9.885209671239766446295193912485, 11.03636167687917677423150794342