L(s) = 1 | + (−0.866 − 0.5i)3-s + (−0.5 − 0.866i)5-s − i·7-s + (0.866 + 0.5i)11-s + 0.999i·15-s + (−0.5 + 0.866i)17-s + (0.866 − 0.5i)19-s + (−0.5 + 0.866i)21-s + (−0.866 + 0.5i)23-s + i·27-s + (0.866 + 0.5i)31-s + (−0.499 − 0.866i)33-s + (−0.866 + 0.5i)35-s + (0.5 + 0.866i)37-s + (0.866 − 0.5i)47-s + ⋯ |
L(s) = 1 | + (−0.866 − 0.5i)3-s + (−0.5 − 0.866i)5-s − i·7-s + (0.866 + 0.5i)11-s + 0.999i·15-s + (−0.5 + 0.866i)17-s + (0.866 − 0.5i)19-s + (−0.5 + 0.866i)21-s + (−0.866 + 0.5i)23-s + i·27-s + (0.866 + 0.5i)31-s + (−0.499 − 0.866i)33-s + (−0.866 + 0.5i)35-s + (0.5 + 0.866i)37-s + (0.866 − 0.5i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.197 + 0.980i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.197 + 0.980i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.5239175543\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5239175543\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 + iT \) |
good | 3 | \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \) |
| 5 | \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 11 | \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \) |
| 13 | \( 1 + T^{2} \) |
| 17 | \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 19 | \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \) |
| 23 | \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \) |
| 29 | \( 1 + T^{2} \) |
| 31 | \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \) |
| 37 | \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 41 | \( 1 + T^{2} \) |
| 43 | \( 1 - T^{2} \) |
| 47 | \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \) |
| 53 | \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 59 | \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \) |
| 61 | \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 67 | \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 79 | \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \) |
| 83 | \( 1 - T^{2} \) |
| 89 | \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 97 | \( 1 + T^{2} \) |
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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
−12.08799371505284644134280319916, −11.64190753115410579160223473130, −10.50254038799050942186063146748, −9.362546294717732787608098910988, −8.215630286176052104965475202038, −7.10972948766242002299187214914, −6.25005304233019844508058994269, −4.86158159448459267204310288019, −3.85082022785688915825160501435, −1.18115060432250576025081445514,
2.76788953705767214556321314744, 4.23739228702478708277672850209, 5.57705880420581847333263358842, 6.37343180237093145708937355264, 7.64951924974791220569551434192, 8.897422967595805715172683466262, 9.938035774668620069351885346279, 11.04069787765799055771426984236, 11.59600633154743071325789949495, 12.22709351788053490089647162831