L(s) = 1 | + (−0.866 − 0.5i)2-s + (−1.73 + i)3-s + (0.499 + 0.866i)4-s + 1.99·6-s + (−0.866 − 2.5i)7-s − 0.999i·8-s + (0.499 − 0.866i)9-s + (−1.73 − 0.999i)12-s + 2i·13-s + (−0.500 + 2.59i)14-s + (−0.5 + 0.866i)16-s + (2.59 − 1.5i)17-s + (−0.866 + 0.499i)18-s + (4 − 6.92i)19-s + (4 + 3.46i)21-s + ⋯ |
L(s) = 1 | + (−0.612 − 0.353i)2-s + (−0.999 + 0.577i)3-s + (0.249 + 0.433i)4-s + 0.816·6-s + (−0.327 − 0.944i)7-s − 0.353i·8-s + (0.166 − 0.288i)9-s + (−0.499 − 0.288i)12-s + 0.554i·13-s + (−0.133 + 0.694i)14-s + (−0.125 + 0.216i)16-s + (0.630 − 0.363i)17-s + (−0.204 + 0.117i)18-s + (0.917 − 1.58i)19-s + (0.872 + 0.755i)21-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.830 + 0.556i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.830 + 0.556i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.635685 - 0.193301i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.635685 - 0.193301i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.866 + 0.5i)T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (0.866 + 2.5i)T \) |
good | 3 | \( 1 + (1.73 - i)T + (1.5 - 2.59i)T^{2} \) |
| 11 | \( 1 + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 - 2iT - 13T^{2} \) |
| 17 | \( 1 + (-2.59 + 1.5i)T + (8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-4 + 6.92i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-7.79 - 4.5i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 - 6T + 29T^{2} \) |
| 31 | \( 1 + (2.5 + 4.33i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-6.92 - 4i)T + (18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + 3T + 41T^{2} \) |
| 43 | \( 1 + 10iT - 43T^{2} \) |
| 47 | \( 1 + (2.59 + 1.5i)T + (23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-5.19 + 3i)T + (26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-6 - 10.3i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-2 + 3.46i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (1.73 - i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 9T + 71T^{2} \) |
| 73 | \( 1 + (8.66 - 5i)T + (36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-2.5 + 4.33i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + 6iT - 83T^{2} \) |
| 89 | \( 1 + (-1.5 + 2.59i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + 5iT - 97T^{2} \) |
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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
−11.43671698703751029585547963874, −10.46495489803820079243864786360, −9.792776053469139254390560571334, −8.921146128640846082345186112196, −7.45882103973415149908678433283, −6.78981061860995961911933690796, −5.38716627155796722449826448907, −4.40629002413611694906270728645, −3.03056681635218462320571195318, −0.805248425416707091552306197107,
1.15686767858077975611303463807, 3.04281427940153653423373120391, 5.16661931313627029139518097470, 5.90117415737540215497126713562, 6.65677790846285754982331692557, 7.76797055868064297030765774033, 8.703086226502672126189803755678, 9.747413280956914619001258924107, 10.65049223475402091071688317960, 11.61470749054750949140944993225