L(s) = 1 | + (0.5 − 0.866i)2-s + (0.500 + 0.866i)4-s + (−0.5 − 2.59i)7-s + 3·8-s + (1.5 − 2.59i)9-s + (−1.5 − 2.59i)11-s + (−2.5 − 0.866i)14-s + (0.500 − 0.866i)16-s + (−3.5 − 6.06i)17-s + (−1.5 − 2.59i)18-s + (−3.5 + 6.06i)19-s − 3·22-s + (3 − 5.19i)23-s + (2.5 + 4.33i)25-s + (2 − 1.73i)28-s − 5·29-s + ⋯ |
L(s) = 1 | + (0.353 − 0.612i)2-s + (0.250 + 0.433i)4-s + (−0.188 − 0.981i)7-s + 1.06·8-s + (0.5 − 0.866i)9-s + (−0.452 − 0.783i)11-s + (−0.668 − 0.231i)14-s + (0.125 − 0.216i)16-s + (−0.848 − 1.47i)17-s + (−0.353 − 0.612i)18-s + (−0.802 + 1.39i)19-s − 0.639·22-s + (0.625 − 1.08i)23-s + (0.5 + 0.866i)25-s + (0.377 − 0.327i)28-s − 0.928·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.266 + 0.963i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.266 + 0.963i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.002686074\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.002686074\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 + (0.5 + 2.59i)T \) |
| 13 | \( 1 \) |
good | 2 | \( 1 + (-0.5 + 0.866i)T + (-1 - 1.73i)T^{2} \) |
| 3 | \( 1 + (-1.5 + 2.59i)T^{2} \) |
| 5 | \( 1 + (-2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (1.5 + 2.59i)T + (-5.5 + 9.52i)T^{2} \) |
| 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 | \( 1 + (-3 + 5.19i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + 5T + 29T^{2} \) |
| 31 | \( 1 + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-4 + 6.92i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + 41T^{2} \) |
| 43 | \( 1 - 2T + 43T^{2} \) |
| 47 | \( 1 + (-3.5 + 6.06i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-1.5 - 2.59i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (3.5 + 6.06i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-3.5 + 6.06i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (1.5 + 2.59i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - 5T + 71T^{2} \) |
| 73 | \( 1 + (-7 - 12.1i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-3 + 5.19i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + 83T^{2} \) |
| 89 | \( 1 + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 - 14T + 97T^{2} \) |
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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
−9.649953399698223935463595591909, −8.786126568384763260614108733579, −7.72937871177289099364753892311, −7.08420183805453174878321351656, −6.32923209422031849582970738260, −4.96905673918061977873504007344, −3.99054887205728175578328385627, −3.41918387351756557942644917843, −2.27110672584278211544112184555, −0.75827170547943211199786953275,
1.77784434914108759578004034314, 2.55110361751634610703236873684, 4.35108388799355082980131351121, 4.92320163728506987877075345213, 5.83694014615232399469479309593, 6.61256860481273696461975222313, 7.36397037398870767500520263020, 8.245483544734073332030914618777, 9.141428851267841831934196126951, 10.06308564520716104313794530665