L(s) = 1 | + (−0.931 − 0.537i)2-s + (−0.866 + 0.5i)3-s + (−0.421 − 0.729i)4-s + (−2.91 − 1.68i)5-s + 1.07·6-s + (0.879 + 2.49i)7-s + 3.05i·8-s + (0.499 − 0.866i)9-s + (1.80 + 3.13i)10-s + (1.21 + 3.08i)11-s + (0.729 + 0.421i)12-s + 4.53·13-s + (0.522 − 2.79i)14-s + 3.36·15-s + (0.802 − 1.38i)16-s + (−0.880 − 1.52i)17-s + ⋯ |
L(s) = 1 | + (−0.658 − 0.380i)2-s + (−0.499 + 0.288i)3-s + (−0.210 − 0.364i)4-s + (−1.30 − 0.752i)5-s + 0.439·6-s + (0.332 + 0.943i)7-s + 1.08i·8-s + (0.166 − 0.288i)9-s + (0.572 + 0.990i)10-s + (0.366 + 0.930i)11-s + (0.210 + 0.121i)12-s + 1.25·13-s + (0.139 − 0.747i)14-s + 0.868·15-s + (0.200 − 0.347i)16-s + (−0.213 − 0.370i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 231 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.444 - 0.895i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 231 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.444 - 0.895i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.351190 + 0.217855i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.351190 + 0.217855i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (0.866 - 0.5i)T \) |
| 7 | \( 1 + (-0.879 - 2.49i)T \) |
| 11 | \( 1 + (-1.21 - 3.08i)T \) |
good | 2 | \( 1 + (0.931 + 0.537i)T + (1 + 1.73i)T^{2} \) |
| 5 | \( 1 + (2.91 + 1.68i)T + (2.5 + 4.33i)T^{2} \) |
| 13 | \( 1 - 4.53T + 13T^{2} \) |
| 17 | \( 1 + (0.880 + 1.52i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (3.52 - 6.10i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (1.67 - 2.90i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 - 3.85iT - 29T^{2} \) |
| 31 | \( 1 + (2.78 - 1.61i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-4.52 + 7.83i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + 6.67T + 41T^{2} \) |
| 43 | \( 1 - 10.7iT - 43T^{2} \) |
| 47 | \( 1 + (4.51 + 2.60i)T + (23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-5.52 - 9.57i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-3.91 + 2.26i)T + (29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (1.72 - 2.98i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (0.151 + 0.262i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - 2.04T + 71T^{2} \) |
| 73 | \( 1 + (5.07 + 8.79i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-0.684 - 0.395i)T + (39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + 1.19T + 83T^{2} \) |
| 89 | \( 1 + (-7.44 - 4.29i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + 5.01iT - 97T^{2} \) |
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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
−12.00999528001176329735624519051, −11.44205507952627257084343034368, −10.53738473326071736785292419658, −9.332673597719403673696937014997, −8.647487687318334928727128099242, −7.80331499240838188008107359675, −6.05989549499932090579346466160, −4.97234527440548590386726204347, −3.94632093006159607476618676150, −1.54782374525725106801023508996,
0.48891420200029042518876076144, 3.52731029075905985736837779768, 4.30798526265481331332560196348, 6.44819646283984698844035605777, 7.03545195969780799839995760265, 8.129770961168062260084743143513, 8.612103426793367258909485779694, 10.30704770961071502745758481667, 11.16248909783669986345109714812, 11.64922890126358930607289484967