L(s) = 1 | + (0.5 + 0.866i)2-s + (1.68 + 0.396i)3-s + (−0.499 + 0.866i)4-s + 1.37·5-s + (0.5 + 1.65i)6-s − 0.999·8-s + (2.68 + 1.33i)9-s + (0.686 + 1.18i)10-s + 4.37·11-s + (−1.18 + 1.26i)12-s + (1 + 1.73i)13-s + (2.31 + 0.543i)15-s + (−0.5 − 0.866i)16-s + (−2.18 − 3.78i)17-s + (0.186 + 2.99i)18-s + (2.5 − 4.33i)19-s + ⋯ |
L(s) = 1 | + (0.353 + 0.612i)2-s + (0.973 + 0.228i)3-s + (−0.249 + 0.433i)4-s + 0.613·5-s + (0.204 + 0.677i)6-s − 0.353·8-s + (0.895 + 0.445i)9-s + (0.216 + 0.375i)10-s + 1.31·11-s + (−0.342 + 0.364i)12-s + (0.277 + 0.480i)13-s + (0.597 + 0.140i)15-s + (−0.125 − 0.216i)16-s + (−0.530 − 0.918i)17-s + (0.0438 + 0.705i)18-s + (0.573 − 0.993i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.373 - 0.927i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.373 - 0.927i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.46010 + 1.66191i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.46010 + 1.66191i\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (-1.68 - 0.396i)T \) |
| 7 | \( 1 \) |
good | 5 | \( 1 - 1.37T + 5T^{2} \) |
| 11 | \( 1 - 4.37T + 11T^{2} \) |
| 13 | \( 1 + (-1 - 1.73i)T + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + (2.18 + 3.78i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-2.5 + 4.33i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + 7.37T + 23T^{2} \) |
| 29 | \( 1 + (1.37 - 2.37i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (-1 + 1.73i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (1 - 1.73i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-5.18 - 8.98i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (4.55 - 7.89i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (1.37 + 2.37i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-3.55 + 6.16i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (7.05 + 12.2i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (7.55 - 13.0i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 - 10.1T + 71T^{2} \) |
| 73 | \( 1 + (2.55 + 4.43i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (6.05 + 10.4i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (2.74 - 4.75i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (-1.62 + 2.81i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-4.55 + 7.89i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−9.699834940105487695726194294647, −9.518699288257019744151421761637, −8.653119239296464640888148963967, −7.74918462882129830111135915140, −6.81539802458336483448266940896, −6.14257766890731202948227007977, −4.82306960102525839619598381578, −4.07863961872967698723859359383, −2.99243127385315125048316031527, −1.73407927815907374175459666774,
1.45368049449275543607068222721, 2.23625729788867751514618077490, 3.68067271713760049751381038426, 4.06701900198906911448336074545, 5.71541173447312992321421994251, 6.34926936412669383505700528566, 7.54442407523677282138486433969, 8.501326700783915964702842922532, 9.212632112042383134348355725637, 9.986673387976032675423288837598