L(s) = 1 | + (−0.5 − 0.866i)2-s + (−0.619 − 1.61i)3-s + (−0.499 + 0.866i)4-s − 1.76·5-s + (−1.09 + 1.34i)6-s + 0.999·8-s + (−2.23 + 2.00i)9-s + (0.880 + 1.52i)10-s + 6.12·11-s + (1.71 + 0.272i)12-s + (0.380 + 0.658i)13-s + (1.09 + 2.84i)15-s + (−0.5 − 0.866i)16-s + (3.42 + 5.92i)17-s + (2.85 + 0.931i)18-s + (−0.971 + 1.68i)19-s + ⋯ |
L(s) = 1 | + (−0.353 − 0.612i)2-s + (−0.357 − 0.933i)3-s + (−0.249 + 0.433i)4-s − 0.787·5-s + (−0.445 + 0.549i)6-s + 0.353·8-s + (−0.744 + 0.668i)9-s + (0.278 + 0.482i)10-s + 1.84·11-s + (0.493 + 0.0785i)12-s + (0.105 + 0.182i)13-s + (0.281 + 0.735i)15-s + (−0.125 − 0.216i)16-s + (0.829 + 1.43i)17-s + (0.672 + 0.219i)18-s + (−0.222 + 0.385i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.417 + 0.908i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.417 + 0.908i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.854525 - 0.548003i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.854525 - 0.548003i\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (0.619 + 1.61i)T \) |
| 7 | \( 1 \) |
good | 5 | \( 1 + 1.76T + 5T^{2} \) |
| 11 | \( 1 - 6.12T + 11T^{2} \) |
| 13 | \( 1 + (-0.380 - 0.658i)T + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + (-3.42 - 5.92i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (0.971 - 1.68i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + 0.421T + 23T^{2} \) |
| 29 | \( 1 + (-0.732 + 1.26i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (-3.85 + 6.67i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-1.44 + 2.49i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-3.47 - 6.01i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-4.33 + 7.49i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (-0.830 - 1.43i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (0.112 + 0.195i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-0.993 + 1.72i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (5.17 + 8.96i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (3.39 - 5.87i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 - 10.7T + 71T^{2} \) |
| 73 | \( 1 + (0.153 + 0.265i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-6.72 - 11.6i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-1.56 + 2.70i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (1.30 - 2.25i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-1.81 + 3.14i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−10.05250257774211129511283110208, −9.087779351970945710306903844760, −8.184936931679901370750976282658, −7.66939033615297648355464918197, −6.55211242284378579004622306134, −5.89467695074466946112461881623, −4.25815264067419351492449065100, −3.59743946881110105349437691509, −2.01170928621604340445318628659, −0.948135128633865970709958598002,
0.860120525913748664708409208595, 3.20201075350889198595479878384, 4.15495510407590828095756576275, 4.92866719286475898525443300046, 6.02873743838603310345288814761, 6.83357082895737276285331335283, 7.74880046678592397260791823992, 8.834820081498663541697626940087, 9.300226592069166644846459702098, 10.11816629835095632867682156541