L(s) = 1 | + (0.787 + 1.36i)2-s + (−1.10 + 1.90i)3-s + (−0.239 + 0.414i)4-s + (−1.88 − 3.25i)5-s − 3.46·6-s + (0.609 + 2.57i)7-s + 2.39·8-s + (−0.924 − 1.60i)9-s + (2.96 − 5.12i)10-s + (1.19 − 2.07i)11-s + (−0.527 − 0.913i)12-s + (−3.03 + 2.85i)14-s + 8.28·15-s + (2.36 + 4.09i)16-s + (0.152 − 0.264i)17-s + (1.45 − 2.52i)18-s + ⋯ |
L(s) = 1 | + (0.556 + 0.964i)2-s + (−0.635 + 1.10i)3-s + (−0.119 + 0.207i)4-s + (−0.840 − 1.45i)5-s − 1.41·6-s + (0.230 + 0.973i)7-s + 0.846·8-s + (−0.308 − 0.533i)9-s + (0.936 − 1.62i)10-s + (0.361 − 0.625i)11-s + (−0.152 − 0.263i)12-s + (−0.809 + 0.763i)14-s + 2.13·15-s + (0.591 + 1.02i)16-s + (0.0369 − 0.0640i)17-s + (0.343 − 0.594i)18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.638 - 0.769i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.638 - 0.769i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.701506455\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.701506455\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 + (-0.609 - 2.57i)T \) |
| 13 | \( 1 \) |
good | 2 | \( 1 + (-0.787 - 1.36i)T + (-1 + 1.73i)T^{2} \) |
| 3 | \( 1 + (1.10 - 1.90i)T + (-1.5 - 2.59i)T^{2} \) |
| 5 | \( 1 + (1.88 + 3.25i)T + (-2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (-1.19 + 2.07i)T + (-5.5 - 9.52i)T^{2} \) |
| 17 | \( 1 + (-0.152 + 0.264i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-1.23 - 2.14i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-4.16 - 7.21i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 - 4.97T + 29T^{2} \) |
| 31 | \( 1 + (-1.82 + 3.16i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-2.82 - 4.89i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + 3.27T + 41T^{2} \) |
| 43 | \( 1 + 9.84T + 43T^{2} \) |
| 47 | \( 1 + (-6.29 - 10.9i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (3.92 - 6.80i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-6.69 + 11.5i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (0.407 + 0.705i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-1.05 + 1.82i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 - 4.60T + 71T^{2} \) |
| 73 | \( 1 + (2.60 - 4.51i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (0.851 + 1.47i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 - 7.65T + 83T^{2} \) |
| 89 | \( 1 + (-6.05 - 10.4i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + 3.06T + 97T^{2} \) |
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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
−9.863857100900882500869792442342, −9.191956224383738510297646620349, −8.294688851110495622585360657834, −7.73438140734562121580822640390, −6.36647046252515233466814784023, −5.52468816510608168821002722354, −5.05147610952427058867949963127, −4.46825727147066776148659041316, −3.50616299729161408483878549266, −1.28818082695700627768865688673,
0.77478350136038007163908758443, 2.11718831100635975946476209417, 3.13858442845253139810232723265, 4.00754991503335719588218971058, 4.88919628773531947540961276214, 6.54561676017072465238998588037, 6.99248806237801918475214578224, 7.39393097911858728522103687009, 8.369536730372624347878665393125, 10.12221920880506219571797260840