| L(s) = 1 | + (−0.978 − 0.207i)2-s + (−0.155 + 1.72i)3-s + (0.913 + 0.406i)4-s + (−0.0873 − 2.23i)5-s + (0.511 − 1.65i)6-s + (1.26 − 2.19i)7-s + (−0.809 − 0.587i)8-s + (−2.95 − 0.537i)9-s + (−0.379 + 2.20i)10-s + (−3.51 − 0.746i)11-s + (−0.843 + 1.51i)12-s + (−4.89 + 1.04i)13-s + (−1.69 + 1.87i)14-s + (3.86 + 0.197i)15-s + (0.669 + 0.743i)16-s + (−3.54 − 2.57i)17-s + ⋯ |
| L(s) = 1 | + (−0.691 − 0.147i)2-s + (−0.0899 + 0.995i)3-s + (0.456 + 0.203i)4-s + (−0.0390 − 0.999i)5-s + (0.208 − 0.675i)6-s + (0.478 − 0.828i)7-s + (−0.286 − 0.207i)8-s + (−0.983 − 0.179i)9-s + (−0.119 + 0.696i)10-s + (−1.05 − 0.224i)11-s + (−0.243 + 0.436i)12-s + (−1.35 + 0.288i)13-s + (−0.452 + 0.502i)14-s + (0.998 + 0.0509i)15-s + (0.167 + 0.185i)16-s + (−0.859 − 0.624i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.672 + 0.740i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.672 + 0.740i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.143172 - 0.323398i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.143172 - 0.323398i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.978 + 0.207i)T \) |
| 3 | \( 1 + (0.155 - 1.72i)T \) |
| 5 | \( 1 + (0.0873 + 2.23i)T \) |
| good | 7 | \( 1 + (-1.26 + 2.19i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (3.51 + 0.746i)T + (10.0 + 4.47i)T^{2} \) |
| 13 | \( 1 + (4.89 - 1.04i)T + (11.8 - 5.28i)T^{2} \) |
| 17 | \( 1 + (3.54 + 2.57i)T + (5.25 + 16.1i)T^{2} \) |
| 19 | \( 1 + (-2.12 - 1.54i)T + (5.87 + 18.0i)T^{2} \) |
| 23 | \( 1 + (5.56 - 6.18i)T + (-2.40 - 22.8i)T^{2} \) |
| 29 | \( 1 + (0.969 + 9.22i)T + (-28.3 + 6.02i)T^{2} \) |
| 31 | \( 1 + (-0.355 + 3.37i)T + (-30.3 - 6.44i)T^{2} \) |
| 37 | \( 1 + (1.69 - 5.21i)T + (-29.9 - 21.7i)T^{2} \) |
| 41 | \( 1 + (-9.70 + 2.06i)T + (37.4 - 16.6i)T^{2} \) |
| 43 | \( 1 + (-4.13 + 7.15i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (0.625 + 5.95i)T + (-45.9 + 9.77i)T^{2} \) |
| 53 | \( 1 + (10.3 - 7.53i)T + (16.3 - 50.4i)T^{2} \) |
| 59 | \( 1 + (-0.997 + 0.211i)T + (53.8 - 23.9i)T^{2} \) |
| 61 | \( 1 + (-1.06 - 0.225i)T + (55.7 + 24.8i)T^{2} \) |
| 67 | \( 1 + (0.101 - 0.963i)T + (-65.5 - 13.9i)T^{2} \) |
| 71 | \( 1 + (-6.61 + 4.80i)T + (21.9 - 67.5i)T^{2} \) |
| 73 | \( 1 + (-0.0809 - 0.249i)T + (-59.0 + 42.9i)T^{2} \) |
| 79 | \( 1 + (0.630 + 6.00i)T + (-77.2 + 16.4i)T^{2} \) |
| 83 | \( 1 + (-8.78 + 3.91i)T + (55.5 - 61.6i)T^{2} \) |
| 89 | \( 1 + (1.70 + 5.25i)T + (-72.0 + 52.3i)T^{2} \) |
| 97 | \( 1 + (-0.391 - 3.72i)T + (-94.8 + 20.1i)T^{2} \) |
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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
−10.57681590978251051472486530372, −9.736481815960573179671416786440, −9.279052708926060918840199805510, −7.993966213987007195387140423585, −7.58032932865352245241113515312, −5.80634797354599893557420557108, −4.84522062321374387553225592955, −3.99861020276243241887680733252, −2.32838746208769854747122259476, −0.25289648509882751601687383278,
2.14898987766711119704117709084, 2.74311785658599851628045351058, 5.04019396541745593716115918899, 6.07214951037128730067828327824, 6.99945082467183852683746680413, 7.74294325164507739676218391603, 8.437359310382783304838874422404, 9.597882458207394554547420035686, 10.70665699413762364321572553737, 11.18771612313535862613746735517
