| L(s) = 1 | + (0.781 + 0.623i)2-s + (0.519 − 1.65i)3-s + (0.222 + 0.974i)4-s + (−0.177 − 2.36i)5-s + (1.43 − 0.967i)6-s + (−2.57 − 0.604i)7-s + (−0.433 + 0.900i)8-s + (−2.45 − 1.71i)9-s + (1.33 − 1.96i)10-s + (−1.84 + 0.724i)11-s + (1.72 + 0.139i)12-s + (2.09 − 0.822i)13-s + (−1.63 − 2.07i)14-s + (−4.00 − 0.937i)15-s + (−0.900 + 0.433i)16-s + (0.646 − 0.199i)17-s + ⋯ |
| L(s) = 1 | + (0.552 + 0.440i)2-s + (0.300 − 0.953i)3-s + (0.111 + 0.487i)4-s + (−0.0793 − 1.05i)5-s + (0.586 − 0.395i)6-s + (−0.973 − 0.228i)7-s + (−0.153 + 0.318i)8-s + (−0.819 − 0.572i)9-s + (0.423 − 0.620i)10-s + (−0.556 + 0.218i)11-s + (0.498 + 0.0401i)12-s + (0.581 − 0.228i)13-s + (−0.437 − 0.555i)14-s + (−1.03 − 0.242i)15-s + (−0.225 + 0.108i)16-s + (0.156 − 0.0483i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.605 + 0.796i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.605 + 0.796i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.637106 - 1.28478i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.637106 - 1.28478i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.781 - 0.623i)T \) |
| 3 | \( 1 + (-0.519 + 1.65i)T \) |
| 7 | \( 1 + (2.57 + 0.604i)T \) |
| good | 5 | \( 1 + (0.177 + 2.36i)T + (-4.94 + 0.745i)T^{2} \) |
| 11 | \( 1 + (1.84 - 0.724i)T + (8.06 - 7.48i)T^{2} \) |
| 13 | \( 1 + (-2.09 + 0.822i)T + (9.52 - 8.84i)T^{2} \) |
| 17 | \( 1 + (-0.646 + 0.199i)T + (14.0 - 9.57i)T^{2} \) |
| 19 | \( 1 + (1.24 + 0.717i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (4.28 + 4.62i)T + (-1.71 + 22.9i)T^{2} \) |
| 29 | \( 1 + (1.73 + 5.61i)T + (-23.9 + 16.3i)T^{2} \) |
| 31 | \( 1 - 0.434iT - 31T^{2} \) |
| 37 | \( 1 + (3.14 + 2.91i)T + (2.76 + 36.8i)T^{2} \) |
| 41 | \( 1 + (-3.09 + 2.11i)T + (14.9 - 38.1i)T^{2} \) |
| 43 | \( 1 + (2.02 + 1.38i)T + (15.7 + 40.0i)T^{2} \) |
| 47 | \( 1 + (3.00 - 3.76i)T + (-10.4 - 45.8i)T^{2} \) |
| 53 | \( 1 + (-3.24 - 3.50i)T + (-3.96 + 52.8i)T^{2} \) |
| 59 | \( 1 + (2.06 - 0.996i)T + (36.7 - 46.1i)T^{2} \) |
| 61 | \( 1 + (3.78 + 0.864i)T + (54.9 + 26.4i)T^{2} \) |
| 67 | \( 1 - 12.0T + 67T^{2} \) |
| 71 | \( 1 + (-2.53 + 0.577i)T + (63.9 - 30.8i)T^{2} \) |
| 73 | \( 1 + (-7.12 - 2.79i)T + (53.5 + 49.6i)T^{2} \) |
| 79 | \( 1 + 1.16T + 79T^{2} \) |
| 83 | \( 1 + (-4.50 + 11.4i)T + (-60.8 - 56.4i)T^{2} \) |
| 89 | \( 1 + (-6.46 + 0.974i)T + (85.0 - 26.2i)T^{2} \) |
| 97 | \( 1 + (4.13 - 2.38i)T + (48.5 - 84.0i)T^{2} \) |
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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
−9.614334603294235955224093167987, −8.711313652501487938384360077454, −8.097647360633203272676747156604, −7.24769386293692419536699799737, −6.31465302543437422573146436471, −5.67304273514547624629661388617, −4.47739816887437539048891997999, −3.42965103078561865410144896500, −2.23027169114775994913880766273, −0.51976765152272994742982460253,
2.29071114306663596202804767633, 3.35143654108696192956131352124, 3.67787812014101000540308103917, 5.08543655901078325299236916310, 5.97160554542898343251361952198, 6.77643369260232882067746573002, 7.976595932836073439177830407107, 9.033477611841916734375145768369, 9.855321441579916558354659995182, 10.45271742650799329669990529978
