| L(s) = 1 | + (−0.643 + 1.25i)2-s + (0.831 + 0.555i)3-s + (−1.17 − 1.61i)4-s + (1.10 + 0.218i)5-s + (−1.23 + 0.690i)6-s + (−0.899 + 2.17i)7-s + (2.79 − 0.436i)8-s + (0.382 + 0.923i)9-s + (−0.983 + 1.24i)10-s + (2.85 + 4.27i)11-s + (−0.0754 − 1.99i)12-s + (−2.75 + 0.548i)13-s + (−2.15 − 2.52i)14-s + (0.793 + 0.793i)15-s + (−1.24 + 3.80i)16-s + (2.30 − 2.30i)17-s + ⋯ |
| L(s) = 1 | + (−0.454 + 0.890i)2-s + (0.480 + 0.320i)3-s + (−0.586 − 0.809i)4-s + (0.492 + 0.0979i)5-s + (−0.503 + 0.281i)6-s + (−0.339 + 0.820i)7-s + (0.988 − 0.154i)8-s + (0.127 + 0.307i)9-s + (−0.311 + 0.393i)10-s + (0.860 + 1.28i)11-s + (−0.0217 − 0.576i)12-s + (−0.765 + 0.152i)13-s + (−0.576 − 0.675i)14-s + (0.204 + 0.204i)15-s + (−0.311 + 0.950i)16-s + (0.558 − 0.558i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.209 - 0.977i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.209 - 0.977i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.687268 + 0.850384i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.687268 + 0.850384i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.643 - 1.25i)T \) |
| 3 | \( 1 + (-0.831 - 0.555i)T \) |
| good | 5 | \( 1 + (-1.10 - 0.218i)T + (4.61 + 1.91i)T^{2} \) |
| 7 | \( 1 + (0.899 - 2.17i)T + (-4.94 - 4.94i)T^{2} \) |
| 11 | \( 1 + (-2.85 - 4.27i)T + (-4.20 + 10.1i)T^{2} \) |
| 13 | \( 1 + (2.75 - 0.548i)T + (12.0 - 4.97i)T^{2} \) |
| 17 | \( 1 + (-2.30 + 2.30i)T - 17iT^{2} \) |
| 19 | \( 1 + (0.562 + 2.82i)T + (-17.5 + 7.27i)T^{2} \) |
| 23 | \( 1 + (-0.814 + 0.337i)T + (16.2 - 16.2i)T^{2} \) |
| 29 | \( 1 + (-1.52 + 2.28i)T + (-11.0 - 26.7i)T^{2} \) |
| 31 | \( 1 - 3.24iT - 31T^{2} \) |
| 37 | \( 1 + (-1.56 + 7.86i)T + (-34.1 - 14.1i)T^{2} \) |
| 41 | \( 1 + (-4.45 + 1.84i)T + (28.9 - 28.9i)T^{2} \) |
| 43 | \( 1 + (-9.42 + 6.29i)T + (16.4 - 39.7i)T^{2} \) |
| 47 | \( 1 + (3.29 - 3.29i)T - 47iT^{2} \) |
| 53 | \( 1 + (7.30 + 10.9i)T + (-20.2 + 48.9i)T^{2} \) |
| 59 | \( 1 + (-11.9 - 2.37i)T + (54.5 + 22.5i)T^{2} \) |
| 61 | \( 1 + (2.93 + 1.96i)T + (23.3 + 56.3i)T^{2} \) |
| 67 | \( 1 + (8.01 + 5.35i)T + (25.6 + 61.8i)T^{2} \) |
| 71 | \( 1 + (5.44 - 13.1i)T + (-50.2 - 50.2i)T^{2} \) |
| 73 | \( 1 + (3.71 + 8.97i)T + (-51.6 + 51.6i)T^{2} \) |
| 79 | \( 1 + (4.99 + 4.99i)T + 79iT^{2} \) |
| 83 | \( 1 + (0.535 + 2.69i)T + (-76.6 + 31.7i)T^{2} \) |
| 89 | \( 1 + (-7.33 - 3.03i)T + (62.9 + 62.9i)T^{2} \) |
| 97 | \( 1 - 17.2iT - 97T^{2} \) |
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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
−12.96099347974940812822893782572, −11.94002503080551003944507801459, −10.35897220112752517692725833251, −9.416200482131835228047776155703, −9.150100347908774554471733068673, −7.66668407699270545269878167856, −6.75009279790110062389495374043, −5.53415942876349741887049358363, −4.36945854175997240784042390281, −2.25392295473586152075259840566,
1.26138425545872511861088855095, 3.02423659035405332498361988944, 4.11882776937877159195245806018, 6.01773599195737668339389523343, 7.43795381668370574948071290057, 8.387094029873244302525835444791, 9.448233985771677320531187567262, 10.16202849881444763863539029893, 11.23629682032852516363924146251, 12.26204994372865758817626910695
