| 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
−12.26204994372865758817626910695, −11.23629682032852516363924146251, −10.16202849881444763863539029893, −9.448233985771677320531187567262, −8.387094029873244302525835444791, −7.43795381668370574948071290057, −6.01773599195737668339389523343, −4.11882776937877159195245806018, −3.02423659035405332498361988944, −1.26138425545872511861088855095,
2.25392295473586152075259840566, 4.36945854175997240784042390281, 5.53415942876349741887049358363, 6.75009279790110062389495374043, 7.66668407699270545269878167856, 9.150100347908774554471733068673, 9.416200482131835228047776155703, 10.35897220112752517692725833251, 11.94002503080551003944507801459, 12.96099347974940812822893782572
