| L(s) = 1 | + (−3.55 + 0.626i)2-s + (0.824 − 2.88i)3-s + (8.48 − 3.09i)4-s + (−5.23 − 6.23i)5-s + (−1.12 + 10.7i)6-s + (−4.71 − 8.16i)7-s + (−15.7 + 9.08i)8-s + (−7.64 − 4.75i)9-s + (22.5 + 18.8i)10-s − 6.50i·11-s + (−1.91 − 27.0i)12-s + (15.4 + 12.9i)13-s + (21.8 + 26.0i)14-s + (−22.2 + 9.94i)15-s + (22.5 − 18.9i)16-s + (6.48 + 7.73i)17-s + ⋯ |
| L(s) = 1 | + (−1.77 + 0.313i)2-s + (0.274 − 0.961i)3-s + (2.12 − 0.772i)4-s + (−1.04 − 1.24i)5-s + (−0.187 + 1.79i)6-s + (−0.673 − 1.16i)7-s + (−1.96 + 1.13i)8-s + (−0.848 − 0.528i)9-s + (2.25 + 1.88i)10-s − 0.591i·11-s + (−0.159 − 2.25i)12-s + (1.18 + 0.995i)13-s + (1.56 + 1.86i)14-s + (−1.48 + 0.663i)15-s + (1.41 − 1.18i)16-s + (0.381 + 0.454i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 171 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.897 - 0.440i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 171 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.897 - 0.440i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.0723265 + 0.311403i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0723265 + 0.311403i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-0.824 + 2.88i)T \) |
| 19 | \( 1 + (2.18 + 18.8i)T \) |
| good | 2 | \( 1 + (3.55 - 0.626i)T + (3.75 - 1.36i)T^{2} \) |
| 5 | \( 1 + (5.23 + 6.23i)T + (-4.34 + 24.6i)T^{2} \) |
| 7 | \( 1 + (4.71 + 8.16i)T + (-24.5 + 42.4i)T^{2} \) |
| 11 | \( 1 + 6.50iT - 121T^{2} \) |
| 13 | \( 1 + (-15.4 - 12.9i)T + (29.3 + 166. i)T^{2} \) |
| 17 | \( 1 + (-6.48 - 7.73i)T + (-50.1 + 284. i)T^{2} \) |
| 23 | \( 1 + (-8.51 - 23.4i)T + (-405. + 340. i)T^{2} \) |
| 29 | \( 1 + (10.7 + 29.6i)T + (-644. + 540. i)T^{2} \) |
| 31 | \( 1 + 39.4T + 961T^{2} \) |
| 37 | \( 1 - 37.5T + 1.36e3T^{2} \) |
| 41 | \( 1 + (-16.1 + 2.84i)T + (1.57e3 - 574. i)T^{2} \) |
| 43 | \( 1 + (17.7 + 6.45i)T + (1.41e3 + 1.18e3i)T^{2} \) |
| 47 | \( 1 + (-4.33 - 11.9i)T + (-1.69e3 + 1.41e3i)T^{2} \) |
| 53 | \( 1 + (-23.5 - 4.14i)T + (2.63e3 + 960. i)T^{2} \) |
| 59 | \( 1 + (-4.48 + 12.3i)T + (-2.66e3 - 2.23e3i)T^{2} \) |
| 61 | \( 1 + (55.8 + 46.8i)T + (646. + 3.66e3i)T^{2} \) |
| 67 | \( 1 + (-7.07 + 40.1i)T + (-4.21e3 - 1.53e3i)T^{2} \) |
| 71 | \( 1 + (8.02 - 1.41i)T + (4.73e3 - 1.72e3i)T^{2} \) |
| 73 | \( 1 + (28.7 + 10.4i)T + (4.08e3 + 3.42e3i)T^{2} \) |
| 79 | \( 1 + (28.4 - 23.8i)T + (1.08e3 - 6.14e3i)T^{2} \) |
| 83 | \( 1 + (-11.0 + 6.37i)T + (3.44e3 - 5.96e3i)T^{2} \) |
| 89 | \( 1 + (-8.83 - 24.2i)T + (-6.06e3 + 5.09e3i)T^{2} \) |
| 97 | \( 1 + (12.8 + 72.8i)T + (-8.84e3 + 3.21e3i)T^{2} \) |
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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
−11.55765163336818072698190631699, −11.04719720687877550201198053810, −9.397705812024971903309907579767, −8.794391328966546763242128526468, −7.894586953317530761592693931979, −7.21497004726489042711184494049, −6.14009185926670504465227887477, −3.75053387507383075379214096069, −1.32412308182198286880280629857, −0.36234799581997023605372326399,
2.68130026518736980005132935909, 3.49721486357779514081926814163, 5.99048794528609824832601275248, 7.33345827462348306749056993216, 8.289997324109650574769332448726, 9.082669472660763538211372583790, 10.14210002079733729180355364318, 10.75918992505545831915773371499, 11.52587732922939311021512699225, 12.55546291183624654853424466822
