| L(s) = 1 | + (−0.640 + 1.26i)2-s + (0.585 − 1.63i)3-s + (−1.17 − 1.61i)4-s + (−3.57 + 2.39i)5-s + (1.67 + 1.78i)6-s + (−0.994 + 2.40i)7-s + (2.79 − 0.450i)8-s + (−2.31 − 1.90i)9-s + (−0.721 − 6.04i)10-s + (−0.714 + 3.59i)11-s + (−3.32 + 0.975i)12-s + (−1.38 + 2.08i)13-s + (−2.38 − 2.79i)14-s + (1.80 + 7.23i)15-s + (−1.22 + 3.80i)16-s + (−0.951 − 0.951i)17-s + ⋯ |
| L(s) = 1 | + (−0.453 + 0.891i)2-s + (0.338 − 0.941i)3-s + (−0.589 − 0.807i)4-s + (−1.60 + 1.06i)5-s + (0.685 + 0.727i)6-s + (−0.375 + 0.907i)7-s + (0.987 − 0.159i)8-s + (−0.771 − 0.636i)9-s + (−0.228 − 1.91i)10-s + (−0.215 + 1.08i)11-s + (−0.959 + 0.281i)12-s + (−0.385 + 0.576i)13-s + (−0.638 − 0.746i)14-s + (0.465 + 1.86i)15-s + (−0.305 + 0.952i)16-s + (−0.230 − 0.230i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.949 - 0.313i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.949 - 0.313i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0617579 + 0.383459i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0617579 + 0.383459i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.640 - 1.26i)T \) |
| 3 | \( 1 + (-0.585 + 1.63i)T \) |
| good | 5 | \( 1 + (3.57 - 2.39i)T + (1.91 - 4.61i)T^{2} \) |
| 7 | \( 1 + (0.994 - 2.40i)T + (-4.94 - 4.94i)T^{2} \) |
| 11 | \( 1 + (0.714 - 3.59i)T + (-10.1 - 4.20i)T^{2} \) |
| 13 | \( 1 + (1.38 - 2.08i)T + (-4.97 - 12.0i)T^{2} \) |
| 17 | \( 1 + (0.951 + 0.951i)T + 17iT^{2} \) |
| 19 | \( 1 + (0.214 - 0.321i)T + (-7.27 - 17.5i)T^{2} \) |
| 23 | \( 1 + (1.14 + 2.75i)T + (-16.2 + 16.2i)T^{2} \) |
| 29 | \( 1 + (2.53 - 0.504i)T + (26.7 - 11.0i)T^{2} \) |
| 31 | \( 1 - 2.03T + 31T^{2} \) |
| 37 | \( 1 + (-4.28 + 2.86i)T + (14.1 - 34.1i)T^{2} \) |
| 41 | \( 1 + (-4.29 + 1.77i)T + (28.9 - 28.9i)T^{2} \) |
| 43 | \( 1 + (1.88 - 9.45i)T + (-39.7 - 16.4i)T^{2} \) |
| 47 | \( 1 + (5.57 - 5.57i)T - 47iT^{2} \) |
| 53 | \( 1 + (-1.47 - 0.292i)T + (48.9 + 20.2i)T^{2} \) |
| 59 | \( 1 + (-3.66 - 5.48i)T + (-22.5 + 54.5i)T^{2} \) |
| 61 | \( 1 + (4.09 - 0.814i)T + (56.3 - 23.3i)T^{2} \) |
| 67 | \( 1 + (-1.50 - 7.58i)T + (-61.8 + 25.6i)T^{2} \) |
| 71 | \( 1 + (-6.27 - 2.59i)T + (50.2 + 50.2i)T^{2} \) |
| 73 | \( 1 + (15.4 - 6.40i)T + (51.6 - 51.6i)T^{2} \) |
| 79 | \( 1 + (-2.31 + 2.31i)T - 79iT^{2} \) |
| 83 | \( 1 + (1.62 + 1.08i)T + (31.7 + 76.6i)T^{2} \) |
| 89 | \( 1 + (-4.04 - 1.67i)T + (62.9 + 62.9i)T^{2} \) |
| 97 | \( 1 - 1.86iT - 97T^{2} \) |
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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
−12.90478707033303574925409953296, −12.05657999892324728372592320687, −11.14587387863118394285377214927, −9.739486696110624632273529002969, −8.617597177088549375321510803706, −7.66426349335698334913425049764, −7.07996728185591750342389933797, −6.18822883972454093339323210842, −4.35229637701160419332270901830, −2.59630951059179241156286785289,
0.37737634967382946800322196181, 3.33462214824595646293917607847, 3.99764782150225510625634983904, 5.05977370205852965602352801587, 7.62310137450850083835270059365, 8.267061074774412296878702548403, 9.100842149453089700792652914464, 10.22040007412204542356836733358, 11.10459543846464210996695478097, 11.80207413569960835170211436916
