Let $E(1)$ be the first Johnson-Wilson spectrum at the prime two. In this paper, we calculate the homotopy groups of the $E(1)$-localized sphere spectrum with a grading over the Picard group of the stable homotopy category of $E(1)$-local spectra.
"The $E(1)$-local Picard graded homotopy groups of the sphere spectrum at the prime two." Hiroshima Math. J. 50 (2) 137 - 150, July 2020. https://doi.org/10.32917/hmj/1595901623