$$\require{cancel}$$
$\textbf{L}(t^n y) = (-1)^n \frac{d^2\bar{y}}{ds^n}.$
Before proceeding further, I strongly recommend that you now apply theorems 14.3.1, 14.4.1, 14.5.1 and 14.6.1 to the several entries in your existing table of Laplace transforms and greatly expand your table of Laplace transforms. For example, you can already add $$(\sin at)/t, \ te^{-at}$$ and $$t^2e^{-t}$$ to the list of functions for which you have calculated the Laplace transforms.