Abstract: In our current study, we aim to examine the precise regularity estimates for the solution $f = f(t, v)$ of the spatially homogeneous Landau-Coulomb equation. Specifically, we focus on the scenario where the initial data $f_0$ belongs to a weighted $L^{p}$ space with $p \geq 3/2$. Our results are summarized as follows:
(i).  If the initial data $f_0$ possesses only polynomial moments, we establish the existence of a global or local solution. Moreover, within the relevant time span, the solution $f(t)$  exhibits the sharp smoothing estimates in {\it any Sobolev spaces}  but with appropriate {\it negative weights}, which  are similar to those for the linear heat equation:
$\|e^{-t\Delta}f_0\|_{H^n} \leq t^{-\frac{n}{2}-\frac{3}{2p}+\frac{3}{4}}\|f_0\|_{L^p}$. 
Furthermore, we substantiate the necessity of these {\it negative weights}(in  smoothing estimates) by constructing typical initial data.
(ii). We establish the propagation of exponential moments in  $L^2$ space without any loss. Consequently, the solution belongs to the Gevrey class, exhibiting an optimal index that depends on the exponential moment for any positive time.