The paper is concerned with the spatially homogeneous Landau equation \[ \begin{cases}\partial_t f=Q(f,f),\\ f_{|t=0}=f_0,\end{cases}(1) \] where \(f=f(t,v)\geq 0\) is the density of particles in a plasma with velocity \(v\in\mathbb{R}^3\) at time \(t\geq 0\), and the Landau collision operator \(Q\) is a bilinear operator given by \[ Q(g,f)=\partial_i\displaystyle\int_{\mathbb{R}^3}a_{ij}(v-v_*)[g_*\partial_jf-f\partial_j g_*]\,dv_*, \] with \(g_*=g(v_*)\), \(\partial_j g_*=\partial_{v_{*j}}g(v_*)\), \(f=f(v)\) and \(\partial_j f=\partial_{v_j}f(v)\). The matrix-valued function \(a\) is nonnegative symmetric and depends on the interaction between particles, given by \(a_{ij}(z)=|z|^{\gamma+2}\Pi_{ij}(z)\) with \(\Pi_{ij}(z)=\left(\delta_{ij}-\displaystyle\frac{z_iz_j}{|z|^2}\right)\) and \(\gamma=(s-4)/s\). The author proves two main theorems which give the rate of convergence to equilibrium of solutions for problem \((1)\). The first one is a polynomial convergence theorem for the relative entropy of solutions for \((1)\) if \(\gamma\in (-2,0)\), proved by using an entropy method with some new a priori estimates. The second one is an exponential convergence theorem of the weak solution of \((1)\) if \(\gamma\in (-1,0)\) with the optimal rate, given by the spectral gap of the associated linearized operator, and proved by using some new decay estimates and the polynomial decay above.