Laplacian Dirichlet eigenvalues on a given domain

Question

Let $\Sigma=[-1,1]\times[0,1]\cup[0,1]\times[-1,1]$ be an L-shape domain, over which I'm solving the Laplacian equation with Dirichlet boundary condition $$-\Delta f=\lambda f$$ I try applying the way solving Laplacian equation over a rectangle as following. First, by separation method assuming $f(x,y)=X(x)Y(y)$, the Laplacian equation now deduce to $$\frac{X''}{X}+\frac{Y''}{Y}=\lambda$$ And to fit in the boundary condition, we choose $X=\sin{n\pi} x, Y=\sin{m\pi}y$. The corresponding $\lambda$ is now $\lambda=(n^2+m^2)\pi^2$. For a general solution we write, $$ f=\sum_{m+n=\lambda}C_{m,n}\sin{{n\pi} x}\sin{{m\pi}y}$$ Here my problem is to justify the method. Is it true that such $\{\sin{{n\pi} x}\sin{{m\pi}y}\}_{m,n\geq 0}$ can serve as a basis of solutions of the Laplacian equation? More generally if true could we find a basis for those $C^2$ function over the domain containing $\{\sin{{n\pi} x}\sin{{m\pi}y}\}_{m,n\geq 0}$? If false, is there any other method to calculate the eigenvalues?