Tikhonov regularization in its most general form is the solution of the problem
$\min_{\mathbf x}\|\mathbf A\mathbf x-\mathbf b\|^2+\alpha^2\|\mathbf L\mathbf x\|^2$
or the problem
$\min_{\mathbf x}\left\|\begin{pmatrix}\mathbf A\\ \alpha \mathbf L\end{pmatrix}\mathbf x-\begin{pmatrix}\mathbf b\\ \mathbf 0\end{pmatrix}\right\|$
Lars Elden gave a method for converting this general Tikhonov problem into an equivalent "standard form" regularization problem. I assume in the sequel that $\mathbf L$ is invertible; for singular $\mathbf L$ (and in fact for rectangular $\mathbf L$ as well), refer to Elden's paper for the required transformations (which involves the use of the QR decomposition).
In particular, if we let $\mathbf y=\mathbf L\mathbf x$, one can see that an equivalent standard Tikhonov problem is
$\min_{\mathbf x}\|\mathbf A\mathbf L^{-1}\mathbf y-\mathbf b\|^2+\alpha^2\|\mathbf y\|^2$
from which you can use the usual formulae for Tikhonov regularization to solve for $\mathbf y$. From this, one obtains the solution to the general problem as $\mathbf x=\mathbf L^{-1}\mathbf y$.