  We investigate the behavior near zero of the integrated density of states
$\ell$ for random Schr\"{o}dinger operators $\Phi(-\Delta) + V^{\omega}$ in
$L^2(\mathbb R^d)$, $d \geq 1$, where $\Phi$ is a complete Bernstein function
such that for some $\alpha \in (0,2]$, one has $ \Phi(\lambda) \asymp
\lambda^{\alpha/2}$, $\lambda \searrow 0$, and $V^{\omega}(x) = \sum_{
\mathbf{i}\in \mathbb{Z}^d} q_{\mathbf{i}}(\omega) W(x-\mathbf{i})$ is a random
nonnegative alloy-type potential with compactly supported single site potential
$W$. We prove that there are constants $C, \widetilde C,D, \widetilde D>0$ such
that
  $$
  -C \leq\liminf_{\lambda \searrow 0} \frac{\lambda^{d/\alpha}}{|\log F_q(D
\lambda)|}{\log \ell(\lambda)}
  \qquad \text{and} \qquad \limsup_{\lambda \searrow 0}
\frac{\lambda^{d/\alpha}}{|\log F_q(\widetilde D \lambda)|}\log \ell(\lambda)
\leq -\widetilde C,
  $$ where $F_q$ is the common cumulative distribution function of the lattice
random variables $q_{\mathbf i}$. In particular, we identify how the behavior
of $\ell$ at zero depends on the lattice configuration. For typical examples of
$F_q$ the constants $D$ and $\widetilde D$ can be eliminated from the statement
above. We combine probabilistic and analytic methods which allow to treat, in a
unified manner, both local and non-local kinetic terms such as the Laplace
operator, its fractional powers and the quasi-relativistic Hamiltonians.
