  We study location of eigenvalues of one-dimensional discrete Schr\"odinger
operators with complex $\ell^{p}$-potentials for $1\leq p\leq \infty$. In the
case of $\ell^{1}$-potentials, the derived bound is shown to be optimal. For
$p>1$, two different spectral bounds are obtained. The method relies on the
Birman-Schwinger principle and various techniques for estimations of the norm
of the Birman-Schwinger operator.
