  The Riemann zeta function $\zeta(s):= \sum_{n=1}^{\infty} 1/n^s$ can be
interpreted as the energy per point of the lattice $\mathbb{Z}$, interacting
pairwisely via the Riesz potential $1/r^s$. Given a parameter $\Delta\in
(0,1]$, this physical model is generalized by considering the energy per point
$E(s,\Delta)$ of a periodic one-dimensional lattice alternating the distances
between the nearest-neighbour particles as $2/(1+\Delta)$ and
$2\Delta/(1+\Delta)$, keeping the lattice density equal to one independently of
$\Delta$. This energy trivially satisfies $E(s,1)=\zeta(s)$ at $\Delta=1$, it
can be easily expressed as a combination of the Riemann and Hurwitz zeta
functions, and extended analytically to the punctured $s$-plane $\mathbb{C}
\setminus \{ 1\}$. In this paper, we perform numerical investigations of the
zeros of the energy $\{ \rho=\rho_x+{\rm i}\rho_y\}$, which are defined by
$E(\rho,\Delta)=0$. The numerical results reveal that in the Riemann limit
$\Delta\to 1^-$ theses zeros include the anticipated critical zeros of the
Riemann zeta function with $\Re(\rho_x)=\frac{1}{2}$ as well as an unexpected
-- comparing to the Riemann Hypothesis -- infinite series of off-critical
zeros. The analytic treatment of these off-critical zeros shows that their
imaginary components are equidistant and their real components diverge
logarithmically to $-\infty$ as $\Delta\to 1^-$, i.e., they become invisible at
the Riemann's $\Delta=1$.
