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The Riemann zeta function $\zeta(s):= \sum_{n=1}^{\infty} 1/n^s$ can beinterpreted as the energy per point of the lattice $\mathbb{Z}$, interactingpairwisely 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 distancesbetween 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 $\Delta=1$, itcan be easily expressed as a combination of the Riemann and Hurwitz zetafunctions, and extended analytically to the punctured $s$-plane $\mathbb{C}\setminus \{ 1\}$. In this paper, we perform numerical investigations of thezeros 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 theRiemann zeta function with $\Re(\rho_x)=\frac{1}{2}$ as well as an unexpected-- comparing to the Riemann Hypothesis -- infinite series of off-criticalzeros. The analytic treatment of these off-critical zeros shows that theirimaginary components are equidistant and their real components divergelogarithmically to $-\infty$ as $\Delta\to 1^-$, i.e., they become invisible atthe Riemann's $\Delta=1$.

On off-critical zeros of lattice energies in the neighborhood of the Riemann zeta function