The second secant zeta function and its anti-periodization evaluated at $1/\sqrt{n}$

The 2-periodic secant zeta function $\psi(r)$ and its $1$-antiperiodization $f(r)=(\psi(r)-\psi(r+1))/2$, arising from a fluid dynamics application, are investigated. In particular, their values at $1/\sqrt{n}$ for positive integers $n$ are determined, using a reformulation of an identity satisfied by $\psi$ as an Abel equation.