The argument of the Riemann $\mathrm{\Xi }$-function off the critical line

We examine the behaviour of the zeros of the real and imaginary parts of $\xi (s)$ on the vertical line $\mathrm{\Re }s=1\mathrm{/}2+\lambda$, for $\lambda \mathrm{\ne }0$. This can be rephrased in terms of studying the zeros of families of entire functions $A(s)=1\mathrm{/}2(\xi (s+\lambda )+\xi (s-\lambda ))$ and $B(s) = \frac{1}{2i} (\xi(s+\lambda) - \xi(s - \lambda))$. We will prove some unconditional analogues of results appearing in \cite{Lag}, specifically that the normalized spacings of the zeros of these functions converges to a limiting distribution consisting of equal spacings of length 1, in contrast to the expected GUE distribution for the same zeros at $\lambda =0$. We will also show that, outside of a small exceptional set, the zeros of $\mathrm{\Re }\xi (s)$ and $\mathrm{\Im }\xi (s)$ interlace on $\mathrm{\Re }s=1\mathrm{/}2+\lambda$. These results will depend on showing that away from the critical line, $\arg \xi (s)$ is well behaved.