  We examine the behaviour of the zeros of the real and imaginary parts of
$\xi(s)$ on the vertical line $\Re s = 1/2+\lambda$, for $\lambda \neq 0$. This
can be rephrased in terms of studying the zeros of families of entire functions
$A(s) = {1/2} (\xi(s+\lambda) + \xi(s - \lambda))$ 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 $\Re \xi(s)$ and $\Im \xi(s)$
interlace on $\Re s = 1/2+\lambda$. These results will depend on showing that
away from the critical line, $\arg \xi(s)$ is well behaved.

Number Theory