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

9 pages