Gram lines and the average of the real part of the Riemann zeta function

The contours $\Im \Lambda (s)=0$ of the function which satisfies
$\zeta (1-s)=\Lambda (s)\zeta (s)$ cross the critical strip on lines which are almost horizontal and straight, and which cut the critical line alternately at Gram points and points where $\zeta (s)$ is imaginary. When suitably averaged the real part of $\zeta (s)$ satisfies a relation which greatly extends a theorem of Titchmarsh, (namely that the average of $\zeta (s)$ over the Gram points has the value 2), to the open right-hand half plane $\sigma >0$.