  For $L(\cdot,\pi)$ in a large class of $L$-functions, assuming the
generalized Riemann hypothesis, we show an explicit bound for the function
$S_1(t,\pi)=\frac{1}{\pi}\int_{1/2}^\infty\log|L(\sigma+it,\pi)|\,d\sigma$,
expressed in terms of its analytic conductor. This enables us to give an
alternative proof of the most recent (conditional) bound for
$S(t,\pi)=\frac{1}{\pi} \,arg\,L(\tfrac12+it,\pi)$, which is the derivative of
$S_1(\cdot,\pi)$ at $t$.
