  This paper studies the connections between the zeros and their distribution
functions for two particular Dirichlet $L$ functions: the Riemann zeta
function, and the Catalan beta function, also known as the Dirichlet beta
function. It is shown that the Riemann hypothesis holds for the Dirichlet beta
function $L_{-4}(s)$ if and only if it holds for $\zeta (s)$- a particular case
of the Generalized Riemann Hypothesis.
