  We generalize our recent construction of the zeros of the Riemann
$\zeta$-function to two infinite classes of $L$-functions, Dirichlet
$L$-functions and those based on level one modular forms. More specifically, we
show that there are an infinite number of zeros on the critical line which are
in one-to-one correspondence with the zeros of the cosine function, and thus
enumerated by an integer $n$. We obtain an exact equation on the critical line
that determines the $n$-th zero of these $L$-functions. We show that the
counting formula on the critical line derived from such an equation agrees with
the known counting formula on the entire critical strip. We provide numerical
evidence supporting our statements, by computing numerical solutions of this
equation, yielding $L$-zeros to high accuracy. We study in detail the
$L$-function for the modular form based on the Ramanujan $\tau$-function, which
is closely related to the bosonic string partition function. The same analysis
for a more general class of $L$-functions is also considered.
