  We consider the non-trivial zeros of the Riemann $\zeta$-function and two
classes of $L$-functions; Dirichlet $L$-functions and those based on level one
modular forms. We show that there are an infinite number of zeros on the
critical line in one-to-one correspondence with the zeros of the cosine
function, and thus enumerated by an integer $n$. From this it follows that the
ordinate of the $n$-th zero satisfies a transcendental equation that depends
only on $n$. Under weak assumptions, we show that the number of solutions of
this equation already saturates the counting formula on the entire critical
strip. We compute numerical solutions of these transcendental equations and
also its asymptotic limit of large ordinate. The starting point is an explicit
formula, yielding an approximate solution for the ordinates of the zeros in
terms of the Lambert $W$-function. Our approach is a novel and simple method,
that takes into account $\arg L$, to numerically compute non-trivial zeros of
$L$-functions. The method is surprisingly accurate, fast and easy to implement.
Employing these numerical solutions, in particular for the $\zeta$-function, we
verify that the leading order asymptotic expansion is accurate enough to
numerically support Montgomery's and Odlyzko's pair correlation conjectures,
and also to reconstruct the prime number counting function. Furthermore, the
numerical solutions of the exact transcendental equation can determine the
ordinates of the zeros to any desired accuracy. We also study in detail
Dirichlet $L$-functions and the $L$-function for the modular form based on the
Ramanujan $\tau$-function, which is closely related to the bosonic string
partition function.
