  In article, we explore the secondary zeta function $Z(s)$, which is defined
as a generalized zeta type of series over imaginary parts of non-trivial zeros
of the Riemann zeta function $\zeta(s)$. This function has been analytically
continued as a meromorphic function in $\mathbb{C}$ with one double pole and an
infinity of simple poles. The secondary zeta function is of interest because it
can naturally represent an analytical formula for non-trivial zeros of the
Riemann zeta function that we will explore, and we show that the non-trivial
zeros can be generated directly from primes by introducing a new form of an
explicit formula written in terms of the prime zeta function. Additionally, we
will also give several new series expansions for $Z(s)$ and numerically compute
these coefficients to high precision, and also develop several new methods to
analytically extend $Z(s)$ to larger domains and develop algorithms to compute
them.
