Is it possible to expand the logarithm of the zeta function
$$ \log\zeta (s)= a_{0}+a_{1}s^{-1}+a_{2}s^{-2}+.... ,$$
with coefficients $ a_{n} = \frac{1}{2\pi i}\oint dz \frac{\log\zeta(z)}{z^{n+1}} $ ?
My idea is to improve the Gram series based on the solution of the integral equation
$$ \log\zeta (s)=s\int_{0}^{\infty}dt \frac{\pi(e^{t})}{e^{st}-1}. $$
Of course I am almost sure that the coefficients $ a_{n} $ must depend on the nontrivial zeros of Riemann zeta $ \zeta (\rho)=0 $.