Crucial steps for robust first quiescent region truncated partial Euler Product based approximations of closely spaced Riemann Zeta function non-trivial zeroes.

First quiescent region $N_1=\sqrt{\frac{t}{2\pi}}$ truncated partial Euler Product based approximations of the Riemann Siegel Z function can be made robust by the following steps (i) spectral filtering and mirror reflection of the partial Euler Product fourier transform, (ii) setting the imaginary parts of fft[1] and fft[$\lceil\frac{n}{2}\rceil+1$] equal to zero, (iii) using the primes over $N_1 +\Delta$ rather than $N_1$ and (iv) averaging the results of inverse fourier transforms of the partial Euler product based Riemann Siegel Z function (of different spectrum widths). These steps reduce the impact of spectral leakage from high frequency components present in the partial Euler Product (but not contained in the Riemann Zeta function) resulting in agreement with the Riemann Zeta function to 6+ decimal places which is required for inspecting closely spaced non-trivial zeroes.