I am willing to know the exact formula/methodology riemann adapted/used for calculations of zeros of zeta function along with that which device/instrument he used for help?
His notes indicate that he was aware of the Riemann-Siegel formula which was deciphered by Carl Siegel in 1932. See the book by H. M. Edwards "Riemann's Zeta Function", Chapter 7. It is very clear.
In short, Riemann did compute the first three zeros by means of what's now known as the Riemann-Siegel formula, as it was rediscovered only years later. Why that was sufficient for his conjecture, is questionable, it's probably more a matter of intuition than anything else , you can check this book for more information :https://www.amazon.com/Riemanns-Zeta-Function-Harold-Edwards/dp/0486417409