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We present the first applications of the recently established by us(arXiv:1304.7895; Ukrainian Math. J. - 2014. -66. - P. 371-383) generalizedLi's criterion equivalent to the Riemann Hypothesis. This criterion is thestatement that the Riemann hypothesis is equivalent to the non-negativity ofthe derivatives 1/((m-1)!)*d^m/dz^m((z+b)^(m-1)*ln(\xi(z))) for z=b+1 of theRiemann xi-function for all real b>-1/2 and all m = 1, 2, 3... We show that forany positive integer n there is such value of b_n (depending on n) that for allm<=n and b>b_n, inequality 1/((m-1)!)*d^m/dz^m((z+b)^(m-1)*ln(\xi(z))) forz=b+1 >=0 does hold true. Assuming RH, we also have found an asymptotic of thegeneralized Li's sums over non-trivial Rieman zeroes for large n, and discusswhat asymptotic of 1/((m-1)!)*d^m/dz^m((z+b)^(m-1)*ln((z-1)*(\zeta(z))) at thepoint z+b+1 is required for the Riemann hypothesis holds true.

First applications of generalized Li's criterion to study the Riemann zeta-function zeroes location