Effective method of computing Li's coefficients and their properties

In this paper we present an effective method for computing certain real coefficients ${\lambda }_n$ which appear in a criterion for the Riemann hypothesis proved by Xian-Jin Li. With the use of this method a sequence of over three-thousand ${\lambda }_n$'s has been calculated. This sequence reveals a peculiar and unexpected behavior: it can be split into a strictly growing $\mathit{\text{trend}}$ and some tiny $\mathit{\text{oscillations}}$ superimposed on this trend.