||We study the value distribution of the Riemann zeta function near the line Re s = 1/2. We find an asymptotic formula for the number of a-values in the rectangle 1/2 + h(1) /(log T)(theta) <= Re s <= 1/2 + h(2)/(log, T)(theta) < T <= Im s <= 2T for fixed h(1), h(2) > 0 and 0 < theta < 1/13. To prove it, we need an extension of the valid range of Lamzouri, Lester and Radziwill's recent results on the discrepancy between the distribution of zeta(s) and its random model. We also propose the secondary main term for the Selberg's central limit theorem by providing sharper estimates on the line Re s = 1/2 + 1/(log T)(theta). (C) 2018 Elsevier Inc. All rights reserved.