In the present paper, we show that, for every δ > 0, the function (log L(s)) (m) , where m ∈ N ∪ {0} and L(s) := ∞ n=1 a(n)n −s is an element of the Selberg class S, takes any value infinitely often in the strip 1 < Re(s) < 1 + δ, provided p≤x |a(p)| 2 ∼ κπ(x) for some κ > 0. In particular, L(s) takes any non-zero value infinitely often in the strip 1 < Re(s) < 1 + δ, and the first derivative of L(s) has infinitely many zeros in the half-plane Re(s) > 1.2010 Mathematics Subject Classification. Primary 11M06, 11M26.