For some random Dirichlet series of order(R) infinite almost surely, every horizontal line is a strong Borel line of order(R) infinite and without exceptional Little functions.
JIN, QIYU and SUN, DAOCHUN (2008) "On the Distribution of Random Dirichlet Series in the Whole Plane," Turkish Journal of Mathematics: Vol. 32: No. 3, Article 1. Available at: https://journals.tubitak.gov.tr/math/vol32/iss3/1