  Assuming the Generalized Riemann Hypothesis, we provide explicit upper bounds
for moduli of $\log{\mathcal{L}(s)}$ and $\mathcal{L}'(s)/\mathcal{L}(s)$ in
the neighbourhood of the 1-line when $\mathcal{L}(s)$ are the Riemann,
Dirichlet and Dedekind zeta-functions. To do this, we generalize Littlewood's
well known conditional result to functions in the Selberg class with a
polynomial Euler product, for which we also establish a suitable convexity
estimate. As an application we provide conditional and effective estimate for
the Mertens function.
