The reference pointed out by GH gives an excellent general write-up. For those interested in completely explicit bounds in the classical case of $GL(2)$ over $\mathbb{Q}$, I've worked these out in a few cases, though none in complete generality. For example if $f$ is a newform in $S_{2}(\Gamma_{0}(D), \chi_{D})$
that does not have complex multiplication and $D$ is a fundamental discriminant, then
Proposition 11 of [this paper][1] shows that
$$
  \frac{3}{\pi [{\rm SL}_{2}(\mathbb{Z}) : \Gamma_{0}(D)]} \iint_{\mathbb{H} / \Gamma_{0}(D)} |f(x+iy)|^{2} y^{2} \, \frac{dx \, dy}{y^{2}} > \frac{3}{208 \pi^{4} \log(D)} \prod_{p | D} \left(\frac{p}{p+1}\right). 
$$
The appendix to Hoffstein and Lockhart's paper explains how one gets a zero-free regions for $L(f \otimes \overline{f}, s)$ near $s = 1$, and a good reference for the classical problem of turning a zero-free region into a lower bound is Hoffstein's paper "On the Siegel-Tatuzawa theorem" (in Acta Arithmetica, 1980/1981).

  [1]: http://front.math.ucdavis.edu/1111.0979