Let $G$ be a reductive algebraic group over a number field $k$. Weil's conjecture on Tamagawa numbers (now a theorem) tells us that the Tamagawa number $\tau(G)$ of $G$ is 1 if $G$ is semisimple and simply-connected. Are there known bounds for $\tau(G)$ for the general case?

Presumably one might be able to use the formula $\tau(G) = |\text{Pic}(G)|/|\text{Sha}(G)|$ and try to estimate the numerator, but I am not familiar with the literature in this respect.