I see this was posted awhile ago, but I would like to offer a counterpoint to Marc's answer: namely, yes there is a relation, at least in certain situations if you twist by quadratic characters, but it is not as clear as you were perhaps hoping. Here is a fundamental case.

Let $G$ be GL(2) and $\chi_1$ and $\chi_2$ be quadratic characters attached to quadratic extensions $E_1$ and $E_2$ of $F$. (You can view $\chi_i$ as a character of $G$ by composing with the determinant.) Then a famous result of Waldspurger (1985, "Sur les valeurs...") gives a formula for $L(1/2,\pi)L(1/2,\pi \otimes \chi_i)$ in terms of the absolute square of a period integral (over a compact torus associated to $E_i$). As a consequence (and one of Waldspurger's motivations) of this, Waldspurger shows that the ratio

$$L(1/2,\pi \otimes \chi_1) / L(1/2,\pi \otimes \chi_2)$$

is the square of an algebraic number (in a suitable field of rationality). If $\pi$ comes from a classical modular form $f$, this ratio is essentially a ratio of Fourier coefficients of the associated half-integral weight form (these ratios of Fourier coefficients were originally studied by Vigneras, and they are related to the $L$-values by a different formula of Waldspurger).

Understanding how central $L$-values vary under twists has various applications, such as the construction of $p$-adic $L$-functions. There are various conjectural generalizations of Waldspurger's period formula to other groups (e.g., Boecherer for GSp(4) and Ichino-Ikeda for SO(2n+1), and (Gan)-Gross-Prasad for less refined nonvanishing conjectures for various classical groups) but not too much is known yet. These conjectures should also give rationality results for ratios of different twists of central values.