  We solve a long standing problem in the theory of Siegel's $E$-functions,
initiated by Lang for Bessel's function $J_0$ in the 60's and considered in
full generality by G. Chudnovsky in the 80's: we prove that irrational values
taken at rational points by $E$-functions with rational Taylor coefficients
have irrationality exponent equal to 2. This result had been obtained before by
Zudilin under stronger assumptions on algebraic independence of $E$-functions,
satisfied by $J_0$ but not by all hypergeometric $E$-functions for instance.
