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Given any non-polynomial G-function F(z)=∑∞k=0Akzk of radius of convergence R and in the kernel a G-operator LF, we consider the G-functions F[s]n(z)=∑∞k=0Ak(k+n)szk for every integers s≥0 and n≥1. These functions can be analytically continued to a domain DF star-shaped at 0 and containing the disk {|z|0 and vF>0. This appears to be the first Diophantine result for values of G-functions evaluated outside their disk of convergence. This theorem encompasses a previous result of the authors in [{\em Linear independence of values of G-functions}, 46 pages, J. Europ. Math. Soc., to appear], where α∈Q¯¯¯¯∗ was assumed to be such that |α|