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The lacunary series are the most classical examples among all the powerseries whose circle of convergence constitutes a natural boundary(\cite[\S~93-94, p.~372-383]{Di}, \cite[\S 7.43, p.~223]{Ti}, ...). In thisNote, we study a family of non-lacunary power series whose coefficients aregiven by means of values of the Gamma function over vertical line. We explainhow to transform these series into lacunary Dirichlet series, which allows usto conclude the existence of their natural boundary. Our results, whichillustrate in what manner the Gamma function may have a unpredictable behaviouron any vertical line, may also be partially understood in the framwork of ourforthcoming work on a class of differential $q$-difference equations, namely,on pantagraph type equations (see \cite{KM} for instance).

La s\'erie enti\`ere $1+\frac z{\Gamma(1+i)}+\frac{z^2}{\Gamma(1+2i)}+\frac{z^3}{\Gamma(1+3i)}+...$ poss\`ede une fronti\`ere naturelle~!