On the nature of eγ and non-vanishing of derivatives of L-series at s = 1/2

In 2011, M.R. Murty and V.K. Murty [10] proved that if L(s,χD)L(s,χD) is the Dirichlet L  -series attached a quadratic character χDχD, and L′(1,χD)=0L′(1,χD)=0, then eγeγ is transcendental. This paper investigates such phenomena in wider collections of L-functions, with a special emphasis on Artin L  -functions. Instead of s=1s=1, we consider s=1/2s=1/2. More precisely, we prove thatexp⁡(L′(1/2,χ)L(1/2,χ)−αγ) is transcendental with some rational number α  . In particular, if we have L(1/2,χ)≠0L(1/2,χ)≠0 and L′(1/2,χ)=0L′(1/2,χ)=0 for some Artin L  -series, we deduce the transcendence of eγeγ.