In 2006, Ihara defined and systematically studied a generalisation of the Euler-Mascheroni constant to all global fields, which he called the Euler-Kronecker constants. In a joint work with Amir Akbary (University of Lethbridge) we study the distribution of these constants when the field runs over the geometric quadratic extensions of a given rational global function field. This study is carried out via the values at 1 of the logarithmic derivatives of Dirichlet L-functions of quadratic characters. Our work is in line with previous works of Granville--Soundararajan, Lamzouri, Mourtada--Murty and Akbary--Hamieh over number fields and Lumley over global function fields. More precisely, using a probabilistic model, we show that these values have a limiting distribution as the genii of the quadratic fields go to infinity. This distribution has a smooth and positive density function. We then prove a discrepancy theorem for the convergence of the frequency of these values and obtain information about the proportion of the small values. Finally, we prove omega results on the extreme values. For imaginary quadratic fields, our results can be recast in terms of Taguchi heights of certain Drinfeld modules, a function field analogue of the Faltings height of Abelian varieties over number fields.