Transcendental values of class group L-functions

Let K be an algebraic number field and f a complex-valued function on the ideal class group of K. Then, f extends in a natural way to the set of all non-zero ideals of the ring of integers of K and we can consider the Dirichlet series \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${L(s,f) =\sum_{{\mathfrak a}} f({\mathfrak a}){\bf N}({\mathfrak a})^{-s}}$$\end{document} which converges for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathfrak R}(s) >1 }$$\end{document}. After extending this function to \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathfrak R}(s)=1}$$\end{document}, and in the case that f does not contain the trivial character, we study the special value L(1, f) when f is algebraic valued and K is an imaginary quadratic field. Applying Kronecker’s limit formula and Baker’s theory of linear forms in logarithms, we derive a variety of results related to the transcendence of this special value.