Abstract: 

 Euler's beautiful formula on the values of the Riemann zeta function at the positive even integers can be seen as the starting point of the investigation of special values of L-functions. In particular, Euler's result shows that all critical zeta values are rational up to multiplication with a particular period, here the period is a power of 2$\pi$i. Conjecturally this is expected to hold for all critical L-values of motives. In this talk, I will explain a joint result with Guido Kings on the algebraicity of critical Hecke L-values up to explicit periods for totally imaginary fields. As an application, I will discuss the construction of p-adic L-functions for such fields at ordinary primes.