Period(d)ness of L-values

In our recent work with Rogers on resolving some of Boyd’s conjectures on two-variate Mahler measures, a new analytical machinery was introduced to write the values L(E, 2) of L-series of elliptic curves as periods in the sense of Kontsevich and Zagier. Here we outline, in slightly more general settings, the novelty of our method with Rogers and provide two illustrative period evaluations of L(E, 2) and L(E, 3) for a conductor 32 elliptic curve E.