L-values of elliptic curves

introduction

Computing $L(E;1)$ is easy: it is either 0 or the period of elliptic curve $E$
Computing $L(E;k)$ for $k\geq 2$ is highly non-trivial. The already mentioned results of Beilinson generalised later by Denninger-Scholl show that any such L-value can be expressed as a period.
Several examples are explicitly given for $k=2$, mainly motivated by showing particular cases of Beilinson's conjectures in K-theory and Boyd's (conjectural) evaluations of Mahler measures.
In spite of the algorithmic nature of Beilinson's method and in view of its complexity, no examples were produced so far for a single $L(E;3)$.
Rogers and Zudilin in 2010-11 created an elementary alternative to Beilinson-Denninger-Scholl to prove some conjectural Mahler evaluations.

[Z2013] Zudilin, Wadim. 2013. “Period(d)ness of L-Values.” In Number Theory and Related Fields, edited by Jonathan M. Borwein, Igor Shparlinski, and Wadim Zudilin, 381–395. Springer Proceedings in Mathematics & Statistics 43. Springer New York. http://link.springer.com/chapter/10.1007/978-1-4614-6642-0_20.