  This is an investigation into the possible existence and consequences of a
Birch-Swinnerton-Dyer-type formula for L-functions of elliptic curves twisted
by Artin representations. We translate expected properties of L-functions into
purely arithmetic predictions for elliptic curves, and show that these force
some peculiar properties of the Tate-Shafarevich group, which do not appear to
be tractable by traditional Selmer group techniques. In particular we exhibit
settings where the different p-primary components of the Tate-Shafarevich group
do not behave independently of one another. We also give examples of
"arithmetically identical" settings for elliptic curves twisted by Artin
representations, where the associated L-values can nonetheless differ, in
contrast to the classical Birch-Swinnerton-Dyer conjecture.
