Critical Values of Symmetric Power L-functions

We consider the critical values of symmetric power L-functions attached to elliptic curves over Q. We show how to calculate a canonical Deligne period, and in several numerical examples, especially for sixth and tenth powers, we examine the factorisation of the rational number apparently obtained when one divides the critical value by the canonical period. This seems to provide some support for the Bloch-Kato conjecture, when we compare it with calculations and bounds for Tamagawa factors and global torsion terms. For large odd powers (5th-9th), we see several examples fitting well with the squareness of the order of the Shafarevich-Tate group.