Suppose we have a GL(1) Hecke character \psi over an imaginary quadratic field, corresponding to a weight 2 modular form (of some level) over the rationals. Then the powers \psi^k will be related to forms of weight k+1. In particular, the symmetric power L-functions of an elliptic curve with complex multiplication will be imprimitive, and will factor as
L({\rm Sym}^k E,s)=\prod_{i=0}^{m/2} L(\psi^{m-2i},s-m/2).
Here we have used the normalisation for which the symmetry law for the completed L-function relates s\rightarrow k+1-s. Also, one needs to be careful about defining \psi^k at bad primes. This preprint paper describes some calculations in the general case of symmetric powers of L-functions of elliptic curves (over the rationals). In particular, we have that
{L({\rm Sym}^{4k+2} E,2k+2)\over (2\pi)^{2k+2}}\biggl({2\pi N\over \Omega_{\rm re}\Omega_{\rm im}}\biggr)^{{2k+2\choose 2}}
should be rational with small denominator. The case of the second symmetric power concerns the modular degree of E. There are no special values here for mth symmetric powers when 4|m. For odd powers, we get that {L({\rm Sym}^{4k+3} E,2k+2)(2\pi N)^{{2k+2\choose 2}}\over \Omega_{\rm re}^{2k+2\choose 2}\Omega_{\rm im}^{{2k+3\choose 2}}}
and {L({\rm Sym}^{4k+1} E,2k+1)(2\pi N)^{{2k+1\choose 2}}\over \Omega_{\rm re}^{2k+2\choose 2}\Omega_{\rm im}^{{2k+1\choose 2}}}
should be rational with small denominator. These are borne out in the data, and fit the conjectures of Deligne (see section 7 here, though you need a suitable graphical browser to get the Access Key) and Bloch-Kato and Beilinson [link?]