Let E be an elliptic curve of conductor N over the rationals. The L-function satisfies \Lambda(E,s)=\pm\Lambda(E,2-s) for
\Lambda(E,s)=({\sqrt N\over 2\pi})^s\Gamma(s)L(E,s).
Given the L-function L(E,s) as an Euler product of degree 2 as \prod_p (1-\alpha_p/p^s)^{-1}(1-\beta_p/p^s)^{-1}, where for primes of good reduction we have \beta_p=\bar\alpha_p and |\alpha_p|=\sqrt p, the kth symmetric power L-functions is defined as
L({\rm Sym}^k E,s)=\prod_{p\>{\rm good}}\prod_{j=0}^k (1-\alpha_p^j/p^s)^{-1}(1-\beta_p^{k-j}/p^s)^{-1}\cdot\prod_{p|N} E_p(s),
where the local factor E_p(s) at bad primes is described in this preprint paper (see below also). The symmetric-power conductor N_k is also computed therein, while the \Gamma-factor \gamma_k(s) is computed by Deligne to be \gamma_{2k-1}(s)=\prod_{i=0}^k\Gamma(s-i)\quad\quad\gamma_{2k}(s)=\Gamma(s/2-\lfloor k/2\rfloor)\prod_{i=0}^k\Gamma(s-i).
Writing C_{2k-1}^2=N_{2k-1}/(2\pi)^{2k} and C_{2k}^2=2N_{2k}/(2\pi)^{2k+1}, we then have that \Lambda_k(E,s)=C_m^s\gamma_k(s)L({\rm Sym}^k E,s) satisfies \Lambda_k(E,s)=\pm\Lambda_k(E,k+1-s).
When the elliptic curve has complex multiplication, the symmetric power L-functions are not primitive, and have a pole at s=1+k/2 when 4|k. The root numbers and Euler factors at bad primes are computed in large generality in this preprint.