Miyawaki lift of type II (awaiting review)

Miyawaki Lift, Type II (conjectural): Let $k$ be an even weight. For each pair of elliptic Hecke cusp eigenforms $$f\in{\mathcal S}_{2k−2}(\SL(2,{\mathbb Z})) \quad\text{quad}\quad g\in{\mathcal S}_{k−2}(\SL(2,{\mathbb Z}))$$ their Miyawaki lift of type II is a Siegel Hecke cusp eigenform $$F\in{\mathcal S}_k(\Sp(6,{\mathbb Z}))$$ whose standard Euler factors and standard $L$-function factor as $$Q_p^{\rm st}(F,X)=Q_p(f,p^{1−k}X)Q_p(f,p^{2−k}X)Q_p^{\rm st}(g,X)$$ and $$L^{\rm st}(F,s)=L(f,s+k−1)L(f,s+k−2)L^{\rm st}(g,s)$$ and whose spinor Euler factors and spinor L-function factor as $$Q_p^{\rm spin}(F,X)=Q_p(g,p^{k−1}X)Q_p(g,p^{k−2}X)Q_p(f\otimes g,X)$$ and $$L^{\rm spin}(F,s)=L(g,s−k+1)L(g,s−k+2)L(f\otimes g,s).$$