Special $L$-values of certain CM weight three Hecke eigenforms

Ramanujan's theory of elliptic functions to alternative bases connects modular forms with hypergeometric series and has led to applications such as the modularity of certain hypergeometric Galois representations. In this paper, we relate special values of $L$-functions of certain CM Hecke eigenforms to Ramanujan's alternative bases via the modularity of hypergeometric Galois representations associated with hypergeometric series ${}_{3}F_{2}\!\left[ \genfrac{}{}{0pt}{}{\frac{1}{2} \ \frac{1}{d} \ \frac{d-1}{d}}{\ 1 \ \ \ \ 1} ;\ t \right]$, $d=2$, $3$, $4$, and $6$, arising from tensor products of CM elliptic curves over real quadratic fields. We also give a complete classification of these type of hypergeometric Galois representations.