  For all k>0 integer, we consider the regularised I-function of the family of
del Pezzo surfaces of degree 8k+4 in P(2,2k+1,2k+1, 4k+1), first constructed by
Johnson and Koll\'ar. We show that this function, which is of hypergeometric
type, is a period of an explicit pencil of curves. Thus the pencil is a
candidate LG mirror of the family of del Pezzo surfaces. The main feature of
these surfaces, which makes the mirror construction especially interesting, is
that the anticanonical system is empty: because of this, our mirrors are not
covered by any other construction known to us. We discuss connections to the
work of Beukers, Cohen and Mellit on hypergeometric functions.
