Exact critical values of a symmetric fourth $L$-function and Zagier's conjecture

Around 1977, Don Zagier conjectured exact critical values of the symmetric fourth $L$-function of the Ramanujan $\Delta$ function, expressing them by explicit rational numbers, power of $\pi$, and the inner product of $\Delta$, based on numerical calculations and Deligne's conjectures.
In this talk, we will give their explicit exact values (with proof), using Siegel modular forms, pullback formulas, and differential operators. This is a joint work with H. Katsurada.
We also talk shortly on some congruence and a theory of differential operators on Siegel modular forms.