Shintani expressed the first derivative at s = 0 of a partial ζfunction of an algebraic number field in terms of the multiple gamma function. Cassou-Noguès constructed a p-adic analogue of the partial ζ-function and calculated the derivative at s = 0. In this paper, we will define a p-adic analogue of the multiple gamma function and give a p-adic analogue of Shintani's formula. This formula has a strong resemblance to the original Shintani's formula. Using this formula, we get a partial result toward Gross' conjecture concerning the order at s = 0 of the p-adic L-function.

We define a "period ring-valued beta function" and give a reciprocity law on its special values. The proof is based on some results of Rohrlich and Coleman concerning Fermat curves. We also have the following application. Stark's conjecture implies that the exponential of the derivatives at s = 0 of partial zeta functions are algebraic numbers which satisfy a reciprocity law under certain conditions. It follows from Euler's formulas and properties of cyclotomic units when the base field is the rational number field. In this paper, we provide an alternative (and partial) proof by using the reciprocity law on the period ring-valued beta function. In other words, the reciprocity law given in this paper is a refinement of the reciprocity law on cyclotomic units.

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In part I of this paper, we studied the p-adic absolute CM-period symbol in the completely split case. We presented a conjecture which predicts the exact value of this symbol with supporting evidences. In this part II, we study the properties of this symbol in the general case and clarify its relation to p-adic periods.