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0. Let K be a totally real algebraic number field. In his paper [20], Siegel obtained explicit arithmetic expressions of the values of a zeta function of K at negative integers by using the method of restricting Hilbert– Eisenstein series for SL2(O) to a diagonal, O denoting the ring of integers of K. Let us consider Hilbert–Eisenstein series of higher level whose 0th Fourier coefficients are special values of L-functions. Then a modified method of Siegel’s gives formulas for the values of L-functions at integers, which is one of the purposes of the present paper. Such Eisenstein series have been considered for example in Shimura [18] and Deligne–Ribet [7]. However, for our purpose it is desirable that the Eisenstein series have many 0 as their 0th coefficients at cusps except for a specific cusp. After constructing such Eisenstein series, we give formulas for values of L-functions of K at integers. As a particular case, they turn out to be formulas for relative class numbers of totally imaginary quadratic extensions of K, where the exact form of fundamental units is not necessary. We also give several numerical examples of special values of L-functions and relative class numbers. Our result is twofold. After Section 5, we take as K a real quadratic field. Under some condition on a character we obtain an elliptic modular form whose 0th coefficient is a product of two L-functions over Q and whose higher coefficients are elementary arithmetic. These modular forms can be applied to the investigation of numbers of representations of a natural number by a positive quadratic form with odd number of variables. We obtain a relation between special values of L-functions and numbers of representations by some such quadratic forms. For example, Gauss’ three-square theorem is an easy consequence of our theorem.

On values of L-functions of totally real algebraic number fields at integers