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A $$p$$-adic measure attached to the zeta functions associated with two elliptic modular forms. II. (English) Zbl 0645.10028
Let $$f=\sum^{\infty}_{n=1} a(n)q^n$$ and $$g=\sum^{\infty}_{n=1}b(n)q^n$$ be holomorphic common eigenforms of all Hecke operators for the congruence subgroup $$\Gamma _0(N)$$ of $$\mathrm{SL}_2(\mathbb Z)$$ with “Nebentypus” character $$\psi$$ and $$\xi$$ and of weight $$k$$ and $$\ell$$, respectively. Define the Rankin product of $$f$$ and $$g$$ by
$\mathcal D_N(s,f,g) = \Bigl(\sum^{\infty}_{n=1}\psi \xi (n)n^{k+\ell - 2s-2}\Bigr)\Bigl(\sum^{\infty}_{n =1}a(n)b(n) n^{-s}\Bigr).$
Supposing $$f$$ and $$g$$ to be ordinary at a prime $$p\geq 5$$, we shall construct a $$p$$-adically analytic $$L$$-function of three variables which interpolate the values $$\frac{\mathcal D_ N(\ell +m,f,g)}{\pi^{\ell +2m+1}<f,f>}$$ for integers $$m$$ with $$0\leq m<k-\ell$$, by regarding all the ingredients $$m$$, $$f$$ and $$g$$ as variables. Here $$<f,f>$$ is the Petersson self-inner product of $$f$$.
[For part I, cf. Invent. Math. 79, 159–195 (1985; Zbl 0573.10020).]
Reviewer: Haruzo Hida

##### MSC:
 11F85 $$p$$-adic theory, local fields 11F67 Special values of automorphic $$L$$-series, periods of automorphic forms, cohomology, modular symbols
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