The author extends P. B. Garrett’s results [Decomposition of Eisenstein series: Rankin triple products, Ann. Math., II. Ser. (to appear)] relating triple L-function in two aspects. Let \(S_ k(\Gamma _ 1)\) be the space of cusp forms of weight k and of degree 1. For Siegel modular forms \(f_ 1,...,f_ m\) (of various degree), let \({\mathbb{Q}}(f_ 1,...,f_ m)\) be the field generated by all the Fourier coefficients of \(f_ 1,...,f_ m\) over \({\mathbb{Q}}\). If \(f\in S_ k(\Gamma _ 1)\) is a normalized Hecke eigenform and p is a prime, we define semi-simple \(M_ p(f)\in GL(2, {\mathbb{C}})\) (up to the conjugacy class) by \(\det (I_ 2-tM_ p(f))=1-a(p,f)t+p^{k-1} t^ 2.\) For normalized eigen cusp forms f, g and h, define the ’triple L-function’ L(s;f,g,h) by \[ L(s;f,g,h)=\prod _{p: prime}\det (I_ s-p^{-s} M_ p(f)\otimes M_ p(g)\otimes M_ p(h))^{-1}\quad. \] If \(f\in S_{k+r}(\Gamma _ 1)\), we denote by \([f]_ r\) the degree two Klingen type Eisenstein series attached to f of type \(\det ^ k\otimes Sym^ r St\), where St is the standard representation of GL(2, \({\mathbb{C}}).\)
Then, the author obtains the following two results. Theorem 1. Let f, g and h be normalized eigen cusp forms of degree one and of weight \(\nu\) (f), \(\nu\) (g) and \(\nu\) (h), respectively, satisfying \(\nu\) (f)\(\geq \nu (g)\geq \nu (h)\). Put \[ \tilde L(s;f,g,h)=\Gamma _ c(s) \Gamma _ c(s- \nu (f)+1) \Gamma _ c(s-\nu (g)+1) \Gamma _ c(s-\nu (h)+1) L(s;f,g,h) \] where \(\Gamma _ c(s)=2(2\pi)^{-s} \Gamma (s)\). Then \~L(s;f,g,h) meromorphically extends to the whole s-plane and satisfies the functional equation \[ \tilde L(s;f,g,h)=-\tilde L(\nu (f)+\nu (g)+\nu (h)-2- s;f,g,h)\quad. \] Moreover, if \(\nu (g)+\nu (h)-\nu (f)>0\) and \(L((\nu (f)+\nu (g)+\nu (h))/2-1;f,g,h)\) is finite, then \[ L((\nu (f)+\nu (g)+\nu (h))/2-1;f,g,h)=0\quad, \] and if \(\nu (g)+\nu (h)-\nu (f)>4\), then \[ \frac{\pi ^{5+\nu (f)-3\nu (g)-3\nu (h)} L(\nu (g)+\nu (h)- 2;f,g,h)}{<f,f>_{\nu (f)} <g,g>_{\nu (g)} <h,h>_{\nu (h)}}\in {\mathbb{Q}}([f]_{2\nu (f)-\nu (g)-\nu (h)}, g, h) \] where \(<, >_ k\) is the Petersson inner product for weight k.
Theorem 2. Let f, g and h be normalized cusp forms of weight k. For an integer m with \(0\leq m\leq k/2-2\), we have \[ \pi ^{5-5k+4m} L(2k-2- m;f,g,h)/(<f,f>_ k <g,g>_ k <h,h>_ k)\in {\mathbb{Q}}(f,g,h). \] Taking \(f=g=h\), one obtains special values of the third L-function of f. It is interesting to observe that triple L-functions and the third L-functions vanish at the center of their functional equation if the center is a critical point. In the proof of Theorem 1, affirmative support to the author’s conjecture [Math. Ann. 274, 335-352 (1986; Zbl 0571.10028)] is given.