We study lower bounds of a general family of L-functions on the $1$ -line. More precisely, we show that for any $F(s)$ in this family, there exist arbitrarily large t such that $F(1+it)\geq e^{\gamma _F} (\log _2 t + \log _3 t)^m + O(1)$ , where m is the order of the pole of $F(s)$ at $s=1$ . This is a generalisation of the result of Aistleitner, Munsch and Mahatab [‘Extreme values of the Riemann zeta function on the $1$ -line’, Int. Math. Res. Not. IMRN2019(22) (2019), 6924–6932]. As a consequence, we get lower bounds for large values of Dedekind zeta-functions and Rankin-Selberg L-functions of the type $L(s,f\times f)$ on the $1$ -line.

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1 线上 L 函数的大值

我们研究一般家庭的下界大号- 上的功能$1$-线。更准确地说，我们证明对于任何$F(s)$在这个家庭中，存在着任意大吨这样$F(1+it)\geq e^{\gamma _F} (\log _2 t + \log _3 t)^m + O(1)$， 在哪里米是极点的阶$F(s)$在$s=1$. 这是 Aistleitner、Munsch 和 Mahatab ['Riemann zeta 函数在$1$-线'，诠释。数学。水库。不是。IMRN2019(22) (2019), 6924–6932]。因此，我们得到 Dedekind zeta 函数和 Rankin-Selberg 的大值的下界大号- 类型的函数$L(s,f\times f)$在$1$-线。