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In this paper, we show the joint universality of the set of the Steuding class  L -functions { L ( s + i  δ   l   τ )}  l =1   m   for almost all ( δ  1 ,…, δ   m  )∈R  m  . From this property, we obtain the result that { L ( s + i  δ   l   τ )}  l = j,k  , where  δ   j   and  δ   k   are two of the above, has a kind of generalized strong recurrence. Roughly speaking, this means that  L ( s + i  δ   j   τ ) can be uniformly approximated by  L ( s + i  δ   k   τ ) in the sense of universality. Moreover, we consider the relation between the generalized strong recurrence and zeros of the Selberg and Steuding class  L -functions  L ( s ). Finally, we obtain the upper bound for the density of universality for the Steuding class  L -functions.

Some topics related to universality of L -functions with an Euler product