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The classical $n$-variable Kloosterman sums over the finite field ${\bf F}_p$give rise to a lisse $\bar {\bf Q}_l$-sheaf ${\rm Kl}_{n+1}$ on ${\bf G}_{m,{\bf F}_p}={\bf P}^1_{{\bf F}_p}-\{0,\infty\}$, which we call the Kloostermansheaf. Let $L_p({\bf G}_{m,{\bf F}_p}, {\rm Sym}^k{\rm Kl}_{n+1}, s)$ be the$L$-function of the $k$-fold symmetric product of ${\rm Kl}_{n+1}$. Weconstruct an explicit virtual scheme $X$ of finite type over ${\rm Spec} {\bfZ}$ such that the $p$-Euler factor of the zeta function of $X$ coincides with$L_p({\bf G}_{m,{\bf F}_p}, {\rm Sym}^k{\rm Kl}_{n+1}, s)$. We also provesimilar results for $\otimes^k {\rm Kl}_{n+1}$ and $\bigwedge^k {\rmKl}_{n+1}$.

L-functions of Symmetric Products of the Kloosterman Sheaf over Z