  The calculation, by L.\ Euler, of the values at positive even integers of the
Riemann zeta function, in terms of powers of $\pi$ and rational numbers, was a
watershed event in the history of number theory and classical analysis. Since
then many important analogs involving $L$-values and periods have been
obtained. In analysis in finite characteristic, a version of Euler's result was
given by L.\ Carlitz \cite{ca2} in the 1930's which involved the period of a
rank 1 Drinfeld module (the Carlitz module) in place of $\pi$. In a very
original work \cite{pe2}, F.\ Pellarin has quite recently established a
"deformation" of Carlitz's result involving certain $L$-series and the
deformation of the Carlitz period given in \cite{at1}. Pellarin works only with
the values of this $L$-series at positive integral points. We show here how the
techniques of \cite{go1} also allow these new $L$-series to be analytically
continued -- with associated trivial zeroes -- and interpolated at finite
primes.
