  In recent work, Griffin, Ono, and Tsai constructs an $L-$series to prove that
the proportion of short Weierstrass elliptic curves over $\mathbb{Q}$ with
trivial Tamagawa product is $0.5054\dots$ and that the average Tamagawa product
is $1.8183\dots$. Following their work, we generalize their $L-$series over
arbitrary number fields $K$ to be \[L_{\mathrm{Tam}}(K;
s):=\sum_{m=1}^{\infty}\frac{P_{\mathrm{Tam}}(K; m)}{m^s},\] where
$P_{\mathrm{Tam}}(K;m)$ is the proportion of short Weierstrass elliptic curves
over $K$ with Tamagawa product $m$. We then construct Markov chains to compute
the exact values of $P_{\mathrm{Tam}}(K;m)$ for all number fields $K$ and
positive integers $m$. As a corollary, we also compute the average Tamagawa
product $L_{\mathrm{Tam}}(K;-1)$. We then use these results to uniformly bound
$P_{\mathrm{Tam}}(K;1)$ and $L_{\mathrm{Tam}}(K,-1)$ in terms of the degree of
$K$. Finally, we show that there exist sequences of $K$ for which
$P_{\mathrm{Tam}}(K;1)$ tends to $0$ and $L_{\mathrm{Tam}}(K;-1)$ to $\infty$,
as well as sequences of $K$ for which $P_{\mathrm{Tam}}(K;1)$ and
$L_{\mathrm{Tam}}(K;-1)$ tend to $1$.
