Let $p$ be a prime number and $d$ a natural number with $p\nmid d$. Furthermore let $\chi$ be a Dirichlet character with conductor $d$ and $G=Gal(\mathbb{Q}(\mu_{p^{\infty}})/\mathbb{Q})$.
In the modern setting a $p$-adic $L$-function is usually seen as an element in the Iwasawa algebra $\Lambda=\mathcal{O}[[G]]$ and one can usually find something like the following result.
There exists an element $\mu_{\chi}\in\Lambda$, such that
$$ \mu_{\chi}(\phi^{-1}\kappa^{n})=(1-\chi\phi(p))p^{-n})L(\chi\phi,n) $$ for all characters $\phi:G\rightarrow\overline{\mathbb{Q}}_p$ and all integers $n\leq0$ and $\mu_{\chi}$ is uniquely determined by this formulas. Here $\kappa$ denotes the cyclotomic character.
Now the only work where I found a formula for positive integers is http://www.numdam.org/article/SB_1998-1999__41__21_0.pdf, Theorem 1.8. Colmez also states a functional equation for $p$-adic $L$-functions. Unfortunately he doesn't state where these results come from.
Therefore my question is if someone knows more about these results, like where this is proven or at least how. The only way I could think of, is using one of the conjectural formulas for the interpolation of motivic $p$-adic $L$-functions. But these are as stated only conjectural.