Prove or disprove that:
$$2 L_{2,1}(s)-2 \zeta (s)+\zeta (s)=\left(1-\frac{1}{2^{s-1}}\right) \zeta (s)$$
$$3 L_{3,1}(s)-3 \zeta (s)+\zeta (s)=\left(1-\frac{1}{3^{s-1}}\right) \zeta (s)$$
Where $L_{2,1}(s)$ and $L_{3,1}(s)$ are Dirichlet L functions and $\zeta(s)$ is the Riemann zeta function.
Mathematica:
Limit[2 DirichletL[2, 1, s] - 2*Zeta[s] + Zeta[s], s -> Pi + Sqrt[2] I]
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Limit[(1 - 1/2^(s - 1)) Zeta[s], s -> Pi + Sqrt[2] I]
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Limit[3 DirichletL[3, 1, s] - 3*Zeta[s] + Zeta[s], s -> Pi + Sqrt[2] I]
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Limit[(1 - 1/3^(s - 1)) Zeta[s], s -> Pi + Sqrt[2] I]
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I assume that $L_{p,1}(s)$ is the Dirichlet $L$-function associated with the principal character $\pmod{p}$.
About the first identity: $$\begin{eqnarray*}\left(1-\frac{1}{2^{s-1}}\right)\zeta(s) &=& -\zeta(s)+2\left(1-\frac{1}{2^s}\right)\zeta(s)=-\zeta(s)+2\prod_{p\neq 2}\left(1-\frac{1}{p^s}\right)^{-1}\\&=&-\sum_{n\geq 1}\frac{1}{n^s}+2\!\!\!\!\sum_{n\equiv 1\!\!\pmod{\!\!2}}\frac{1}{n^s}=-\zeta(s)+L_{2,1}(s).\tag{1}\end{eqnarray*}$$ In a similar way, $$\begin{eqnarray*}\left(1-\frac{1}{3^{s-1}}\right)\zeta(s)=-2\zeta(s)+3\prod_{p\neq 3}\left(1-\frac{1}{p^s}\right)^{-1}=-2\zeta(s)+L_{3,1}(s),\tag{2}\end{eqnarray*}$$ so everything follows from the Euler product for the $\zeta$ function.