A criterion is given for studying (explicit) Baker type lower bounds of linear forms in numbers $1,{\Theta }_1,...,{\Theta }_m\in ℂ*$ over the ring $\mathbb{Z}$ of an imaginary quadratic field . This work deals with the simultaneous auxiliary functions case.

We prove that 7. 398 537 is an irrationality measure of $\zeta \left(2\right)={\pi }^2/6$. We employ double integrals of suitable rational functions invariant under a group of birational transformations of ${ℂ}^2$. The numerical results are obtained with the aid of a semi-infinite linear programming method.

Let ${\left(t_k\right)}_{k\ge 0}$ be the Thue–Morse sequence on $\{0,1\}$ defined by $t_0=0$, $t_{2k}=t_k$ and $t_{2k+1}=1-t_k$ for $k\ge 0$. Let $b\ge 2$ be an integer. We establish that the irrationality exponent of the Thue–Morse–Mahler number ${\sum }_{k\ge 0}{t}_k{b}^{-k}$ is equal to $2$.