Archimedean Modular Symbols for Rankin-Selberg Convolutions of ${\mathrm{G}\mathrm{L}}_n(\mathbb{C})\times {\mathrm{G}\mathrm{L}}_n(\mathbb{C})$ and ${\mathrm{G}\mathrm{L}}_n(\mathbb{C})\times {\mathrm{G}\mathrm{L}}_{n+1}(\mathbb{C})$

We explicit calculate the archimedean modular symbol for Rankin-Selberg convolutions of ${\mathrm{G}\mathrm{L}}_n(\mathbb{C})\times {\mathrm{G}\mathrm{L}}_n(\mathbb{C})$ and ${\mathrm{G}\mathrm{L}}_n(\mathbb{C})\times {\mathrm{G}\mathrm{L}}_{n+1}(\mathbb{C})$, for all generic cohomological representations.