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As a consequence of Euler's refectin and Gauss's multiplication formulas, all values $\Gamma(a)$ with $24a\in\mathbb{Z}$ or $60a\in\mathbb{Z}$ can be expressed algebraically in terms of these values: $$\Gamma\!\left(\frac12\right)=\sqrt{\pi},\quad \Gamma\!\left(\frac13\right), \quad \Gamma\!\left(\frac14\right), \quad \Gamma\!\left(\frac15\right), \quad \Gamma\!\left(\frac25\right), \quad \Gamma\!\left(\frac18\right),$$ $$\Gamma\!\left(\frac1{15}\right),\quad \Gamma\!\left(\frac1{20}\right), \quad \Gamma\!\left(\frac1{24}\right), \quad \Gamma\!\left(\frac1{60}\right), \quad \Gamma\!\left(\frac7{60}\right).$$ This is worked out in this paper, along with basic expressions for $a\in(0,1)$. For example, $$\Gamma\!\left(\frac7{10}\right)=\sqrt{\pi}\,2^{3/5}\,\Gamma\left(\frac15\right)^{\!-1}\,\Gamma\!\left(\frac2{5}\right),$$ $$\Gamma\!\left(\frac1{12}\right)=\frac{3^{3/8}\,\sqrt{\sqrt{3}+1}}{\sqrt{\pi}\,2^{1/4}}\,\Gamma\!\left(\frac13\right)\,\Gamma\!\left(\frac14\right),$$ $$\Gamma\!\left(\frac{11}{15}\right)=2\pi\cdot 3^{3/10}\,\Gamma\!\left(\frac15\right)\,\Gamma\left(\frac25\right)^{\!-1}\,\Gamma\left(\frac1{15}\right)^{\!-1},$$ $$\Gamma\!\left(\frac{11}{20}\right)=2^{1/5}\,\sqrt{5+\sqrt5}\;\Gamma\!\left(\frac15\right)\,\Gamma\!\left(\frac25\right)\,\Gamma\left(\frac1{20}\right)^{\!-1},$$ $$\Gamma\!\left(\frac{49}{60}\right)=\frac{\sqrt{\pi}\,\sqrt3\,\sqrt{\sqrt3+1}\,\sqrt{5+\sqrt5}\,\sqrt{\sqrt5+\sqrt3}}{5^{1/24}}\,\Gamma\!\left(\frac13\right)\,\Gamma\!\left(\frac1{60}\right)^{\!-1}.$$ Further algebraic independence of the listed 11 "basic" values (and more general answers) are yet to be settled.