Magnetic properties and heat capacity of the three-dimensional frustrated<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>S</mml:mi><mml:mo>=</mml:mo><mml:mfrac><mml:mn>1</mml:mn><mml:mn>2</mml:mn></mml:mfrac></mml:mrow></mml:math>antiferromagnet<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mrow><mml:mi mathvariant="normal">PbCuTe</mml:mi></mml:mrow><mml:mn>2</mml:mn></mml:msub><mml:msub><mml:mrow><mml:mi mathvariant="normal">O</mml:mi></mml:mrow><mml:mn>6</mml:mn></mml:msub></mml:math>

We report magnetic susceptibility $(\ensuremath{\chi})$ and heat capacity $({C}_{p})$ measurements along with ab initio electronic structure calculations on ${\mathrm{PbCuTe}}_{2}{\mathrm{O}}_{6}$, a compound made up of a three-dimensional (3D) network of corner-shared triangular units. The presence of antiferromagnetic interactions is inferred from a Curie-Weiss temperature $({\ensuremath{\theta}}_{\mathrm{CW}})$ of about $\ensuremath{-}22$ K from the $\ensuremath{\chi}(T)$ data. The magnetic heat capacity ${C}_{m}$ data show a broad maximum at ${T}^{\mathrm{max}}\ensuremath{\simeq}1.15$ K (i.e., ${T}^{\mathrm{max}}/{\ensuremath{\theta}}_{\mathrm{CW}}\ensuremath{\simeq}0.05)$, which is analogous to the the observed broad maximum in the ${C}_{m}/T$ data of a hyper-kagome system, ${\mathrm{Na}}_{4}{\mathrm{Ir}}_{3}{\mathrm{O}}_{8}$. In addition, ${C}_{m}$ data exhibit a weak kink at ${T}^{*}\ensuremath{\simeq}0.87$ K. While the ${T}^{\mathrm{max}}$ is nearly unchanged, the ${T}^{*}$ is systematically suppressed in an increasing magnetic field $(H)$ up to 80 kOe. For $H\ensuremath{\ge}80$ kOe, the ${C}_{m}$ data at low temperatures exhibit a characteristic power-law $(T{}^{\ensuremath{\alpha}})$ behavior with an exponent $\ensuremath{\alpha}$ slightly less than 2. Hopping integrals obtained from the electronic structure calculations show the presence of strongly frustrated 3D spin interactions along with non-negligible unfrustrated couplings. Our results suggest that ${\mathrm{PbCuTe}}_{2}{\mathrm{O}}_{6}$ is a candidate material for realizing a 3D quantum spin liquid state at high magnetic fields.