Gosper-type Lambert series identities of level 14

We derive two Gosper-type Lambert series identities of level $14$ which involve the $q$-constant ${\mathrm{\Pi }}_q$ using a special case of Bailey’s ${}_6{\psi }_6$ summation formula and certain propeties of $\eta$-quotients and generalized $\eta$-quotients on the congruence subgroup ${\mathrm{\Gamma }}_0(14)$.