The authors study the distribution of zeros of the Lerch zeta-function \[ L(\lambda,\alpha, s):= \sum^\infty_{n=0} e^{2\pi i\lambda n}(n+\alpha)^{-s}, \] defined by R. Lipschitz in 1857 and further studied by M. Lerch thirty years later, and of its derivative \({\partial\over\partial s} L(\lambda,\alpha, s)\). Let me cite one of the authors' result: If \(0< \lambda\), \(\alpha\leq 1\), then \[ \sum_{|\gamma|\leq T}(\beta- 1/2)= {T\over 2\pi} \log{\alpha\over \sqrt{\lambda(1- \{\lambda\})}}+ O(\log T) \] as \(T\to\infty\), where the sum is extended over the non-trivial zeros \(\rho= \beta+ i\gamma\) of the function \(s\mapsto L(\lambda, \alpha,s)\). The authors summarize their conclusions as follows: ``Our observations show some analogies to the Riemann zeta-function (existence and number of trivial and non-trivial zeros) and some differences (asymmetrical distribution of the non-trivial zeros for almost all \(L(\lambda,\alpha, s)\))''.