The Lobachevsky function Λ or Л is essentially the same function with a change of variable:  though the name "Lobachevsky function" is not quite historically accurate, as Lobachevsky's formulas for hyperbolic volume used the slightly different function  For rational values of formula_116 (that is, for formula_117 for some integers "p" and "q"), the function formula_118 can be understood to represent a periodic orbit of an element in the cyclic group, and thus formula_119 can be expressed as a simple sum involving the Hurwitz zeta function. This allows relations between certain Dirichlet L-functions to be easily computed. A series acceleration for the Clausen function is given by  which holds for formula_121. Here, formula_122 is the Riemann zeta function. A more rapidly convergent form is given by  Convergence is aided by the fact that formula_124 approaches zero rapidly for large values of "n". Both forms are obtainable through the types of resummation techniques used to obtain rational zeta series. (ref. Borwein, et al., 2000, below). Recall the Barnes G-function and Catalan constant "K". Some special values include  In general, from the Barnes G-function reflection formula,  Equivalently, using Euler's reflection formula for the gamma function, then,  Some special values for higher order Clausen functions include  where formula_139 is Catalan's constant, formula_140 is the Dirichlet beta function, formula_141 is the Dirichlet eta function (also called the alternating zeta function), and formula_142 is the Riemann zeta function. The following integrals are easily proven from the series representations of the Clausen function:  Fourier-analytic methods can be used to find the first moments of the square of the function formula_148 on the interval formula_149:  Here formula_153 denotes the Multiple zeta function. A large number of trigonometric and logarithmo-trigonometric integrals can be evaluated in terms of the Clausen function, and various common mathematical constants like formula_154 (Catalan's constant), formula_155, and the special cases of the zeta function, formula_156 and formula_157. The examples listed below follow directly from the integral representation of the Clausen function, and the proofs require little more than basic trigonometry, integration by parts, and occasional term-by-term integration of the Fourier series definitions of the Clausen functions.