模λ函数在由下式生成的模群的主同余子群Γ(2)的作用下保持不变:[3]:115
![{\displaystyle \tau \mapsto \tau +2\ ;\ \tau \mapsto {\frac {\tau }{1-2\tau }}\ .}](//wikimedia.org/api/rest_v1/media/math/render/svg/716cade925b82fc42e9424caa0b3fe85e8acdc60)
模群自身的生成元则以如下方式作用于模λ函数之上:[3]:109
![{\displaystyle \tau \mapsto \tau +1\ :\ \lambda \mapsto {\frac {\lambda }{\lambda -1}}\,;}](//wikimedia.org/api/rest_v1/media/math/render/svg/27a6863e75dbe8f7d16e24a8b41695df17149f78)
![{\displaystyle \tau \mapsto -{\frac {1}{\tau }}\ :\ \lambda \mapsto 1-\lambda \ .}](//wikimedia.org/api/rest_v1/media/math/render/svg/a93bfdca68636bf1a102f8ff9d814a797452858b)
λ函数为亚可比模量(Jacobi modulus)的平方[3]:108,即
;亦可以戴德金η函数与Θ函数表达:
![{\displaystyle \lambda (\tau )={\Bigg (}{\frac {{\sqrt {2}}\,\eta ({\tfrac {\tau }{2}})\eta ^{2}(2\tau )}{\eta ^{3}(\tau )}}{\Bigg )}^{8}={\frac {16}{\left({\frac {\eta (\tau /2)}{\eta (2\tau )}}\right)^{8}+16}}={\frac {\theta _{2}^{4}(0,\tau )}{\theta _{3}^{4}(0,\tau )}}}](//wikimedia.org/api/rest_v1/media/math/render/svg/ea8e2e2d593715d59cf72e72a50a434545821a34)
![{\displaystyle {\frac {1}{{\big (}\lambda (\tau ){\big )}^{1/4}}}-{\big (}\lambda (\tau ){\big )}^{1/4}={\frac {1}{2}}\left({\frac {\eta ({\tfrac {\tau }{4}})}{\eta (\tau )}}\right)^{4}=2\,{\frac {\theta _{4}^{2}(0,{\tfrac {\tau }{2}})}{\theta _{2}^{2}(0,{\tfrac {\tau }{2}})}}}](//wikimedia.org/api/rest_v1/media/math/render/svg/bc02886842a2567477f470eb4cea6df840c85ff0)
其中:[3]:63
![{\displaystyle \theta _{2}(0,\tau )=\sum _{n=-\infty }^{\infty }q^{\left({n+{\frac {1}{2}}}\right)^{2}}}](//wikimedia.org/api/rest_v1/media/math/render/svg/82be315e0b33481762ae20a66f250342b26ccf94)
![{\displaystyle \theta _{3}(0,\tau )=\sum _{n=-\infty }^{\infty }q^{n^{2}}}](//wikimedia.org/api/rest_v1/media/math/render/svg/f2d2ed596042232319d3995a050fbb1570b409d5)
![{\displaystyle \theta _{4}(0,\tau )=\sum _{n=-\infty }^{\infty }(-1)^{n}q^{n^{2}}}](//wikimedia.org/api/rest_v1/media/math/render/svg/f9d950651393a39d2197a55b715cc937584d0ce4)
![{\displaystyle q=e^{\pi i\tau }}](//wikimedia.org/api/rest_v1/media/math/render/svg/b04b5fc2797318028727bd06733bb52300c99e9b)
λ函数亦可以魏尔斯特拉斯椭圆函数在定义其的格子的棱边中点和面心处的函数值表达;若令
为满足
的基本周期二元组:
![{\displaystyle e_{1}=\wp \left({\frac {\omega _{1}}{2}}\right),e_{2}=\wp \left({\frac {\omega _{2}}{2}}\right),e_{3}=\wp \left({\frac {\omega _{1}+\omega _{2}}{2}}\right)}](//wikimedia.org/api/rest_v1/media/math/render/svg/961a179f9f0f2a4845bb0389d259bf8ecfb9d4b3)
则有:[3]:108
![{\displaystyle \lambda ={\frac {e_{3}-e_{2}}{e_{1}-e_{2}}}\,.}](//wikimedia.org/api/rest_v1/media/math/render/svg/5e3313c93b7fc35f4b371635972d2471fad46885)
魏尔斯特拉斯函数在上述三点的值各不相同,这意味着λ函数取不到值0或1。[3]:108
其与克莱因j函数的关系为:[3]:117[4]
![{\displaystyle j(\tau )={\frac {256(1-\lambda (1-\lambda ))^{3}}{(\lambda (1-\lambda ))^{2}}}={\frac {256(1-\lambda +\lambda ^{2})^{3}}{\lambda ^{2}(1-\lambda )^{2}}}\ .}](//wikimedia.org/api/rest_v1/media/math/render/svg/198fd73303c696f78bb9066d8319801929d0f8dd)
日本数学会. 数学百科辞典. 科学出版社. 1984 [2021-07-15]. (原始内容存档于2021-10-23).
Selberg, A.; Chowla, S. "On Epstein's Zeta-Function.". J. reine angew. Math. 1967, 227: 86–110.