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By degree: 1   2   3   4  
By motivic weight: 0   1   2   3  
By conductor: 1-99   100-999   1000-9999   10000-100000   >100000
By lowest zero
A plot of classical modular form L-functions by analytic conductor, first zero and weight
Some interesting L-functions in degree 1, 2, 3, 4
A random L-function in degree 1, 2, 3, 4

By origin

Dirichlet character Classical modular form Elliptic curve over $\mathbb{Q}$ Artin representation
Hilbert modular form Bianchi modular form Elliptic curve over $\mathbb{Q}(\sqrt{d})$ Genus 2 curve over $\mathbb{Q}$

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e.g. 2-1-1.1-c11-0-0 or 4-1-1.1-r0e4-c4.72c12.47-0