| L(s) = 1 | − 9-s + 2·16-s + 2·19-s + 2·31-s + 12·37-s + 6·43-s − 49-s − 2·67-s − 2·73-s + 81-s − 2·97-s + 2·103-s − 4·109-s + 127-s + 131-s + 137-s + 139-s − 2·144-s + 149-s + 151-s + 157-s + 163-s + 167-s − 169-s − 2·171-s + 173-s + 179-s + ⋯ |
| L(s) = 1 | − 9-s + 2·16-s + 2·19-s + 2·31-s + 12·37-s + 6·43-s − 49-s − 2·67-s − 2·73-s + 81-s − 2·97-s + 2·103-s − 4·109-s + 127-s + 131-s + 137-s + 139-s − 2·144-s + 149-s + 151-s + 157-s + 163-s + 167-s − 169-s − 2·171-s + 173-s + 179-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{24} \cdot 7^{48} \cdot 13^{24}\right)^{s/2} \, \Gamma_{\C}(s)^{24} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{24} \cdot 7^{48} \cdot 13^{24}\right)^{s/2} \, \Gamma_{\C}(s)^{24} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(2.951219320\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.951219320\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T^{2} - T^{6} - T^{8} + T^{12} - T^{16} - T^{18} + T^{22} + T^{24} \) |
| 7 | \( 1 + T^{2} - T^{6} - T^{8} + T^{12} - T^{16} - T^{18} + T^{22} + T^{24} \) |
| 13 | \( 1 + T^{2} - T^{6} - T^{8} + T^{12} - T^{16} - T^{18} + T^{22} + T^{24} \) |
| good | 2 | \( ( 1 - T^{4} + T^{8} - T^{12} + T^{16} - T^{20} + T^{24} )^{2} \) |
| 5 | \( 1 + T^{4} - T^{12} - T^{16} + T^{24} - T^{32} - T^{36} + T^{44} + T^{48} \) |
| 11 | \( 1 + T^{4} - T^{12} - T^{16} + T^{24} - T^{32} - T^{36} + T^{44} + T^{48} \) |
| 17 | \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )^{4}( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} )^{4} \) |
| 19 | \( ( 1 - T + T^{3} - T^{4} + T^{6} - T^{8} + T^{9} - T^{11} + T^{12} )^{2}( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} )^{2} \) |
| 23 | \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} )^{4} \) |
| 29 | \( ( 1 - T + T^{3} - T^{4} + T^{6} - T^{8} + T^{9} - T^{11} + T^{12} )^{2}( 1 + T - T^{3} - T^{4} + T^{6} - T^{8} - T^{9} + T^{11} + T^{12} )^{2} \) |
| 31 | \( ( 1 - T + T^{3} - T^{4} + T^{6} - T^{8} + T^{9} - T^{11} + T^{12} )^{2}( 1 + T^{2} - T^{6} - T^{8} + T^{12} - T^{16} - T^{18} + T^{22} + T^{24} ) \) |
| 37 | \( ( 1 - T + T^{2} )^{12}( 1 + T^{2} - T^{6} - T^{8} + T^{12} - T^{16} - T^{18} + T^{22} + T^{24} ) \) |
| 41 | \( 1 + T^{4} - T^{12} - T^{16} + T^{24} - T^{32} - T^{36} + T^{44} + T^{48} \) |
| 43 | \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )^{4}( 1 - T + T^{3} - T^{4} + T^{6} - T^{8} + T^{9} - T^{11} + T^{12} )^{2} \) |
| 47 | \( 1 + T^{4} - T^{12} - T^{16} + T^{24} - T^{32} - T^{36} + T^{44} + T^{48} \) |
| 53 | \( ( 1 - T + T^{3} - T^{4} + T^{6} - T^{8} + T^{9} - T^{11} + T^{12} )^{2}( 1 + T - T^{3} - T^{4} + T^{6} - T^{8} - T^{9} + T^{11} + T^{12} )^{2} \) |
| 59 | \( ( 1 - T^{4} + T^{8} - T^{12} + T^{16} - T^{20} + T^{24} )^{2} \) |
| 61 | \( ( 1 - T^{2} + T^{4} )^{6}( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} )^{2} \) |
| 67 | \( ( 1 + T - T^{3} - T^{4} + T^{6} - T^{8} - T^{9} + T^{11} + T^{12} )^{2}( 1 + T^{2} - T^{6} - T^{8} + T^{12} - T^{16} - T^{18} + T^{22} + T^{24} ) \) |
| 71 | \( 1 + T^{4} - T^{12} - T^{16} + T^{24} - T^{32} - T^{36} + T^{44} + T^{48} \) |
| 73 | \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} )^{2}( 1 + T - T^{3} - T^{4} + T^{6} - T^{8} - T^{9} + T^{11} + T^{12} )^{2} \) |
| 79 | \( ( 1 + T^{2} - T^{6} - T^{8} + T^{12} - T^{16} - T^{18} + T^{22} + T^{24} )^{2} \) |
| 83 | \( ( 1 - T^{4} + T^{8} - T^{12} + T^{16} - T^{20} + T^{24} )^{2} \) |
| 89 | \( ( 1 - T^{4} + T^{8} - T^{12} + T^{16} - T^{20} + T^{24} )^{2} \) |
| 97 | \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} )^{2}( 1 + T - T^{3} - T^{4} + T^{6} - T^{8} - T^{9} + T^{11} + T^{12} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{48} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−2.13944282631257454673267406759, −1.82947076807366493047818492033, −1.81104594810244583942966847628, −1.71175462952358521220175170977, −1.69336640879571032071954416015, −1.61638763587599945610963593872, −1.52624268565872018925878806647, −1.46158305537243108427254716072, −1.44009107809052135572790220226, −1.40059813044444084220214615126, −1.32664542743279764674669519733, −1.32035461672153633639479012433, −1.26345127330152227728515305039, −1.24483059936557328445780828123, −1.20681533098195492385747929647, −1.17021787070133487296656500028, −1.13154392337542971087670130153, −1.09978296947111078261497139065, −0.916236871481454022994742601021, −0.916018502266066632276688399408, −0.904295930860383391477701547039, −0.66200065159909362692704804930, −0.63798162286424823449295512912, −0.48325125750361904956094797860, −0.45047638482190693969591226831,
0.45047638482190693969591226831, 0.48325125750361904956094797860, 0.63798162286424823449295512912, 0.66200065159909362692704804930, 0.904295930860383391477701547039, 0.916018502266066632276688399408, 0.916236871481454022994742601021, 1.09978296947111078261497139065, 1.13154392337542971087670130153, 1.17021787070133487296656500028, 1.20681533098195492385747929647, 1.24483059936557328445780828123, 1.26345127330152227728515305039, 1.32035461672153633639479012433, 1.32664542743279764674669519733, 1.40059813044444084220214615126, 1.44009107809052135572790220226, 1.46158305537243108427254716072, 1.52624268565872018925878806647, 1.61638763587599945610963593872, 1.69336640879571032071954416015, 1.71175462952358521220175170977, 1.81104594810244583942966847628, 1.82947076807366493047818492033, 2.13944282631257454673267406759
Plot not available for L-functions of degree greater than 10.
