| L(s) = 1 | − 2·4-s − 9-s + 3·16-s + 2·19-s + 4·31-s + 2·36-s − 49-s + 26·61-s − 2·64-s − 4·76-s + 2·79-s + 81-s + 2·109-s + 2·121-s − 8·124-s + 127-s + 131-s + 137-s + 139-s − 3·144-s + 149-s + 151-s + 157-s + 163-s + 167-s − 2·169-s − 2·171-s + ⋯ |
| L(s) = 1 | − 2·4-s − 9-s + 3·16-s + 2·19-s + 4·31-s + 2·36-s − 49-s + 26·61-s − 2·64-s − 4·76-s + 2·79-s + 81-s + 2·109-s + 2·121-s − 8·124-s + 127-s + 131-s + 137-s + 139-s − 3·144-s + 149-s + 151-s + 157-s + 163-s + 167-s − 2·169-s − 2·171-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{24} \cdot 5^{48} \cdot 7^{48}\right)^{s/2} \, \Gamma_{\C}(s)^{24} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{24} \cdot 5^{48} \cdot 7^{48}\right)^{s/2} \, \Gamma_{\C}(s)^{24} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(15.60915073\) |
| \(L(\frac12)\) |
\(\approx\) |
\(15.60915073\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T^{2} - T^{6} - T^{8} + T^{12} - T^{16} - T^{18} + T^{22} + T^{24} \) |
| 5 | \( 1 \) |
| 7 | \( 1 + T^{2} - T^{6} - T^{8} + T^{12} - T^{16} - T^{18} + T^{22} + T^{24} \) |
| good | 2 | \( ( 1 + T^{2} - T^{6} - T^{8} + T^{12} - T^{16} - T^{18} + T^{22} + T^{24} )^{2} \) |
| 11 | \( ( 1 - T + T^{3} - T^{4} + T^{6} - T^{8} + T^{9} - T^{11} + T^{12} )^{2}( 1 + T - T^{3} - T^{4} + T^{6} - T^{8} - T^{9} + T^{11} + T^{12} )^{2} \) |
| 13 | \( ( 1 + T^{2} - T^{6} - T^{8} + T^{12} - T^{16} - T^{18} + T^{22} + T^{24} )^{2} \) |
| 17 | \( ( 1 + T^{2} - T^{6} - T^{8} + T^{12} - T^{16} - T^{18} + T^{22} + T^{24} )^{2} \) |
| 19 | \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )^{4}( 1 + T - T^{3} - T^{4} + T^{6} - T^{8} - T^{9} + T^{11} + T^{12} )^{2} \) |
| 23 | \( ( 1 + T^{2} - T^{6} - T^{8} + T^{12} - T^{16} - T^{18} + T^{22} + T^{24} )^{2} \) |
| 29 | \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )^{4}( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} )^{4} \) |
| 31 | \( ( 1 - T + T^{3} - T^{4} + T^{6} - T^{8} + T^{9} - T^{11} + T^{12} )^{4} \) |
| 37 | \( ( 1 - T^{2} + T^{4} )^{6}( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} )^{2} \) |
| 41 | \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )^{4}( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} )^{4} \) |
| 43 | \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} )^{4} \) |
| 47 | \( ( 1 + T^{2} - T^{6} - T^{8} + T^{12} - T^{16} - T^{18} + T^{22} + T^{24} )^{2} \) |
| 53 | \( ( 1 + T^{2} - T^{6} - T^{8} + T^{12} - T^{16} - T^{18} + T^{22} + T^{24} )^{2} \) |
| 59 | \( ( 1 - T + T^{3} - T^{4} + T^{6} - T^{8} + T^{9} - T^{11} + T^{12} )^{2}( 1 + T - T^{3} - T^{4} + T^{6} - T^{8} - T^{9} + T^{11} + T^{12} )^{2} \) |
| 61 | \( ( 1 - T )^{24}( 1 - T + T^{3} - T^{4} + T^{6} - T^{8} + T^{9} - T^{11} + T^{12} )^{2} \) |
| 67 | \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} )^{2}( 1 + T^{2} - T^{6} - T^{8} + T^{12} - T^{16} - T^{18} + T^{22} + T^{24} ) \) |
| 71 | \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )^{4}( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} )^{4} \) |
| 73 | \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} )^{2}( 1 + T^{2} - T^{6} - T^{8} + T^{12} - T^{16} - T^{18} + T^{22} + T^{24} ) \) |
| 79 | \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )^{4}( 1 + T - T^{3} - T^{4} + T^{6} - T^{8} - T^{9} + T^{11} + T^{12} )^{2} \) |
| 83 | \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} )^{4} \) |
| 89 | \( ( 1 - T + T^{3} - T^{4} + T^{6} - T^{8} + T^{9} - T^{11} + T^{12} )^{2}( 1 + T - T^{3} - T^{4} + T^{6} - T^{8} - T^{9} + T^{11} + T^{12} )^{2} \) |
| 97 | \( ( 1 + T^{2} - T^{6} - T^{8} + T^{12} - T^{16} - T^{18} + T^{22} + T^{24} )^{2} \) |
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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
−1.85532721287296615303300038292, −1.65661580641095062242087669628, −1.61179772260006406098153003669, −1.58919346438634634828461735723, −1.45751313645617828397654049947, −1.45562218232704193621540482917, −1.43037872556857910893514857325, −1.33773511231653310813865059051, −1.29159883436679084329789261202, −1.25575362270486390265670674630, −1.17768351112663763186241676015, −1.09212850668704230254950859038, −1.07551611303513596067166678743, −0.942150851897989654065633968903, −0.905971922826405355903103998532, −0.815750279121388337909963361285, −0.814543577406772891704453398776, −0.795056512642020629885345788155, −0.78358928242562494509975653641, −0.77105249181775731605265546194, −0.70181475319526073540770129858, −0.66832242621897235069686670292, −0.57023901673805594183742727782, −0.51462947742194570898640725518, −0.40992055149658821856707915502,
0.40992055149658821856707915502, 0.51462947742194570898640725518, 0.57023901673805594183742727782, 0.66832242621897235069686670292, 0.70181475319526073540770129858, 0.77105249181775731605265546194, 0.78358928242562494509975653641, 0.795056512642020629885345788155, 0.814543577406772891704453398776, 0.815750279121388337909963361285, 0.905971922826405355903103998532, 0.942150851897989654065633968903, 1.07551611303513596067166678743, 1.09212850668704230254950859038, 1.17768351112663763186241676015, 1.25575362270486390265670674630, 1.29159883436679084329789261202, 1.33773511231653310813865059051, 1.43037872556857910893514857325, 1.45562218232704193621540482917, 1.45751313645617828397654049947, 1.58919346438634634828461735723, 1.61179772260006406098153003669, 1.65661580641095062242087669628, 1.85532721287296615303300038292
Plot not available for L-functions of degree greater than 10.
