| L(s) = 1 | + 3·5-s − 8-s − 3·23-s + 6·25-s + 3·27-s − 3·40-s + 3·41-s − 3·43-s + 3·61-s + 64-s + 3·67-s + 3·89-s + 3·101-s + 12·107-s + 3·109-s − 9·115-s + 7·125-s + 127-s + 131-s + 9·135-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + ⋯ |
| L(s) = 1 | + 3·5-s − 8-s − 3·23-s + 6·25-s + 3·27-s − 3·40-s + 3·41-s − 3·43-s + 3·61-s + 64-s + 3·67-s + 3·89-s + 3·101-s + 12·107-s + 3·109-s − 9·115-s + 7·125-s + 127-s + 131-s + 9·135-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 5^{24} \cdot 181^{24}\right)^{s/2} \, \Gamma_{\C}(s)^{24} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 5^{24} \cdot 181^{24}\right)^{s/2} \, \Gamma_{\C}(s)^{24} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(27.00705576\) |
| \(L(\frac12)\) |
\(\approx\) |
\(27.00705576\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T^{3} - T^{9} - T^{12} - T^{15} + T^{21} + T^{24} \) |
| 5 | \( ( 1 - T + T^{3} - T^{4} + T^{5} - T^{7} + T^{8} )^{3} \) |
| 181 | \( 1 - T^{3} + T^{9} - T^{12} + T^{15} - T^{21} + T^{24} \) |
| good | 3 | \( ( 1 - T^{3} + T^{6} )^{4}( 1 + T^{3} - T^{9} - T^{12} - T^{15} + T^{21} + T^{24} ) \) |
| 7 | \( ( 1 - T^{3} + T^{6} )^{8} \) |
| 11 | \( ( 1 - T^{3} + T^{9} - T^{12} + T^{15} - T^{21} + T^{24} )( 1 + T^{3} - T^{9} - T^{12} - T^{15} + T^{21} + T^{24} ) \) |
| 13 | \( ( 1 - T^{3} + T^{9} - T^{12} + T^{15} - T^{21} + T^{24} )( 1 + T^{3} - T^{9} - T^{12} - T^{15} + T^{21} + T^{24} ) \) |
| 17 | \( ( 1 - T^{3} + T^{6} )^{4}( 1 + T^{3} + T^{6} )^{4} \) |
| 19 | \( ( 1 - T )^{24}( 1 + T )^{24} \) |
| 23 | \( ( 1 + T - T^{3} - T^{4} - T^{5} + T^{7} + T^{8} )^{3}( 1 + T^{3} - T^{9} - T^{12} - T^{15} + T^{21} + T^{24} ) \) |
| 29 | \( ( 1 - T^{3} + T^{9} - T^{12} + T^{15} - T^{21} + T^{24} )^{2} \) |
| 31 | \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{6}( 1 + T + T^{2} + T^{3} + T^{4} )^{6} \) |
| 37 | \( ( 1 - T^{3} + T^{9} - T^{12} + T^{15} - T^{21} + T^{24} )( 1 + T^{3} - T^{9} - T^{12} - T^{15} + T^{21} + T^{24} ) \) |
| 41 | \( ( 1 - T + T^{3} - T^{4} + T^{5} - T^{7} + T^{8} )^{3}( 1 - T^{3} + T^{9} - T^{12} + T^{15} - T^{21} + T^{24} ) \) |
| 43 | \( ( 1 + T - T^{3} - T^{4} - T^{5} + T^{7} + T^{8} )^{3}( 1 + T^{3} - T^{9} - T^{12} - T^{15} + T^{21} + T^{24} ) \) |
| 47 | \( ( 1 - T^{3} + T^{6} )^{4}( 1 + T^{3} - T^{9} - T^{12} - T^{15} + T^{21} + T^{24} ) \) |
| 53 | \( ( 1 - T^{3} + T^{9} - T^{12} + T^{15} - T^{21} + T^{24} )( 1 + T^{3} - T^{9} - T^{12} - T^{15} + T^{21} + T^{24} ) \) |
| 59 | \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{6}( 1 + T + T^{2} + T^{3} + T^{4} )^{6} \) |
| 61 | \( ( 1 - T + T^{3} - T^{4} + T^{5} - T^{7} + T^{8} )^{3}( 1 - T^{3} + T^{9} - T^{12} + T^{15} - T^{21} + T^{24} ) \) |
| 67 | \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{6}( 1 + T - T^{3} - T^{4} - T^{5} + T^{7} + T^{8} )^{3} \) |
| 71 | \( ( 1 - T + T^{3} - T^{4} + T^{5} - T^{7} + T^{8} )^{3}( 1 + T - T^{3} - T^{4} - T^{5} + T^{7} + T^{8} )^{3} \) |
| 73 | \( ( 1 - T^{3} + T^{6} )^{4}( 1 + T^{3} + T^{6} )^{4} \) |
| 79 | \( ( 1 - T^{3} + T^{9} - T^{12} + T^{15} - T^{21} + T^{24} )( 1 + T^{3} - T^{9} - T^{12} - T^{15} + T^{21} + T^{24} ) \) |
| 83 | \( ( 1 + T^{3} - T^{9} - T^{12} - T^{15} + T^{21} + T^{24} )^{2} \) |
| 89 | \( ( 1 - T + T^{3} - T^{4} + T^{5} - T^{7} + T^{8} )^{3}( 1 - T^{3} + T^{9} - T^{12} + T^{15} - T^{21} + T^{24} ) \) |
| 97 | \( ( 1 - T^{3} + T^{9} - T^{12} + T^{15} - T^{21} + T^{24} )( 1 + T^{3} - T^{9} - T^{12} - T^{15} + T^{21} + T^{24} ) \) |
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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
−1.68250455473291768805099027230, −1.64084919936106449230104712741, −1.60998905762221953347227949818, −1.60712696702123553133435767750, −1.59649002387813853163787727308, −1.52935532956760452560763400680, −1.51751380728509135123464962760, −1.49684660252552202063979051502, −1.41831744440605824089144794619, −1.40785943744441735520353904583, −1.32809586603355764336835417078, −1.28907982548226851223185463157, −1.12229650853664067971727237086, −0.918708451955647259711458770550, −0.916485023980014134025995152554, −0.889082364214780483242060201355, −0.876084114073326286585695609624, −0.834158962971403494605669329040, −0.796272406823512472722899510459, −0.77698246645841822538652771116, −0.77430469053924771370771014175, −0.56025392672855759207227583780, −0.47019344937324784795721163511, −0.45416868822729459147581761814, −0.45161360091346675685309464460,
0.45161360091346675685309464460, 0.45416868822729459147581761814, 0.47019344937324784795721163511, 0.56025392672855759207227583780, 0.77430469053924771370771014175, 0.77698246645841822538652771116, 0.796272406823512472722899510459, 0.834158962971403494605669329040, 0.876084114073326286585695609624, 0.889082364214780483242060201355, 0.916485023980014134025995152554, 0.918708451955647259711458770550, 1.12229650853664067971727237086, 1.28907982548226851223185463157, 1.32809586603355764336835417078, 1.40785943744441735520353904583, 1.41831744440605824089144794619, 1.49684660252552202063979051502, 1.51751380728509135123464962760, 1.52935532956760452560763400680, 1.59649002387813853163787727308, 1.60712696702123553133435767750, 1.60998905762221953347227949818, 1.64084919936106449230104712741, 1.68250455473291768805099027230
Plot not available for L-functions of degree greater than 10.
