| L(s) = 1 | − 2·2-s + 2·3-s + 9·4-s − 4·6-s − 14·8-s + 3·9-s + 18·12-s + 34·16-s − 6·18-s − 28·24-s − 4·25-s + 2·27-s − 2·31-s − 42·32-s + 27·36-s + 2·41-s − 2·47-s + 68·48-s − 4·49-s + 8·50-s − 4·54-s + 4·62-s + 65·64-s − 42·72-s − 2·73-s − 8·75-s + 81-s + ⋯ |
| L(s) = 1 | − 2·2-s + 2·3-s + 9·4-s − 4·6-s − 14·8-s + 3·9-s + 18·12-s + 34·16-s − 6·18-s − 28·24-s − 4·25-s + 2·27-s − 2·31-s − 42·32-s + 27·36-s + 2·41-s − 2·47-s + 68·48-s − 4·49-s + 8·50-s − 4·54-s + 4·62-s + 65·64-s − 42·72-s − 2·73-s − 8·75-s + 81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{24} \cdot 23^{24} \cdot 29^{24}\right)^{s/2} \, \Gamma_{\C}(s)^{24} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{24} \cdot 23^{24} \cdot 29^{24}\right)^{s/2} \, \Gamma_{\C}(s)^{24} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.06735506523\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.06735506523\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( ( 1 - T + T^{3} - T^{4} + T^{6} - T^{8} + T^{9} - T^{11} + T^{12} )^{2} \) |
| 23 | \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} )^{2} \) |
| 29 | \( 1 + T^{2} - T^{6} - T^{8} + T^{12} - T^{16} - T^{18} + T^{22} + T^{24} \) |
| good | 2 | \( ( 1 - T^{2} + T^{4} )^{6}( 1 + T - T^{3} - T^{4} + T^{6} - T^{8} - T^{9} + T^{11} + T^{12} )^{2} \) |
| 5 | \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )^{4}( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} )^{4} \) |
| 7 | \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )^{4}( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} )^{4} \) |
| 11 | \( ( 1 - T^{4} + T^{8} - T^{12} + T^{16} - T^{20} + T^{24} )^{2} \) |
| 13 | \( ( 1 + T^{2} - T^{6} - T^{8} + T^{12} - T^{16} - T^{18} + T^{22} + T^{24} )^{2} \) |
| 17 | \( ( 1 + T^{4} )^{12} \) |
| 19 | \( ( 1 - T^{4} + T^{8} - T^{12} + T^{16} - T^{20} + T^{24} )^{2} \) |
| 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 | \( ( 1 - T^{4} + T^{8} - T^{12} + T^{16} - T^{20} + T^{24} )^{2} \) |
| 41 | \( ( 1 - T + T^{3} - T^{4} + T^{6} - T^{8} + T^{9} - T^{11} + T^{12} )^{2}( 1 + T^{2} - T^{6} - T^{8} + T^{12} - T^{16} - T^{18} + T^{22} + T^{24} ) \) |
| 43 | \( ( 1 - T^{4} + T^{8} - T^{12} + T^{16} - T^{20} + T^{24} )^{2} \) |
| 47 | \( ( 1 + T - T^{3} - T^{4} + T^{6} - T^{8} - T^{9} + T^{11} + T^{12} )^{2}( 1 + T^{2} - T^{6} - T^{8} + T^{12} - T^{16} - T^{18} + T^{22} + T^{24} ) \) |
| 53 | \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} )^{4} \) |
| 59 | \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} )^{4} \) |
| 61 | \( ( 1 - T^{4} + T^{8} - T^{12} + T^{16} - T^{20} + T^{24} )^{2} \) |
| 67 | \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} )^{4} \) |
| 71 | \( ( 1 + T^{2} - T^{6} - T^{8} + T^{12} - T^{16} - T^{18} + T^{22} + T^{24} )^{2} \) |
| 73 | \( ( 1 + T - T^{3} - T^{4} + T^{6} - T^{8} - T^{9} + T^{11} + T^{12} )^{2}( 1 + T^{2} - T^{6} - T^{8} + T^{12} - T^{16} - T^{18} + T^{22} + T^{24} ) \) |
| 79 | \( ( 1 - T^{4} + T^{8} - T^{12} + T^{16} - T^{20} + T^{24} )^{2} \) |
| 83 | \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} )^{4} \) |
| 89 | \( ( 1 - T^{4} + T^{8} - T^{12} + T^{16} - T^{20} + T^{24} )^{2} \) |
| 97 | \( ( 1 - T^{4} + T^{8} - T^{12} + T^{16} - T^{20} + T^{24} )^{2} \) |
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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
−1.99534114763329552592234682278, −1.99016940123758317434931147849, −1.92066128757736793348212094665, −1.88205690466628513365784688854, −1.88102875090444196826947201403, −1.72143489761037131323511587508, −1.63664517257999090804666446787, −1.61786924811916406230213698516, −1.47971851647448996515718058460, −1.47227113317181140138268875290, −1.45365314213183018153841147944, −1.33804740205544358664665139221, −1.32291495589009466147394620293, −1.30561391527506769766683580055, −1.29090988496769093430522470024, −1.24253000687596509081482811039, −1.22837660014012224975507460414, −1.21234371746410461182917854769, −1.19991699225866404565140561636, −1.08856002052976390898478681196, −0.838728400105434187613977298708, −0.819071870434695818069593075108, −0.54848036853844730473019349729, −0.34728250528033734823058044462, −0.03593774278688144680402585589,
0.03593774278688144680402585589, 0.34728250528033734823058044462, 0.54848036853844730473019349729, 0.819071870434695818069593075108, 0.838728400105434187613977298708, 1.08856002052976390898478681196, 1.19991699225866404565140561636, 1.21234371746410461182917854769, 1.22837660014012224975507460414, 1.24253000687596509081482811039, 1.29090988496769093430522470024, 1.30561391527506769766683580055, 1.32291495589009466147394620293, 1.33804740205544358664665139221, 1.45365314213183018153841147944, 1.47227113317181140138268875290, 1.47971851647448996515718058460, 1.61786924811916406230213698516, 1.63664517257999090804666446787, 1.72143489761037131323511587508, 1.88102875090444196826947201403, 1.88205690466628513365784688854, 1.92066128757736793348212094665, 1.99016940123758317434931147849, 1.99534114763329552592234682278
Plot not available for L-functions of degree greater than 10.
