| L(s) = 1 | − 2·2-s + 3·4-s − 2·8-s − 13·9-s + 10·13-s + 16-s + 2·17-s + 26·18-s + 2·25-s − 20·26-s − 4·34-s − 39·36-s + 2·49-s − 4·50-s + 30·52-s − 2·53-s + 6·68-s + 26·72-s + 91·81-s + 2·89-s − 4·98-s + 6·100-s − 4·101-s − 20·104-s + 4·106-s − 130·117-s + 8·121-s + ⋯ |
| L(s) = 1 | − 2·2-s + 3·4-s − 2·8-s − 13·9-s + 10·13-s + 16-s + 2·17-s + 26·18-s + 2·25-s − 20·26-s − 4·34-s − 39·36-s + 2·49-s − 4·50-s + 30·52-s − 2·53-s + 6·68-s + 26·72-s + 91·81-s + 2·89-s − 4·98-s + 6·100-s − 4·101-s − 20·104-s + 4·106-s − 130·117-s + 8·121-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 7^{48} \cdot 17^{24}\right)^{s/2} \, \Gamma_{\C}(s)^{24} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 7^{48} \cdot 17^{24}\right)^{s/2} \, \Gamma_{\C}(s)^{24} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.5604730263\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5604730263\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( ( 1 + T - T^{3} - T^{4} + T^{6} - T^{8} - T^{9} + T^{11} + T^{12} )^{2} \) |
| 7 | \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} )^{2} \) |
| 17 | \( ( 1 - T + T^{3} - T^{4} + T^{6} - T^{8} + T^{9} - T^{11} + T^{12} )^{2} \) |
| good | 3 | \( ( 1 + T^{2} )^{12}( 1 + T^{2} - T^{6} - T^{8} + T^{12} - T^{16} - T^{18} + T^{22} + T^{24} ) \) |
| 5 | \( ( 1 - T + T^{3} - T^{4} + T^{6} - T^{8} + T^{9} - T^{11} + T^{12} )^{2}( 1 + T - T^{3} - T^{4} + T^{6} - T^{8} - T^{9} + T^{11} + T^{12} )^{2} \) |
| 11 | \( ( 1 - T^{2} + T^{4} )^{6}( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} )^{2} \) |
| 13 | \( ( 1 - T + T^{2} )^{12}( 1 + T - T^{3} - T^{4} + T^{6} - T^{8} - T^{9} + T^{11} + T^{12} )^{2} \) |
| 19 | \( ( 1 - T + T^{2} )^{12}( 1 + T + T^{2} )^{12} \) |
| 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 | \( ( 1 + T^{2} - T^{6} - T^{8} + T^{12} - T^{16} - T^{18} + T^{22} + T^{24} )^{2} \) |
| 37 | \( ( 1 - T + T^{3} - T^{4} + T^{6} - T^{8} + T^{9} - T^{11} + T^{12} )^{2}( 1 + T - T^{3} - T^{4} + T^{6} - T^{8} - T^{9} + T^{11} + T^{12} )^{2} \) |
| 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 | \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )^{4}( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} )^{4} \) |
| 47 | \( ( 1 - T + T^{3} - T^{4} + T^{6} - T^{8} + T^{9} - T^{11} + T^{12} )^{2}( 1 + T - T^{3} - T^{4} + T^{6} - T^{8} - T^{9} + T^{11} + T^{12} )^{2} \) |
| 53 | \( ( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} )^{4}( 1 - T + T^{3} - T^{4} + T^{6} - T^{8} + T^{9} - T^{11} + T^{12} )^{2} \) |
| 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 | \( ( 1 - T + T^{3} - T^{4} + T^{6} - T^{8} + T^{9} - T^{11} + T^{12} )^{2}( 1 + T - T^{3} - T^{4} + T^{6} - T^{8} - T^{9} + T^{11} + T^{12} )^{2} \) |
| 67 | \( ( 1 - T + T^{2} )^{12}( 1 + T + T^{2} )^{12} \) |
| 71 | \( ( 1 + T^{2} - T^{6} - T^{8} + T^{12} - T^{16} - T^{18} + T^{22} + T^{24} )^{2} \) |
| 73 | \( ( 1 - T + T^{3} - T^{4} + T^{6} - T^{8} + T^{9} - T^{11} + T^{12} )^{2}( 1 + T - T^{3} - T^{4} + T^{6} - T^{8} - T^{9} + T^{11} + T^{12} )^{2} \) |
| 79 | \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} )^{2}( 1 + T^{2} - T^{6} - T^{8} + T^{12} - T^{16} - T^{18} + T^{22} + T^{24} ) \) |
| 83 | \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )^{4}( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} )^{4} \) |
| 89 | \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )^{4}( 1 + T - T^{3} - T^{4} + T^{6} - T^{8} - T^{9} + T^{11} + T^{12} )^{2} \) |
| 97 | \( ( 1 - T )^{24}( 1 + T )^{24} \) |
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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
−1.71231673097765192992542225312, −1.69633659702278923578541159927, −1.68348336970180104727869907784, −1.63815865515787329196836207150, −1.57287402742528229654710940885, −1.54004002385790361494389007272, −1.53432394649444765626658820204, −1.45199320801222298966065369871, −1.36233289576536783533925602750, −1.35483410091838821716398889191, −1.11637839363043069283824711332, −1.09895376787393161775256216249, −1.03774895530056787745488036532, −1.01103626063230301384191423904, −0.969150968131458083727713899063, −0.901598226761024685228963230808, −0.884543582747539532695854431845, −0.812594350950895735525040373257, −0.78403436064583527412418920883, −0.75281708539361017126704520878, −0.70386911688408942165290017403, −0.58968798134225306611113522994, −0.54431201849975845724625344343, −0.47456260015251222676412502069, −0.13920567056649789409934277120,
0.13920567056649789409934277120, 0.47456260015251222676412502069, 0.54431201849975845724625344343, 0.58968798134225306611113522994, 0.70386911688408942165290017403, 0.75281708539361017126704520878, 0.78403436064583527412418920883, 0.812594350950895735525040373257, 0.884543582747539532695854431845, 0.901598226761024685228963230808, 0.969150968131458083727713899063, 1.01103626063230301384191423904, 1.03774895530056787745488036532, 1.09895376787393161775256216249, 1.11637839363043069283824711332, 1.35483410091838821716398889191, 1.36233289576536783533925602750, 1.45199320801222298966065369871, 1.53432394649444765626658820204, 1.54004002385790361494389007272, 1.57287402742528229654710940885, 1.63815865515787329196836207150, 1.68348336970180104727869907784, 1.69633659702278923578541159927, 1.71231673097765192992542225312
Plot not available for L-functions of degree greater than 10.
