| L(s) = 1 | − 2-s + 3-s − 6-s − 7-s − 8-s + 9-s − 11-s + 14-s + 16-s − 18-s + 2·19-s − 21-s + 22-s − 24-s − 10·25-s − 29-s − 31-s − 33-s − 2·38-s + 42-s − 47-s + 48-s + 10·50-s + 56-s + 2·57-s + 58-s + 2·61-s + ⋯ |
| L(s) = 1 | − 2-s + 3-s − 6-s − 7-s − 8-s + 9-s − 11-s + 14-s + 16-s − 18-s + 2·19-s − 21-s + 22-s − 24-s − 10·25-s − 29-s − 31-s − 33-s − 2·38-s + 42-s − 47-s + 48-s + 10·50-s + 56-s + 2·57-s + 58-s + 2·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{40} \cdot 167^{20}\right)^{s/2} \, \Gamma_{\C}(s)^{20} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{40} \cdot 167^{20}\right)^{s/2} \, \Gamma_{\C}(s)^{20} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.03797477876\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.03797477876\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T + T^{3} - T^{4} + T^{6} - T^{7} + T^{9} - T^{10} + T^{11} - T^{13} + T^{14} - T^{16} + T^{17} - T^{19} + T^{20} \) |
| 167 | \( ( 1 + T + T^{2} )^{10} \) |
| good | 2 | \( ( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} )^{2}( 1 - T + T^{3} - T^{4} + T^{6} - T^{7} + T^{9} - T^{10} + T^{11} - T^{13} + T^{14} - T^{16} + T^{17} - T^{19} + T^{20} ) \) |
| 5 | \( ( 1 - T + T^{2} )^{10}( 1 + T + T^{2} )^{10} \) |
| 7 | \( ( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} )^{2}( 1 - T + T^{3} - T^{4} + T^{6} - T^{7} + T^{9} - T^{10} + T^{11} - T^{13} + T^{14} - T^{16} + T^{17} - T^{19} + T^{20} ) \) |
| 11 | \( ( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} )^{2}( 1 - T + T^{3} - T^{4} + T^{6} - T^{7} + T^{9} - T^{10} + T^{11} - T^{13} + T^{14} - T^{16} + T^{17} - T^{19} + T^{20} ) \) |
| 13 | \( ( 1 - T + T^{2} )^{10}( 1 + T + T^{2} )^{10} \) |
| 17 | \( ( 1 - T )^{20}( 1 + T )^{20} \) |
| 19 | \( ( 1 - T + T^{3} - T^{4} + T^{6} - T^{7} + T^{9} - T^{10} + T^{11} - T^{13} + T^{14} - T^{16} + T^{17} - T^{19} + T^{20} )^{2} \) |
| 23 | \( ( 1 - T + T^{2} )^{10}( 1 + T + T^{2} )^{10} \) |
| 29 | \( ( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} )^{2}( 1 - T + T^{3} - T^{4} + T^{6} - T^{7} + T^{9} - T^{10} + T^{11} - T^{13} + T^{14} - T^{16} + T^{17} - T^{19} + T^{20} ) \) |
| 31 | \( ( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} )^{2}( 1 - T + T^{3} - T^{4} + T^{6} - T^{7} + T^{9} - T^{10} + T^{11} - T^{13} + T^{14} - T^{16} + T^{17} - T^{19} + T^{20} ) \) |
| 37 | \( ( 1 - T )^{20}( 1 + T )^{20} \) |
| 41 | \( ( 1 - T + T^{2} )^{10}( 1 + T + T^{2} )^{10} \) |
| 43 | \( ( 1 - T + T^{2} )^{10}( 1 + T + T^{2} )^{10} \) |
| 47 | \( ( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} )^{2}( 1 - T + T^{3} - T^{4} + T^{6} - T^{7} + T^{9} - T^{10} + T^{11} - T^{13} + T^{14} - T^{16} + T^{17} - T^{19} + T^{20} ) \) |
| 53 | \( ( 1 - T )^{20}( 1 + T )^{20} \) |
| 59 | \( ( 1 - T + T^{2} )^{10}( 1 + T + T^{2} )^{10} \) |
| 61 | \( ( 1 - T + T^{3} - T^{4} + T^{6} - T^{7} + T^{9} - T^{10} + T^{11} - T^{13} + T^{14} - T^{16} + T^{17} - T^{19} + T^{20} )^{2} \) |
| 67 | \( ( 1 - T + T^{2} )^{10}( 1 + T + T^{2} )^{10} \) |
| 71 | \( ( 1 - T )^{20}( 1 + T )^{20} \) |
| 73 | \( ( 1 - T )^{20}( 1 + T )^{20} \) |
| 79 | \( ( 1 - T + T^{2} )^{10}( 1 + T + T^{2} )^{10} \) |
| 83 | \( ( 1 - T + T^{2} )^{10}( 1 + T + T^{2} )^{10} \) |
| 89 | \( ( 1 - T + T^{3} - T^{4} + T^{6} - T^{7} + T^{9} - T^{10} + T^{11} - T^{13} + T^{14} - T^{16} + T^{17} - T^{19} + T^{20} )^{2} \) |
| 97 | \( ( 1 - T + T^{3} - T^{4} + T^{6} - T^{7} + T^{9} - T^{10} + T^{11} - T^{13} + T^{14} - T^{16} + T^{17} - T^{19} + T^{20} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{40} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−2.17530743654655514447522587235, −2.14396614264764320861934669105, −2.12411195596493788798965879338, −2.12209605220003440653709383077, −2.11404663261894957064802076386, −2.10374374832451426694324189210, −2.09485811408538418310023457044, −2.04941161487163597099571154830, −1.78216389281521211174669203058, −1.70323546421535368409084072573, −1.60730189065608048597452288240, −1.59288647317470438617910280532, −1.56395743915187067037306007022, −1.54964037857960168108665446835, −1.45925005899777842929935703909, −1.40346178543609988657623059464, −1.37880262668379416661136073264, −1.20605069411994647896840475068, −1.01877374194905198550975234450, −0.878050014798597375235898463768, −0.825473545068127294897526248215, −0.78719637920725477006538219327, −0.65698921081762489730133640662, −0.39511257561634008756486624954, −0.15495342712190889598668668894,
0.15495342712190889598668668894, 0.39511257561634008756486624954, 0.65698921081762489730133640662, 0.78719637920725477006538219327, 0.825473545068127294897526248215, 0.878050014798597375235898463768, 1.01877374194905198550975234450, 1.20605069411994647896840475068, 1.37880262668379416661136073264, 1.40346178543609988657623059464, 1.45925005899777842929935703909, 1.54964037857960168108665446835, 1.56395743915187067037306007022, 1.59288647317470438617910280532, 1.60730189065608048597452288240, 1.70323546421535368409084072573, 1.78216389281521211174669203058, 2.04941161487163597099571154830, 2.09485811408538418310023457044, 2.10374374832451426694324189210, 2.11404663261894957064802076386, 2.12209605220003440653709383077, 2.12411195596493788798965879338, 2.14396614264764320861934669105, 2.17530743654655514447522587235
Plot not available for L-functions of degree greater than 10.
