| L(s) = 1 | − 15·5-s + 6·7-s − 12·11-s + 18·13-s + 21·17-s + 57·19-s − 87·23-s + 150·25-s − 138·29-s + 117·31-s − 90·35-s + 150·37-s + 180·43-s − 684·47-s − 537·49-s − 87·53-s + 180·55-s − 714·59-s − 513·61-s − 270·65-s − 174·67-s − 768·71-s − 252·73-s − 72·77-s + 207·79-s − 1.68e3·83-s − 315·85-s + ⋯ |
| L(s) = 1 | − 1.34·5-s + 0.323·7-s − 0.328·11-s + 0.384·13-s + 0.299·17-s + 0.688·19-s − 0.788·23-s + 6/5·25-s − 0.883·29-s + 0.677·31-s − 0.434·35-s + 0.666·37-s + 0.638·43-s − 2.12·47-s − 1.56·49-s − 0.225·53-s + 0.441·55-s − 1.57·59-s − 1.07·61-s − 0.515·65-s − 0.317·67-s − 1.28·71-s − 0.404·73-s − 0.106·77-s + 0.294·79-s − 2.23·83-s − 0.401·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{9} \cdot 3^{9} \cdot 5^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{9} \cdot 3^{9} \cdot 5^{3}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 3 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 + p T )^{3} \) |
| good | 7 | $S_4\times C_2$ | \( 1 - 6 T + 573 T^{2} - 20 p^{3} T^{3} + 573 p^{3} T^{4} - 6 p^{6} T^{5} + p^{9} T^{6} \) |
| 11 | $S_4\times C_2$ | \( 1 + 12 T + 2733 T^{2} + 11024 T^{3} + 2733 p^{3} T^{4} + 12 p^{6} T^{5} + p^{9} T^{6} \) |
| 13 | $S_4\times C_2$ | \( 1 - 18 T + 5391 T^{2} - 55708 T^{3} + 5391 p^{3} T^{4} - 18 p^{6} T^{5} + p^{9} T^{6} \) |
| 17 | $S_4\times C_2$ | \( 1 - 21 T + 7458 T^{2} - 253565 T^{3} + 7458 p^{3} T^{4} - 21 p^{6} T^{5} + p^{9} T^{6} \) |
| 19 | $S_4\times C_2$ | \( 1 - 3 p T + 8592 T^{2} - 1114405 T^{3} + 8592 p^{3} T^{4} - 3 p^{7} T^{5} + p^{9} T^{6} \) |
| 23 | $S_4\times C_2$ | \( 1 + 87 T + 28512 T^{2} + 1461623 T^{3} + 28512 p^{3} T^{4} + 87 p^{6} T^{5} + p^{9} T^{6} \) |
| 29 | $S_4\times C_2$ | \( 1 + 138 T + 20103 T^{2} + 123916 p T^{3} + 20103 p^{3} T^{4} + 138 p^{6} T^{5} + p^{9} T^{6} \) |
| 31 | $S_4\times C_2$ | \( 1 - 117 T + 20988 T^{2} + 2423207 T^{3} + 20988 p^{3} T^{4} - 117 p^{6} T^{5} + p^{9} T^{6} \) |
| 37 | $S_4\times C_2$ | \( 1 - 150 T + 95907 T^{2} - 17945796 T^{3} + 95907 p^{3} T^{4} - 150 p^{6} T^{5} + p^{9} T^{6} \) |
| 41 | $S_4\times C_2$ | \( 1 + 68055 T^{2} - 17332488 T^{3} + 68055 p^{3} T^{4} + p^{9} T^{6} \) |
| 43 | $S_4\times C_2$ | \( 1 - 180 T + 22293 T^{2} + 3328688 T^{3} + 22293 p^{3} T^{4} - 180 p^{6} T^{5} + p^{9} T^{6} \) |
| 47 | $S_4\times C_2$ | \( 1 + 684 T + 347853 T^{2} + 110757224 T^{3} + 347853 p^{3} T^{4} + 684 p^{6} T^{5} + p^{9} T^{6} \) |
| 53 | $S_4\times C_2$ | \( 1 + 87 T + 277926 T^{2} - 2139897 T^{3} + 277926 p^{3} T^{4} + 87 p^{6} T^{5} + p^{9} T^{6} \) |
| 59 | $S_4\times C_2$ | \( 1 + 714 T + 669321 T^{2} + 272637508 T^{3} + 669321 p^{3} T^{4} + 714 p^{6} T^{5} + p^{9} T^{6} \) |
| 61 | $S_4\times C_2$ | \( 1 + 513 T + 466794 T^{2} + 175503629 T^{3} + 466794 p^{3} T^{4} + 513 p^{6} T^{5} + p^{9} T^{6} \) |
| 67 | $S_4\times C_2$ | \( 1 + 174 T + 677229 T^{2} + 48960124 T^{3} + 677229 p^{3} T^{4} + 174 p^{6} T^{5} + p^{9} T^{6} \) |
| 71 | $S_4\times C_2$ | \( 1 + 768 T + 869433 T^{2} + 382859112 T^{3} + 869433 p^{3} T^{4} + 768 p^{6} T^{5} + p^{9} T^{6} \) |
| 73 | $S_4\times C_2$ | \( 1 + 252 T + 967647 T^{2} + 141584400 T^{3} + 967647 p^{3} T^{4} + 252 p^{6} T^{5} + p^{9} T^{6} \) |
| 79 | $S_4\times C_2$ | \( 1 - 207 T + 981660 T^{2} - 306571995 T^{3} + 981660 p^{3} T^{4} - 207 p^{6} T^{5} + p^{9} T^{6} \) |
| 83 | $S_4\times C_2$ | \( 1 + 1689 T + 2368140 T^{2} + 1880402401 T^{3} + 2368140 p^{3} T^{4} + 1689 p^{6} T^{5} + p^{9} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 + 312 T + 1492215 T^{2} + 319203960 T^{3} + 1492215 p^{3} T^{4} + 312 p^{6} T^{5} + p^{9} T^{6} \) |
| 97 | $S_4\times C_2$ | \( 1 + 1080 T + 1410147 T^{2} + 2105654384 T^{3} + 1410147 p^{3} T^{4} + 1080 p^{6} T^{5} + p^{9} T^{6} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{6} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.682647081309511959101621614569, −8.391060083245493301120027648766, −8.224105947794659174390729348985, −8.037881906616622396619564626332, −7.60223600228903689059165489150, −7.57814896900267243031974835516, −7.39708394952974130625071613689, −6.66697153965319629591050197700, −6.62420315222244241818644742127, −6.47827803710107923763731921809, −5.81608789978201845783401510059, −5.60712167993410584464833593888, −5.51633475074722403115979584822, −4.78737943043371031191158694242, −4.68350837537307571290715642326, −4.55460112202936376234151788747, −3.91745810339942319399504836320, −3.77875790195348105766777160890, −3.51959144883890842562687995851, −2.91353802485379134713463961787, −2.74381412182205848640981571588, −2.51157914312916136308767971296, −1.49158526900901127653196770842, −1.34616444004392114695325295248, −1.29590027269441164641487579845, 0, 0, 0,
1.29590027269441164641487579845, 1.34616444004392114695325295248, 1.49158526900901127653196770842, 2.51157914312916136308767971296, 2.74381412182205848640981571588, 2.91353802485379134713463961787, 3.51959144883890842562687995851, 3.77875790195348105766777160890, 3.91745810339942319399504836320, 4.55460112202936376234151788747, 4.68350837537307571290715642326, 4.78737943043371031191158694242, 5.51633475074722403115979584822, 5.60712167993410584464833593888, 5.81608789978201845783401510059, 6.47827803710107923763731921809, 6.62420315222244241818644742127, 6.66697153965319629591050197700, 7.39708394952974130625071613689, 7.57814896900267243031974835516, 7.60223600228903689059165489150, 8.037881906616622396619564626332, 8.224105947794659174390729348985, 8.391060083245493301120027648766, 8.682647081309511959101621614569
