| L(s) = 1 | + 8·5-s + 24·7-s + 64·11-s + 52·13-s + 80·17-s + 24·19-s + 16·23-s − 228·25-s + 240·29-s + 24·31-s + 192·35-s − 152·37-s + 136·41-s − 64·43-s + 528·47-s − 260·49-s + 496·53-s + 512·55-s + 832·59-s − 104·61-s + 416·65-s + 1.00e3·67-s + 1.58e3·71-s − 232·73-s + 1.53e3·77-s + 464·79-s + 1.88e3·83-s + ⋯ |
| L(s) = 1 | + 0.715·5-s + 1.29·7-s + 1.75·11-s + 1.10·13-s + 1.14·17-s + 0.289·19-s + 0.145·23-s − 1.82·25-s + 1.53·29-s + 0.139·31-s + 0.927·35-s − 0.675·37-s + 0.518·41-s − 0.226·43-s + 1.63·47-s − 0.758·49-s + 1.28·53-s + 1.25·55-s + 1.83·59-s − 0.218·61-s + 0.793·65-s + 1.82·67-s + 2.64·71-s − 0.371·73-s + 2.27·77-s + 0.660·79-s + 2.49·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{8} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{8} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(24.78679893\) |
| \(L(\frac12)\) |
\(\approx\) |
\(24.78679893\) |
| \(L(\frac{5}{2})\) |
|
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 | | \( 1 \) |
| 13 | $C_1$ | \( ( 1 - p T )^{4} \) |
| good | 5 | $C_2 \wr S_4$ | \( 1 - 8 T + 292 T^{2} - 664 p T^{3} + 43158 T^{4} - 664 p^{4} T^{5} + 292 p^{6} T^{6} - 8 p^{9} T^{7} + p^{12} T^{8} \) |
| 7 | $C_2 \wr S_4$ | \( 1 - 24 T + 836 T^{2} - 17240 T^{3} + 384614 T^{4} - 17240 p^{3} T^{5} + 836 p^{6} T^{6} - 24 p^{9} T^{7} + p^{12} T^{8} \) |
| 11 | $C_2 \wr S_4$ | \( 1 - 64 T + 5812 T^{2} - 22272 p T^{3} + 11826038 T^{4} - 22272 p^{4} T^{5} + 5812 p^{6} T^{6} - 64 p^{9} T^{7} + p^{12} T^{8} \) |
| 17 | $C_2 \wr S_4$ | \( 1 - 80 T + 13604 T^{2} - 1085680 T^{3} + 91125254 T^{4} - 1085680 p^{3} T^{5} + 13604 p^{6} T^{6} - 80 p^{9} T^{7} + p^{12} T^{8} \) |
| 19 | $C_2 \wr S_4$ | \( 1 - 24 T + 924 p T^{2} + 166664 T^{3} + 135130134 T^{4} + 166664 p^{3} T^{5} + 924 p^{7} T^{6} - 24 p^{9} T^{7} + p^{12} T^{8} \) |
| 23 | $C_2 \wr S_4$ | \( 1 - 16 T + 1124 p T^{2} + 1198000 T^{3} + 303678566 T^{4} + 1198000 p^{3} T^{5} + 1124 p^{7} T^{6} - 16 p^{9} T^{7} + p^{12} T^{8} \) |
| 29 | $C_2 \wr S_4$ | \( 1 - 240 T + 46580 T^{2} - 1616656 T^{3} + 222480726 T^{4} - 1616656 p^{3} T^{5} + 46580 p^{6} T^{6} - 240 p^{9} T^{7} + p^{12} T^{8} \) |
| 31 | $C_2 \wr S_4$ | \( 1 - 24 T + 74596 T^{2} - 1338520 T^{3} + 3095804166 T^{4} - 1338520 p^{3} T^{5} + 74596 p^{6} T^{6} - 24 p^{9} T^{7} + p^{12} T^{8} \) |
| 37 | $C_2 \wr S_4$ | \( 1 + 152 T + 108364 T^{2} + 19657480 T^{3} + 6426482870 T^{4} + 19657480 p^{3} T^{5} + 108364 p^{6} T^{6} + 152 p^{9} T^{7} + p^{12} T^{8} \) |
| 41 | $C_2 \wr S_4$ | \( 1 - 136 T + 238004 T^{2} - 25382680 T^{3} + 23453322054 T^{4} - 25382680 p^{3} T^{5} + 238004 p^{6} T^{6} - 136 p^{9} T^{7} + p^{12} T^{8} \) |
| 43 | $C_2 \wr S_4$ | \( 1 + 64 T + 124940 T^{2} - 36824000 T^{3} + 3430262006 T^{4} - 36824000 p^{3} T^{5} + 124940 p^{6} T^{6} + 64 p^{9} T^{7} + p^{12} T^{8} \) |
| 47 | $C_2 \wr S_4$ | \( 1 - 528 T + 461156 T^{2} - 164452432 T^{3} + 74136791814 T^{4} - 164452432 p^{3} T^{5} + 461156 p^{6} T^{6} - 528 p^{9} T^{7} + p^{12} T^{8} \) |
| 53 | $C_2 \wr S_4$ | \( 1 - 496 T + 125044 T^{2} - 50463120 T^{3} + 37381353590 T^{4} - 50463120 p^{3} T^{5} + 125044 p^{6} T^{6} - 496 p^{9} T^{7} + p^{12} T^{8} \) |
| 59 | $C_2 \wr S_4$ | \( 1 - 832 T + 561908 T^{2} - 347780224 T^{3} + 186629122230 T^{4} - 347780224 p^{3} T^{5} + 561908 p^{6} T^{6} - 832 p^{9} T^{7} + p^{12} T^{8} \) |
| 61 | $C_2 \wr S_4$ | \( 1 + 104 T + 554732 T^{2} - 13810248 T^{3} + 151346444950 T^{4} - 13810248 p^{3} T^{5} + 554732 p^{6} T^{6} + 104 p^{9} T^{7} + p^{12} T^{8} \) |
| 67 | $C_2 \wr S_4$ | \( 1 - 1000 T + 576596 T^{2} - 440604296 T^{3} + 335528696406 T^{4} - 440604296 p^{3} T^{5} + 576596 p^{6} T^{6} - 1000 p^{9} T^{7} + p^{12} T^{8} \) |
| 71 | $C_2 \wr S_4$ | \( 1 - 1584 T + 2079428 T^{2} - 1769394672 T^{3} + 1230328599846 T^{4} - 1769394672 p^{3} T^{5} + 2079428 p^{6} T^{6} - 1584 p^{9} T^{7} + p^{12} T^{8} \) |
| 73 | $C_2 \wr S_4$ | \( 1 + 232 T + 499004 T^{2} + 153948312 T^{3} + 348409025510 T^{4} + 153948312 p^{3} T^{5} + 499004 p^{6} T^{6} + 232 p^{9} T^{7} + p^{12} T^{8} \) |
| 79 | $C_2 \wr S_4$ | \( 1 - 464 T + 1809180 T^{2} - 694995088 T^{3} + 1297758133510 T^{4} - 694995088 p^{3} T^{5} + 1809180 p^{6} T^{6} - 464 p^{9} T^{7} + p^{12} T^{8} \) |
| 83 | $C_2 \wr S_4$ | \( 1 - 1888 T + 3261012 T^{2} - 3236627872 T^{3} + 3005103527254 T^{4} - 3236627872 p^{3} T^{5} + 3261012 p^{6} T^{6} - 1888 p^{9} T^{7} + p^{12} T^{8} \) |
| 89 | $C_2 \wr S_4$ | \( 1 - 696 T + 1587188 T^{2} - 284316712 T^{3} + 1030022604294 T^{4} - 284316712 p^{3} T^{5} + 1587188 p^{6} T^{6} - 696 p^{9} T^{7} + p^{12} T^{8} \) |
| 97 | $C_2 \wr S_4$ | \( 1 + 1384 T + 3636572 T^{2} + 3308513752 T^{3} + 4856816482886 T^{4} + 3308513752 p^{3} T^{5} + 3636572 p^{6} T^{6} + 1384 p^{9} T^{7} + p^{12} T^{8} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−6.91960115631296674853282942340, −6.39127793509568987511955750715, −6.27206908632830761245517557884, −6.22283948171720820048647484808, −6.03983799561361520583713113091, −5.51124411070281282600756941149, −5.38162199360294807574451485343, −5.30539236484873130123925566431, −5.14949637852563971892128286567, −4.61910207589177802904177553471, −4.28472526226685755765029436404, −4.26385576286615525660430149851, −4.11386551824491915649282197097, −3.53880457819101299164598473982, −3.43111341708932709763242459306, −3.28391896076351167366701656373, −3.05618424305418570402623088123, −2.17270381538392888027719614275, −2.14009417788221070099622618803, −1.96796826112039404786676908803, −1.87455464236533134018589203407, −1.14304356090261569639856889461, −0.942739226360293276552620262783, −0.75997603083148103787568162006, −0.61414330926368696447925532216,
0.61414330926368696447925532216, 0.75997603083148103787568162006, 0.942739226360293276552620262783, 1.14304356090261569639856889461, 1.87455464236533134018589203407, 1.96796826112039404786676908803, 2.14009417788221070099622618803, 2.17270381538392888027719614275, 3.05618424305418570402623088123, 3.28391896076351167366701656373, 3.43111341708932709763242459306, 3.53880457819101299164598473982, 4.11386551824491915649282197097, 4.26385576286615525660430149851, 4.28472526226685755765029436404, 4.61910207589177802904177553471, 5.14949637852563971892128286567, 5.30539236484873130123925566431, 5.38162199360294807574451485343, 5.51124411070281282600756941149, 6.03983799561361520583713113091, 6.22283948171720820048647484808, 6.27206908632830761245517557884, 6.39127793509568987511955750715, 6.91960115631296674853282942340
