| L(s) = 1 | − 3·3-s + 486·5-s + 1.33e4·7-s − 1.80e4·9-s − 5.59e4·11-s + 1.58e5·13-s − 1.45e3·15-s − 6.29e5·17-s + 3.90e5·19-s − 3.99e4·21-s + 9.24e5·23-s − 5.21e6·25-s + 2.31e6·27-s − 9.83e6·29-s − 1.36e6·31-s + 1.67e5·33-s + 6.47e6·35-s − 2.28e6·37-s − 4.74e5·39-s − 1.28e7·41-s + 2.23e7·43-s − 8.78e6·45-s − 5.88e7·47-s + 3.92e7·49-s + 1.88e6·51-s + 8.77e6·53-s − 2.72e7·55-s + ⋯ |
| L(s) = 1 | − 0.0213·3-s + 0.347·5-s + 2.09·7-s − 0.918·9-s − 1.15·11-s + 1.53·13-s − 0.00743·15-s − 1.82·17-s + 0.688·19-s − 0.0448·21-s + 0.688·23-s − 2.66·25-s + 0.836·27-s − 2.58·29-s − 0.265·31-s + 0.0246·33-s + 0.729·35-s − 0.200·37-s − 0.0328·39-s − 0.712·41-s + 0.998·43-s − 0.319·45-s − 1.76·47-s + 0.973·49-s + 0.0390·51-s + 0.152·53-s − 0.400·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 28094464 ^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr =\mathstrut & -\, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 28094464 ^{s/2} \, \Gamma_{\C}(s+9/2)^{3} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{11}{2})\) |
|
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 | | \( 1 \) |
| 19 | $C_1$ | \( ( 1 - p^{4} T )^{3} \) |
| good | 3 | $S_4\times C_2$ | \( 1 + p T + 2009 p^{2} T^{2} - 81574 p^{3} T^{3} + 2009 p^{11} T^{4} + p^{19} T^{5} + p^{27} T^{6} \) |
| 5 | $S_4\times C_2$ | \( 1 - 486 T + 5450544 T^{2} - 1910152084 T^{3} + 5450544 p^{9} T^{4} - 486 p^{18} T^{5} + p^{27} T^{6} \) |
| 7 | $S_4\times C_2$ | \( 1 - 13317 T + 2817432 p^{2} T^{2} - 20382956301 p^{2} T^{3} + 2817432 p^{11} T^{4} - 13317 p^{18} T^{5} + p^{27} T^{6} \) |
| 11 | $S_4\times C_2$ | \( 1 + 5088 p T + 162303954 p T^{2} + 110373793234486 T^{3} + 162303954 p^{10} T^{4} + 5088 p^{19} T^{5} + p^{27} T^{6} \) |
| 13 | $S_4\times C_2$ | \( 1 - 158181 T + 36345696879 T^{2} - 3315103030674770 T^{3} + 36345696879 p^{9} T^{4} - 158181 p^{18} T^{5} + p^{27} T^{6} \) |
| 17 | $S_4\times C_2$ | \( 1 + 629091 T + 322401568650 T^{2} + 106985872063124811 T^{3} + 322401568650 p^{9} T^{4} + 629091 p^{18} T^{5} + p^{27} T^{6} \) |
| 23 | $S_4\times C_2$ | \( 1 - 924627 T + 42546695811 p T^{2} - 526819541184348842 T^{3} + 42546695811 p^{10} T^{4} - 924627 p^{18} T^{5} + p^{27} T^{6} \) |
| 29 | $S_4\times C_2$ | \( 1 + 9839019 T + 64313088760239 T^{2} + \)\(29\!\cdots\!62\)\( T^{3} + 64313088760239 p^{9} T^{4} + 9839019 p^{18} T^{5} + p^{27} T^{6} \) |
| 31 | $S_4\times C_2$ | \( 1 + 1364628 T + 6528423117837 T^{2} + \)\(10\!\cdots\!36\)\( T^{3} + 6528423117837 p^{9} T^{4} + 1364628 p^{18} T^{5} + p^{27} T^{6} \) |
| 37 | $S_4\times C_2$ | \( 1 + 2289090 T + 309609164810043 T^{2} + \)\(75\!\cdots\!08\)\( T^{3} + 309609164810043 p^{9} T^{4} + 2289090 p^{18} T^{5} + p^{27} T^{6} \) |
| 41 | $S_4\times C_2$ | \( 1 + 12899580 T + 933941039944047 T^{2} + \)\(77\!\cdots\!84\)\( T^{3} + 933941039944047 p^{9} T^{4} + 12899580 p^{18} T^{5} + p^{27} T^{6} \) |
| 43 | $S_4\times C_2$ | \( 1 - 22378638 T + 1149456564786162 T^{2} - \)\(23\!\cdots\!16\)\( T^{3} + 1149456564786162 p^{9} T^{4} - 22378638 p^{18} T^{5} + p^{27} T^{6} \) |
| 47 | $S_4\times C_2$ | \( 1 + 58896366 T + 3778439737379502 T^{2} + \)\(11\!\cdots\!64\)\( T^{3} + 3778439737379502 p^{9} T^{4} + 58896366 p^{18} T^{5} + p^{27} T^{6} \) |
| 53 | $S_4\times C_2$ | \( 1 - 8770629 T + 2855843385985791 T^{2} + \)\(18\!\cdots\!78\)\( T^{3} + 2855843385985791 p^{9} T^{4} - 8770629 p^{18} T^{5} + p^{27} T^{6} \) |
| 59 | $S_4\times C_2$ | \( 1 + 16426299 T + 24576394162259001 T^{2} + \)\(43\!\cdots\!10\)\( p T^{3} + 24576394162259001 p^{9} T^{4} + 16426299 p^{18} T^{5} + p^{27} T^{6} \) |
| 61 | $S_4\times C_2$ | \( 1 + 126843780 T + 39342501567258564 T^{2} + \)\(30\!\cdots\!18\)\( T^{3} + 39342501567258564 p^{9} T^{4} + 126843780 p^{18} T^{5} + p^{27} T^{6} \) |
| 67 | $S_4\times C_2$ | \( 1 - 288075309 T + 92688504651310905 T^{2} - \)\(15\!\cdots\!54\)\( T^{3} + 92688504651310905 p^{9} T^{4} - 288075309 p^{18} T^{5} + p^{27} T^{6} \) |
| 71 | $S_4\times C_2$ | \( 1 + 78122274 T + 123167736240843729 T^{2} + \)\(73\!\cdots\!48\)\( T^{3} + 123167736240843729 p^{9} T^{4} + 78122274 p^{18} T^{5} + p^{27} T^{6} \) |
| 73 | $S_4\times C_2$ | \( 1 + 557941845 T + 181729649630455470 T^{2} + \)\(43\!\cdots\!53\)\( T^{3} + 181729649630455470 p^{9} T^{4} + 557941845 p^{18} T^{5} + p^{27} T^{6} \) |
| 79 | $S_4\times C_2$ | \( 1 - 320222022 T + 382092522680784861 T^{2} - \)\(76\!\cdots\!36\)\( T^{3} + 382092522680784861 p^{9} T^{4} - 320222022 p^{18} T^{5} + p^{27} T^{6} \) |
| 83 | $S_4\times C_2$ | \( 1 + 430491462 T + 580489855508328153 T^{2} + \)\(15\!\cdots\!28\)\( T^{3} + 580489855508328153 p^{9} T^{4} + 430491462 p^{18} T^{5} + p^{27} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 - 437689644 T + 775511286211782651 T^{2} - \)\(33\!\cdots\!72\)\( T^{3} + 775511286211782651 p^{9} T^{4} - 437689644 p^{18} T^{5} + p^{27} T^{6} \) |
| 97 | $S_4\times C_2$ | \( 1 + 384952146 T + 823894073491956771 T^{2} + \)\(18\!\cdots\!72\)\( T^{3} + 823894073491956771 p^{9} T^{4} + 384952146 p^{18} T^{5} + p^{27} T^{6} \) |
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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
−9.410482204117489619450120186631, −8.762883188109152266818828113462, −8.695183102923497423849332912142, −8.625173626941770178646861860564, −7.963154638604528357513149035261, −7.78565710847215670737485811206, −7.75673546725043661331658236416, −7.22683943106290994498446640956, −6.74651059566235257740774503896, −6.45561929438944279370373692768, −5.80261486038264291643386361073, −5.67906628799293007328118720988, −5.66921584938113332497274919926, −4.90920894277287068067574674774, −4.87208641118675051084632606761, −4.47820270625196236892403912919, −3.96264392647228253086341911481, −3.50167099145132836533283411681, −3.42780528488184082166071824905, −2.59011535316144148392973321829, −2.43347923564431516901543027955, −1.91755321804519125412657613028, −1.69665684215440275897273433312, −1.38939146576847322176011743322, −1.00582399824836014698384334355, 0, 0, 0,
1.00582399824836014698384334355, 1.38939146576847322176011743322, 1.69665684215440275897273433312, 1.91755321804519125412657613028, 2.43347923564431516901543027955, 2.59011535316144148392973321829, 3.42780528488184082166071824905, 3.50167099145132836533283411681, 3.96264392647228253086341911481, 4.47820270625196236892403912919, 4.87208641118675051084632606761, 4.90920894277287068067574674774, 5.66921584938113332497274919926, 5.67906628799293007328118720988, 5.80261486038264291643386361073, 6.45561929438944279370373692768, 6.74651059566235257740774503896, 7.22683943106290994498446640956, 7.75673546725043661331658236416, 7.78565710847215670737485811206, 7.963154638604528357513149035261, 8.625173626941770178646861860564, 8.695183102923497423849332912142, 8.762883188109152266818828113462, 9.410482204117489619450120186631
