| L(s) = 1 | − 6·2-s + 5·3-s + 24·4-s + 5-s − 30·6-s + 26·7-s − 80·8-s − 9-s − 6·10-s + 4·11-s + 120·12-s − 129·13-s − 156·14-s + 5·15-s + 240·16-s + 51·17-s + 6·18-s + 24·20-s + 130·21-s − 24·22-s − 47·23-s − 400·24-s − 18·25-s + 774·26-s + 28·27-s + 624·28-s + 125·29-s + ⋯ |
| L(s) = 1 | − 2.12·2-s + 0.962·3-s + 3·4-s + 0.0894·5-s − 2.04·6-s + 1.40·7-s − 3.53·8-s − 0.0370·9-s − 0.189·10-s + 0.109·11-s + 2.88·12-s − 2.75·13-s − 2.97·14-s + 0.0860·15-s + 15/4·16-s + 0.727·17-s + 0.0785·18-s + 0.268·20-s + 1.35·21-s − 0.232·22-s − 0.426·23-s − 3.40·24-s − 0.143·25-s + 5.83·26-s + 0.199·27-s + 4.21·28-s + 0.800·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{3} \cdot 19^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{3} \cdot 19^{6}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.7786897229\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7786897229\) |
| \(L(\frac{5}{2})\) |
|
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 | $C_1$ | \( ( 1 + p T )^{3} \) |
| 19 | | \( 1 \) |
| good | 3 | $S_4\times C_2$ | \( 1 - 5 T + 26 T^{2} - 163 T^{3} + 26 p^{3} T^{4} - 5 p^{6} T^{5} + p^{9} T^{6} \) |
| 5 | $S_4\times C_2$ | \( 1 - T + 19 T^{2} + 1688 T^{3} + 19 p^{3} T^{4} - p^{6} T^{5} + p^{9} T^{6} \) |
| 7 | $S_4\times C_2$ | \( 1 - 26 T + 1191 T^{2} - 17932 T^{3} + 1191 p^{3} T^{4} - 26 p^{6} T^{5} + p^{9} T^{6} \) |
| 11 | $S_4\times C_2$ | \( 1 - 4 T + 62 p T^{2} + 39332 T^{3} + 62 p^{4} T^{4} - 4 p^{6} T^{5} + p^{9} T^{6} \) |
| 13 | $S_4\times C_2$ | \( 1 + 129 T + 10575 T^{2} + 555454 T^{3} + 10575 p^{3} T^{4} + 129 p^{6} T^{5} + p^{9} T^{6} \) |
| 17 | $S_4\times C_2$ | \( 1 - 3 p T + 12507 T^{2} - 394854 T^{3} + 12507 p^{3} T^{4} - 3 p^{7} T^{5} + p^{9} T^{6} \) |
| 23 | $S_4\times C_2$ | \( 1 + 47 T + 35053 T^{2} + 1076204 T^{3} + 35053 p^{3} T^{4} + 47 p^{6} T^{5} + p^{9} T^{6} \) |
| 29 | $S_4\times C_2$ | \( 1 - 125 T + 23767 T^{2} - 8534000 T^{3} + 23767 p^{3} T^{4} - 125 p^{6} T^{5} + p^{9} T^{6} \) |
| 31 | $S_4\times C_2$ | \( 1 + 50 T + 37223 T^{2} + 6788948 T^{3} + 37223 p^{3} T^{4} + 50 p^{6} T^{5} + p^{9} T^{6} \) |
| 37 | $S_4\times C_2$ | \( 1 + 188 T + 151301 T^{2} + 18957524 T^{3} + 151301 p^{3} T^{4} + 188 p^{6} T^{5} + p^{9} T^{6} \) |
| 41 | $S_4\times C_2$ | \( 1 + 475 T + 225622 T^{2} + 56464759 T^{3} + 225622 p^{3} T^{4} + 475 p^{6} T^{5} + p^{9} T^{6} \) |
| 43 | $S_4\times C_2$ | \( 1 - 73 T + 126797 T^{2} + 4494434 T^{3} + 126797 p^{3} T^{4} - 73 p^{6} T^{5} + p^{9} T^{6} \) |
| 47 | $S_4\times C_2$ | \( 1 - 241 T + 294481 T^{2} - 44975956 T^{3} + 294481 p^{3} T^{4} - 241 p^{6} T^{5} + p^{9} T^{6} \) |
| 53 | $S_4\times C_2$ | \( 1 + 29 T + 407983 T^{2} + 5339846 T^{3} + 407983 p^{3} T^{4} + 29 p^{6} T^{5} + p^{9} T^{6} \) |
| 59 | $S_4\times C_2$ | \( 1 - 1065 T + 972522 T^{2} - 473334231 T^{3} + 972522 p^{3} T^{4} - 1065 p^{6} T^{5} + p^{9} T^{6} \) |
| 61 | $S_4\times C_2$ | \( 1 - 981 T + 949611 T^{2} - 464210144 T^{3} + 949611 p^{3} T^{4} - 981 p^{6} T^{5} + p^{9} T^{6} \) |
| 67 | $S_4\times C_2$ | \( 1 + 877 T + 841242 T^{2} + 528520259 T^{3} + 841242 p^{3} T^{4} + 877 p^{6} T^{5} + p^{9} T^{6} \) |
| 71 | $S_4\times C_2$ | \( 1 + 2135 T + 2443777 T^{2} + 354926 p^{2} T^{3} + 2443777 p^{3} T^{4} + 2135 p^{6} T^{5} + p^{9} T^{6} \) |
| 73 | $S_4\times C_2$ | \( 1 + 667 T + 1261314 T^{2} + 519793475 T^{3} + 1261314 p^{3} T^{4} + 667 p^{6} T^{5} + p^{9} T^{6} \) |
| 79 | $S_4\times C_2$ | \( 1 + 1671 T + 1726197 T^{2} + 1253980738 T^{3} + 1726197 p^{3} T^{4} + 1671 p^{6} T^{5} + p^{9} T^{6} \) |
| 83 | $S_4\times C_2$ | \( 1 - 588 T + 867318 T^{2} - 834896496 T^{3} + 867318 p^{3} T^{4} - 588 p^{6} T^{5} + p^{9} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 + 693 T + 1798515 T^{2} + 924765786 T^{3} + 1798515 p^{3} T^{4} + 693 p^{6} T^{5} + p^{9} T^{6} \) |
| 97 | $S_4\times C_2$ | \( 1 - 985 T + 305282 T^{2} + 810153335 T^{3} + 305282 p^{3} T^{4} - 985 p^{6} T^{5} + p^{9} T^{6} \) |
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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
−8.826946109203603545460033086782, −8.455871987749453685295253402868, −8.431980621267862536267449236255, −8.174320995878936278126452930216, −7.71001907080689022765428444101, −7.63606448346370457727599939252, −7.21523822658206295354740905614, −7.00880172371570034038367843109, −6.92896959956176413490800306373, −6.40528478358162235551937590370, −5.80782119594235096576740438077, −5.56912303080697666217213197015, −5.37965840513156294929560171619, −4.78161085424270451298843821940, −4.50455033251725668621558512183, −4.35779865734621765354042703571, −3.37496205973337046700033942411, −3.18528066279893655900914981490, −2.92315843640839320069384792721, −2.35828635850231993145017987838, −2.11463113496928364922812481963, −1.72842677778250286578406442581, −1.44341910669980912926532697313, −0.72575322944354107543684981241, −0.23990912904581069583215100151,
0.23990912904581069583215100151, 0.72575322944354107543684981241, 1.44341910669980912926532697313, 1.72842677778250286578406442581, 2.11463113496928364922812481963, 2.35828635850231993145017987838, 2.92315843640839320069384792721, 3.18528066279893655900914981490, 3.37496205973337046700033942411, 4.35779865734621765354042703571, 4.50455033251725668621558512183, 4.78161085424270451298843821940, 5.37965840513156294929560171619, 5.56912303080697666217213197015, 5.80782119594235096576740438077, 6.40528478358162235551937590370, 6.92896959956176413490800306373, 7.00880172371570034038367843109, 7.21523822658206295354740905614, 7.63606448346370457727599939252, 7.71001907080689022765428444101, 8.174320995878936278126452930216, 8.431980621267862536267449236255, 8.455871987749453685295253402868, 8.826946109203603545460033086782
