| L(s) = 1 | − 2·2-s + 4-s + 4·7-s + 8-s − 11-s + 3·13-s − 8·14-s − 5·16-s − 14·17-s + 3·19-s + 2·22-s − 8·23-s − 6·26-s + 4·28-s + 5·29-s − 31-s + 8·32-s + 28·34-s + 5·37-s − 6·38-s − 41-s − 5·43-s − 44-s + 16·46-s − 9·47-s − 6·49-s + 3·52-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 1/2·4-s + 1.51·7-s + 0.353·8-s − 0.301·11-s + 0.832·13-s − 2.13·14-s − 5/4·16-s − 3.39·17-s + 0.688·19-s + 0.426·22-s − 1.66·23-s − 1.17·26-s + 0.755·28-s + 0.928·29-s − 0.179·31-s + 1.41·32-s + 4.80·34-s + 0.821·37-s − 0.973·38-s − 0.156·41-s − 0.762·43-s − 0.150·44-s + 2.35·46-s − 1.31·47-s − 6/7·49-s + 0.416·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{6} \cdot 5^{6} \cdot 19^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{6} \cdot 5^{6} \cdot 19^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| 5 | | \( 1 \) |
| 19 | $C_1$ | \( ( 1 - T )^{3} \) |
| good | 2 | $A_4\times C_2$ | \( 1 + p T + 3 T^{2} + 3 T^{3} + 3 p T^{4} + p^{3} T^{5} + p^{3} T^{6} \) |
| 7 | $A_4\times C_2$ | \( 1 - 4 T + 22 T^{2} - 55 T^{3} + 22 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 11 | $A_4\times C_2$ | \( 1 + T + 29 T^{2} + 23 T^{3} + 29 p T^{4} + p^{2} T^{5} + p^{3} T^{6} \) |
| 13 | $A_4\times C_2$ | \( 1 - 3 T + 3 T^{2} + 25 T^{3} + 3 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 17 | $A_4\times C_2$ | \( 1 + 14 T + 112 T^{2} + 555 T^{3} + 112 p T^{4} + 14 p^{2} T^{5} + p^{3} T^{6} \) |
| 23 | $A_4\times C_2$ | \( 1 + 8 T + 60 T^{2} + 243 T^{3} + 60 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 29 | $A_4\times C_2$ | \( 1 - 5 T + 91 T^{2} - 285 T^{3} + 91 p T^{4} - 5 p^{2} T^{5} + p^{3} T^{6} \) |
| 31 | $A_4\times C_2$ | \( 1 + T + 63 T^{2} + 115 T^{3} + 63 p T^{4} + p^{2} T^{5} + p^{3} T^{6} \) |
| 37 | $A_4\times C_2$ | \( 1 - 5 T + p T^{2} + 25 T^{3} + p^{2} T^{4} - 5 p^{2} T^{5} + p^{3} T^{6} \) |
| 41 | $A_4\times C_2$ | \( 1 + T + p T^{2} - 73 T^{3} + p^{2} T^{4} + p^{2} T^{5} + p^{3} T^{6} \) |
| 43 | $A_4\times C_2$ | \( 1 + 5 T + 16 T^{2} + 113 T^{3} + 16 p T^{4} + 5 p^{2} T^{5} + p^{3} T^{6} \) |
| 47 | $A_4\times C_2$ | \( 1 + 9 T + 77 T^{2} + 535 T^{3} + 77 p T^{4} + 9 p^{2} T^{5} + p^{3} T^{6} \) |
| 53 | $A_4\times C_2$ | \( 1 + 31 T + 462 T^{2} + 4191 T^{3} + 462 p T^{4} + 31 p^{2} T^{5} + p^{3} T^{6} \) |
| 59 | $A_4\times C_2$ | \( 1 - 6 T + 137 T^{2} - 508 T^{3} + 137 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 61 | $A_4\times C_2$ | \( 1 - 3 T + 173 T^{2} - 367 T^{3} + 173 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 67 | $A_4\times C_2$ | \( 1 + 13 T + 3 p T^{2} + 1573 T^{3} + 3 p^{2} T^{4} + 13 p^{2} T^{5} + p^{3} T^{6} \) |
| 71 | $A_4\times C_2$ | \( 1 + 7 T + 212 T^{2} + 947 T^{3} + 212 p T^{4} + 7 p^{2} T^{5} + p^{3} T^{6} \) |
| 73 | $A_4\times C_2$ | \( 1 - T + 189 T^{2} - 199 T^{3} + 189 p T^{4} - p^{2} T^{5} + p^{3} T^{6} \) |
| 79 | $A_4\times C_2$ | \( 1 + 18 T + 254 T^{2} + 31 p T^{3} + 254 p T^{4} + 18 p^{2} T^{5} + p^{3} T^{6} \) |
| 83 | $A_4\times C_2$ | \( 1 + 3 T - 21 T^{2} + 629 T^{3} - 21 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 89 | $A_4\times C_2$ | \( 1 - 20 T + 318 T^{2} - 3435 T^{3} + 318 p T^{4} - 20 p^{2} T^{5} + p^{3} T^{6} \) |
| 97 | $A_4\times C_2$ | \( 1 + 13 T + 3 p T^{2} + 2353 T^{3} + 3 p^{2} T^{4} + 13 p^{2} T^{5} + p^{3} T^{6} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{6} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.88402029411891461375957048541, −7.69575610460000418804933686697, −7.68439110695480863051572785298, −6.87611681630669854246862555928, −6.79485734368531162122401664726, −6.76387261025963788761210058710, −6.41430848517713743336462074270, −6.25255667395217256205803466843, −5.82639603082197173989983162781, −5.72629553907618129570247335183, −5.04914632462716171722170655456, −4.97231607681381963456764674940, −4.83934229253721202854902250315, −4.38479606423847842225979730704, −4.29757330439102968084705968159, −4.28659609587802807998646897064, −3.56618856496280352079943848112, −3.45571752067920037746993685238, −2.80238360452357796373664268487, −2.74835090251108993470981599065, −2.28156573865239013751306909602, −1.93306333656746624600391945297, −1.65945841947794841284978273456, −1.42537893837014131200467735448, −1.13541503679291288213918840563, 0, 0, 0,
1.13541503679291288213918840563, 1.42537893837014131200467735448, 1.65945841947794841284978273456, 1.93306333656746624600391945297, 2.28156573865239013751306909602, 2.74835090251108993470981599065, 2.80238360452357796373664268487, 3.45571752067920037746993685238, 3.56618856496280352079943848112, 4.28659609587802807998646897064, 4.29757330439102968084705968159, 4.38479606423847842225979730704, 4.83934229253721202854902250315, 4.97231607681381963456764674940, 5.04914632462716171722170655456, 5.72629553907618129570247335183, 5.82639603082197173989983162781, 6.25255667395217256205803466843, 6.41430848517713743336462074270, 6.76387261025963788761210058710, 6.79485734368531162122401664726, 6.87611681630669854246862555928, 7.68439110695480863051572785298, 7.69575610460000418804933686697, 7.88402029411891461375957048541
