| L(s) = 1 | + 3·11-s + 3·13-s + 6·17-s − 3·19-s − 3·23-s − 6·27-s + 12·29-s − 12·31-s − 9·37-s − 9·41-s − 12·43-s − 15·47-s − 9·53-s − 24·59-s + 6·61-s − 6·67-s − 18·79-s − 30·83-s − 6·97-s + 18·101-s − 6·103-s − 18·107-s + 6·109-s + 127-s + 131-s + 137-s + 139-s + ⋯ |
| L(s) = 1 | + 0.904·11-s + 0.832·13-s + 1.45·17-s − 0.688·19-s − 0.625·23-s − 1.15·27-s + 2.22·29-s − 2.15·31-s − 1.47·37-s − 1.40·41-s − 1.82·43-s − 2.18·47-s − 1.23·53-s − 3.12·59-s + 0.768·61-s − 0.733·67-s − 2.02·79-s − 3.29·83-s − 0.609·97-s + 1.79·101-s − 0.591·103-s − 1.74·107-s + 0.574·109-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{9} \cdot 5^{6} \cdot 7^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{9} \cdot 5^{6} \cdot 7^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]
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 | 2 | | \( 1 \) |
| 5 | | \( 1 \) |
| 7 | | \( 1 \) |
| good | 3 | $D_{6}$ | \( 1 + 2 p T^{3} + p^{3} T^{6} \) |
| 11 | $S_4\times C_2$ | \( 1 - 3 T + 9 T^{2} - 2 p T^{3} + 9 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 13 | $S_4\times C_2$ | \( 1 - 3 T + 15 T^{2} - 10 T^{3} + 15 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{3} \) |
| 19 | $S_4\times C_2$ | \( 1 + 3 T + 33 T^{2} + 130 T^{3} + 33 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 23 | $S_4\times C_2$ | \( 1 + 3 T + 54 T^{2} + 145 T^{3} + 54 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 29 | $S_4\times C_2$ | \( 1 - 12 T + 108 T^{2} - 670 T^{3} + 108 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 31 | $S_4\times C_2$ | \( 1 + 12 T + 105 T^{2} + 616 T^{3} + 105 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 37 | $S_4\times C_2$ | \( 1 + 9 T + 3 p T^{2} + 570 T^{3} + 3 p^{2} T^{4} + 9 p^{2} T^{5} + p^{3} T^{6} \) |
| 41 | $S_4\times C_2$ | \( 1 + 9 T + 78 T^{2} + 357 T^{3} + 78 p T^{4} + 9 p^{2} T^{5} + p^{3} T^{6} \) |
| 43 | $S_4\times C_2$ | \( 1 + 12 T + 168 T^{2} + 1054 T^{3} + 168 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 47 | $S_4\times C_2$ | \( 1 + 15 T + 45 T^{2} - 178 T^{3} + 45 p T^{4} + 15 p^{2} T^{5} + p^{3} T^{6} \) |
| 53 | $S_4\times C_2$ | \( 1 + 9 T + 87 T^{2} + 330 T^{3} + 87 p T^{4} + 9 p^{2} T^{5} + p^{3} T^{6} \) |
| 59 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{3} \) |
| 61 | $S_4\times C_2$ | \( 1 - 6 T + 96 T^{2} - 188 T^{3} + 96 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 67 | $S_4\times C_2$ | \( 1 + 6 T + 132 T^{2} + 812 T^{3} + 132 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{3} \) |
| 73 | $S_4\times C_2$ | \( 1 + 111 T^{2} + 336 T^{3} + 111 p T^{4} + p^{3} T^{6} \) |
| 79 | $S_4\times C_2$ | \( 1 + 18 T + 3 p T^{2} + 2076 T^{3} + 3 p^{2} T^{4} + 18 p^{2} T^{5} + p^{3} T^{6} \) |
| 83 | $S_4\times C_2$ | \( 1 + 30 T + 540 T^{2} + 5884 T^{3} + 540 p T^{4} + 30 p^{2} T^{5} + p^{3} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 + 240 T^{2} - 42 T^{3} + 240 p T^{4} + p^{3} T^{6} \) |
| 97 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{3} \) |
| 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
−6.95076985810327842827697665655, −6.88383605283557482840410177491, −6.64370343262737014760255135982, −6.56116399493601395218027674107, −6.12312866526596185438585062472, −6.03090415825285562239079418529, −5.69683583668051450750959583680, −5.58123855309416976855172413501, −5.39794404452737445596817290448, −5.01977515465552347950942138204, −4.65000120376688709514104758974, −4.53068871564629245015496639504, −4.50605483425947620889353946016, −4.02729330976029053457416522680, −3.61460504184586802721992292259, −3.55512383381911318835181623820, −3.29745541261982033314216981557, −3.09748118765441318341241593770, −3.04979195054700510813541569726, −2.31641052355648800063846422747, −2.16724842429477228298554308863, −1.64315861987382242879915248734, −1.54059850481879428186539457725, −1.25168602122973013636741962071, −1.25059595800345646662094253437, 0, 0, 0,
1.25059595800345646662094253437, 1.25168602122973013636741962071, 1.54059850481879428186539457725, 1.64315861987382242879915248734, 2.16724842429477228298554308863, 2.31641052355648800063846422747, 3.04979195054700510813541569726, 3.09748118765441318341241593770, 3.29745541261982033314216981557, 3.55512383381911318835181623820, 3.61460504184586802721992292259, 4.02729330976029053457416522680, 4.50605483425947620889353946016, 4.53068871564629245015496639504, 4.65000120376688709514104758974, 5.01977515465552347950942138204, 5.39794404452737445596817290448, 5.58123855309416976855172413501, 5.69683583668051450750959583680, 6.03090415825285562239079418529, 6.12312866526596185438585062472, 6.56116399493601395218027674107, 6.64370343262737014760255135982, 6.88383605283557482840410177491, 6.95076985810327842827697665655
