| L(s) = 1 | − 4·4-s − 4·7-s + 8·13-s + 4·16-s − 8·19-s − 6·25-s + 16·28-s − 8·31-s − 16·37-s + 4·43-s + 10·49-s − 32·52-s + 8·61-s + 16·64-s + 4·67-s + 48·73-s + 32·76-s + 24·79-s − 32·91-s + 16·97-s + 24·100-s − 8·103-s − 36·109-s − 16·112-s + 32·124-s + 127-s + 131-s + ⋯ |
| L(s) = 1 | − 2·4-s − 1.51·7-s + 2.21·13-s + 16-s − 1.83·19-s − 6/5·25-s + 3.02·28-s − 1.43·31-s − 2.63·37-s + 0.609·43-s + 10/7·49-s − 4.43·52-s + 1.02·61-s + 2·64-s + 0.488·67-s + 5.61·73-s + 3.67·76-s + 2.70·79-s − 3.35·91-s + 1.62·97-s + 12/5·100-s − 0.788·103-s − 3.44·109-s − 1.51·112-s + 2.87·124-s + 0.0887·127-s + 0.0873·131-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 7^{4} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 7^{4} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.098854421\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.098854421\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 7 | $C_1$ | \( ( 1 + T )^{4} \) |
| 11 | | \( 1 \) |
| good | 2 | $C_2^2$ | \( ( 1 + p T^{2} + p^{2} T^{4} )^{2} \) |
| 5 | $C_2^2$ | \( ( 1 + 3 T^{2} + p^{2} T^{4} )^{2} \) |
| 13 | $D_{4}$ | \( ( 1 - 4 T + 16 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 17 | $D_4\times C_2$ | \( 1 + 38 T^{2} + 715 T^{4} + 38 p^{2} T^{6} + p^{4} T^{8} \) |
| 19 | $D_{4}$ | \( ( 1 + 4 T + 28 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 23 | $D_4\times C_2$ | \( 1 + 32 T^{2} + 1090 T^{4} + 32 p^{2} T^{6} + p^{4} T^{8} \) |
| 29 | $D_4\times C_2$ | \( 1 + 56 T^{2} + 2242 T^{4} + 56 p^{2} T^{6} + p^{4} T^{8} \) |
| 31 | $D_{4}$ | \( ( 1 + 4 T + 10 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 37 | $D_{4}$ | \( ( 1 + 8 T + 34 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 41 | $C_2^2$ | \( ( 1 + 50 T^{2} + p^{2} T^{4} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - T + p T^{2} )^{4} \) |
| 47 | $D_4\times C_2$ | \( 1 + 158 T^{2} + 10435 T^{4} + 158 p^{2} T^{6} + p^{4} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 - 28 T^{2} + 2230 T^{4} - 28 p^{2} T^{6} + p^{4} T^{8} \) |
| 59 | $D_4\times C_2$ | \( 1 + 158 T^{2} + 12307 T^{4} + 158 p^{2} T^{6} + p^{4} T^{8} \) |
| 61 | $D_{4}$ | \( ( 1 - 4 T + 112 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 67 | $D_{4}$ | \( ( 1 - 2 T + 79 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 + 140 T^{2} + p^{2} T^{4} )^{2} \) |
| 73 | $D_{4}$ | \( ( 1 - 24 T + 276 T^{2} - 24 p T^{3} + p^{2} T^{4} )^{2} \) |
| 79 | $D_{4}$ | \( ( 1 - 12 T + 138 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 83 | $D_4\times C_2$ | \( 1 + 174 T^{2} + 19331 T^{4} + 174 p^{2} T^{6} + p^{4} T^{8} \) |
| 89 | $D_4\times C_2$ | \( 1 + 198 T^{2} + 23627 T^{4} + 198 p^{2} T^{6} + p^{4} T^{8} \) |
| 97 | $D_{4}$ | \( ( 1 - 8 T + 84 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−5.32686132755265527637734014207, −5.27433428975415827722478909487, −5.23677958911659639294878099410, −5.21296378410675720868445627944, −4.81912011049572721152787517580, −4.34264998265708567756820058587, −4.22565761302153338441261506745, −4.21679330721637160676173851604, −4.19038943914720570282610990245, −3.72394903147817739148788824092, −3.68060357551865997577721371184, −3.47056802819728049973090436880, −3.46370126450760537986432899182, −3.28662973785159314069378584008, −2.85822768431748660076986591692, −2.59082491088377181815303397769, −2.27081008589983561823246724631, −2.00542259431317959656581270429, −1.99070503419576130764316728749, −1.76219906738499970597008545326, −1.44121537315351609743723016258, −0.867129393274427662215298563327, −0.58881439203640554331968930877, −0.43367795918564847594465123156, −0.42762777990318806100516691929,
0.42762777990318806100516691929, 0.43367795918564847594465123156, 0.58881439203640554331968930877, 0.867129393274427662215298563327, 1.44121537315351609743723016258, 1.76219906738499970597008545326, 1.99070503419576130764316728749, 2.00542259431317959656581270429, 2.27081008589983561823246724631, 2.59082491088377181815303397769, 2.85822768431748660076986591692, 3.28662973785159314069378584008, 3.46370126450760537986432899182, 3.47056802819728049973090436880, 3.68060357551865997577721371184, 3.72394903147817739148788824092, 4.19038943914720570282610990245, 4.21679330721637160676173851604, 4.22565761302153338441261506745, 4.34264998265708567756820058587, 4.81912011049572721152787517580, 5.21296378410675720868445627944, 5.23677958911659639294878099410, 5.27433428975415827722478909487, 5.32686132755265527637734014207
