| L(s) = 1 | + 4-s − 4·5-s + 9-s − 4·20-s + 10·25-s + 6·29-s + 36-s + 2·41-s − 4·45-s + 49-s − 64-s − 4·89-s + 10·100-s + 4·101-s + 2·109-s + 6·116-s − 4·121-s − 20·125-s + 127-s + 131-s + 137-s + 139-s − 24·145-s + 149-s + 151-s + 157-s + 163-s + ⋯ |
| L(s) = 1 | + 4-s − 4·5-s + 9-s − 4·20-s + 10·25-s + 6·29-s + 36-s + 2·41-s − 4·45-s + 49-s − 64-s − 4·89-s + 10·100-s + 4·101-s + 2·109-s + 6·116-s − 4·121-s − 20·125-s + 127-s + 131-s + 137-s + 139-s − 24·145-s + 149-s + 151-s + 157-s + 163-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 5^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 5^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.7660550645\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7660550645\) |
| \(L(1)\) |
|
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 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 3 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 5 | $C_1$ | \( ( 1 + T )^{4} \) |
| 7 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| good | 11 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 13 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 23 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 29 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{4}( 1 - T + T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 41 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{4}( 1 + T + T^{2} )^{2} \) |
| 43 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 47 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 53 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 73 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 83 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 89 | $C_2$ | \( ( 1 + T + T^{2} )^{4} \) |
| 97 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.08306263283818787325872646738, −6.86580027711492765219290296501, −6.80200915222145220340710965467, −6.72608306929107193347953499911, −6.52896801410802086717922928816, −6.08966369229074821195465246188, −6.02831129149565488126926598592, −5.52849866737003095186485462603, −5.25932466250393343784622561448, −4.86005201798596134187974432446, −4.65159310679076219922108652710, −4.52957847528257497226065847747, −4.45499199969828857576395630992, −4.24942676342106737749656615011, −4.03295640683285972302453678291, −3.57078485597832804668927800380, −3.52041502107327942795861360787, −3.10502512536619287865005350163, −2.82055832952102812995392234485, −2.70895887295229795491606934733, −2.60121535867422162821335173420, −2.02612074199945558743499952184, −1.18522564287751680934537943335, −1.06996741021812534367689563741, −0.825824073747193117095482650300,
0.825824073747193117095482650300, 1.06996741021812534367689563741, 1.18522564287751680934537943335, 2.02612074199945558743499952184, 2.60121535867422162821335173420, 2.70895887295229795491606934733, 2.82055832952102812995392234485, 3.10502512536619287865005350163, 3.52041502107327942795861360787, 3.57078485597832804668927800380, 4.03295640683285972302453678291, 4.24942676342106737749656615011, 4.45499199969828857576395630992, 4.52957847528257497226065847747, 4.65159310679076219922108652710, 4.86005201798596134187974432446, 5.25932466250393343784622561448, 5.52849866737003095186485462603, 6.02831129149565488126926598592, 6.08966369229074821195465246188, 6.52896801410802086717922928816, 6.72608306929107193347953499911, 6.80200915222145220340710965467, 6.86580027711492765219290296501, 7.08306263283818787325872646738
