| L(s) = 1 | + 2-s + 3-s − 5-s + 6-s − 7-s − 8-s − 10-s − 11-s − 14-s − 15-s − 16-s − 21-s − 22-s − 24-s + 25-s − 27-s + 2·29-s − 30-s + 31-s − 33-s + 35-s + 40-s − 42-s − 48-s + 50-s − 53-s − 54-s + ⋯ |
| L(s) = 1 | + 2-s + 3-s − 5-s + 6-s − 7-s − 8-s − 10-s − 11-s − 14-s − 15-s − 16-s − 21-s − 22-s − 24-s + 25-s − 27-s + 2·29-s − 30-s + 31-s − 33-s + 35-s + 40-s − 42-s − 48-s + 50-s − 53-s − 54-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 28224 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 28224 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.6320305171\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6320305171\) |
| \(L(1)\) |
|
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 | $C_2$ | \( 1 - T + T^{2} \) |
| 3 | $C_2$ | \( 1 - T + T^{2} \) |
| 7 | $C_2$ | \( 1 + T + T^{2} \) |
| good | 5 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| 11 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| 13 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 17 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 31 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 41 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 43 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 47 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 53 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| 59 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 73 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 79 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−13.16302320037969944480696072567, −13.00038542653060252709731992770, −12.45124170982219093015462280986, −11.83714033401652427485785498122, −11.74342602875900181326235820481, −10.67912812258416350923297324413, −10.44604020380389069179961353883, −9.606095076322175874873032937294, −9.240445997654200465171850406238, −8.642314643704530090554462014690, −7.983189307617049929617508331280, −7.943921129814027549837558082507, −6.85306887210919170952066197257, −6.45840513587427204111383799514, −5.72205468344983056887725200940, −4.90885335948892201490429824040, −4.41569849154479348210501520321, −3.63772716762778935296786994848, −2.94693018909009501599165498144, −2.75321242507591566263979590084,
2.75321242507591566263979590084, 2.94693018909009501599165498144, 3.63772716762778935296786994848, 4.41569849154479348210501520321, 4.90885335948892201490429824040, 5.72205468344983056887725200940, 6.45840513587427204111383799514, 6.85306887210919170952066197257, 7.943921129814027549837558082507, 7.983189307617049929617508331280, 8.642314643704530090554462014690, 9.240445997654200465171850406238, 9.606095076322175874873032937294, 10.44604020380389069179961353883, 10.67912812258416350923297324413, 11.74342602875900181326235820481, 11.83714033401652427485785498122, 12.45124170982219093015462280986, 13.00038542653060252709731992770, 13.16302320037969944480696072567
