| L(s) = 1 | − 2·3-s + 2·5-s + 2·9-s + 2·11-s + 2·13-s − 4·15-s − 4·17-s + 6·19-s + 2·25-s − 6·27-s − 6·29-s + 16·31-s − 4·33-s − 6·37-s − 4·39-s + 10·43-s + 4·45-s − 16·47-s + 10·49-s + 8·51-s + 10·53-s + 4·55-s − 12·57-s − 6·59-s + 18·61-s + 4·65-s − 10·67-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 0.894·5-s + 2/3·9-s + 0.603·11-s + 0.554·13-s − 1.03·15-s − 0.970·17-s + 1.37·19-s + 2/5·25-s − 1.15·27-s − 1.11·29-s + 2.87·31-s − 0.696·33-s − 0.986·37-s − 0.640·39-s + 1.52·43-s + 0.596·45-s − 2.33·47-s + 10/7·49-s + 1.12·51-s + 1.37·53-s + 0.539·55-s − 1.58·57-s − 0.781·59-s + 2.30·61-s + 0.496·65-s − 1.22·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 16384 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16384 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9253180416\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9253180416\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| good | 3 | $C_2^2$ | \( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 5 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 7 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 10 T + 50 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 18 T + 162 T^{2} - 18 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 10 T + 50 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 42 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 - 130 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 162 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
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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
−13.57763943423367984700554367560, −13.30409326191450323749036487387, −12.39725935628457991775816094547, −12.04703488585734106578020538026, −11.34988060586784876578537077178, −11.33777772939555990236655141902, −10.50284001106027731166125190930, −10.01906219176728977678061627474, −9.544068541528414162031497236083, −9.013444924239594305077968375781, −8.352243291068938215378300579181, −7.57636266886867046204462624117, −6.75957304751256734756669518283, −6.49880138617070707658762768363, −5.66975054063335894590245814235, −5.44908684486335744558365099104, −4.53151267740980108681903213984, −3.78289150363022985981756278865, −2.56170305732102056983156404427, −1.29307975640473051161799074505,
1.29307975640473051161799074505, 2.56170305732102056983156404427, 3.78289150363022985981756278865, 4.53151267740980108681903213984, 5.44908684486335744558365099104, 5.66975054063335894590245814235, 6.49880138617070707658762768363, 6.75957304751256734756669518283, 7.57636266886867046204462624117, 8.352243291068938215378300579181, 9.013444924239594305077968375781, 9.544068541528414162031497236083, 10.01906219176728977678061627474, 10.50284001106027731166125190930, 11.33777772939555990236655141902, 11.34988060586784876578537077178, 12.04703488585734106578020538026, 12.39725935628457991775816094547, 13.30409326191450323749036487387, 13.57763943423367984700554367560
