| L(s) = 1 | − 2·3-s + 3·9-s + 2·11-s + 2·17-s + 6·19-s + 6·25-s − 4·27-s − 4·33-s − 2·41-s + 18·43-s − 10·49-s − 4·51-s − 12·57-s − 4·59-s + 14·67-s + 16·73-s − 12·75-s + 5·81-s − 14·83-s − 12·89-s + 26·97-s + 6·99-s + 20·107-s − 14·113-s + 6·121-s + 4·123-s + 127-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 9-s + 0.603·11-s + 0.485·17-s + 1.37·19-s + 6/5·25-s − 0.769·27-s − 0.696·33-s − 0.312·41-s + 2.74·43-s − 1.42·49-s − 0.560·51-s − 1.58·57-s − 0.520·59-s + 1.71·67-s + 1.87·73-s − 1.38·75-s + 5/9·81-s − 1.53·83-s − 1.27·89-s + 2.63·97-s + 0.603·99-s + 1.93·107-s − 1.31·113-s + 6/11·121-s + 0.360·123-s + 0.0887·127-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 831744 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 831744 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.591002758\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.591002758\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | $C_1$ | \( ( 1 + T )^{2} \) |
| 19 | $C_2$ | \( 1 - 6 T + p T^{2} \) |
| good | 5 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 13 | $C_2^2$ | \( 1 + 8 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 - 12 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 18 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 37 | $C_2^2$ | \( 1 - 52 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 - 8 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 + 84 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 66 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 + 66 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 18 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 18 T + p T^{2} )( 1 - 8 T + p T^{2} ) \) |
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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
−8.096536419308353420569845427253, −7.64063965282913115343698428657, −7.33729092640203614112775561586, −6.76621247402802627893108854917, −6.48282255579477677511893397531, −5.93735148620914014433772665478, −5.51537799117374680135155047251, −5.10702131188837437287336451119, −4.65616899847906745823131130644, −4.11118167916778834217031103955, −3.50616789235451339867534692835, −3.00308720947754150127181816892, −2.17109459328236276447614845162, −1.25281326373268093244779376707, −0.76008109706863821147326681628,
0.76008109706863821147326681628, 1.25281326373268093244779376707, 2.17109459328236276447614845162, 3.00308720947754150127181816892, 3.50616789235451339867534692835, 4.11118167916778834217031103955, 4.65616899847906745823131130644, 5.10702131188837437287336451119, 5.51537799117374680135155047251, 5.93735148620914014433772665478, 6.48282255579477677511893397531, 6.76621247402802627893108854917, 7.33729092640203614112775561586, 7.64063965282913115343698428657, 8.096536419308353420569845427253
