| L(s) = 1 | − 4·2-s + 10·4-s − 20·8-s + 9-s + 35·16-s − 4·18-s + 2·23-s − 25-s − 56·32-s + 10·36-s − 8·46-s + 4·50-s + 84·64-s + 4·71-s − 20·72-s + 4·79-s + 20·92-s − 10·100-s − 2·113-s − 2·121-s + 127-s − 120·128-s + 131-s + 137-s + 139-s − 16·142-s + 35·144-s + ⋯ |
| L(s) = 1 | − 4·2-s + 10·4-s − 20·8-s + 9-s + 35·16-s − 4·18-s + 2·23-s − 25-s − 56·32-s + 10·36-s − 8·46-s + 4·50-s + 84·64-s + 4·71-s − 20·72-s + 4·79-s + 20·92-s − 10·100-s − 2·113-s − 2·121-s + 127-s − 120·128-s + 131-s + 137-s + 139-s − 16·142-s + 35·144-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.3257770476\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3257770476\) |
| \(L(1)\) |
|
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 | $C_1$ | \( ( 1 + T )^{4} \) |
| 3 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 7 | | \( 1 \) |
| good | 5 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 11 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 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$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 23 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{4}( 1 + T + T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 31 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 37 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 47 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 53 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 61 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 67 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 71 | $C_2$ | \( ( 1 - T + T^{2} )^{4} \) |
| 73 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - T + T^{2} )^{4} \) |
| 83 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
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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
−6.57886958433010366035368021234, −6.28682503289757458251499592501, −6.24105101734465867043662240252, −5.72354796510526699632570690900, −5.66814827901407948103492125458, −5.26430270649537209195397563684, −5.12627555295153848654115432309, −5.07625010715186454669430892054, −5.02613504011627037139446590308, −4.13848183942837512099984201199, −4.02478695232244343387917314903, −3.96926537487367800893808822558, −3.56784654393501426486002407151, −3.50234537701979501605034463044, −2.95883535136583688738117303430, −2.88151665973368974295905410272, −2.83194160759533188908704808198, −2.34246982427643622087069820684, −2.19544701259194308732220256223, −1.94375408824720839235719744808, −1.75789353772282664738240592557, −1.32034395543754513661483588273, −1.21458692847495199011085181740, −0.792385404286588407875381672765, −0.56693079551653595504041530247,
0.56693079551653595504041530247, 0.792385404286588407875381672765, 1.21458692847495199011085181740, 1.32034395543754513661483588273, 1.75789353772282664738240592557, 1.94375408824720839235719744808, 2.19544701259194308732220256223, 2.34246982427643622087069820684, 2.83194160759533188908704808198, 2.88151665973368974295905410272, 2.95883535136583688738117303430, 3.50234537701979501605034463044, 3.56784654393501426486002407151, 3.96926537487367800893808822558, 4.02478695232244343387917314903, 4.13848183942837512099984201199, 5.02613504011627037139446590308, 5.07625010715186454669430892054, 5.12627555295153848654115432309, 5.26430270649537209195397563684, 5.66814827901407948103492125458, 5.72354796510526699632570690900, 6.24105101734465867043662240252, 6.28682503289757458251499592501, 6.57886958433010366035368021234
