| L(s) = 1 | − 2·13-s + 4·19-s − 2·31-s + 2·43-s + 2·49-s − 2·61-s − 2·67-s + 4·73-s + 2·79-s + 2·97-s + 2·103-s + 4·109-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 3·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + ⋯ |
| L(s) = 1 | − 2·13-s + 4·19-s − 2·31-s + 2·43-s + 2·49-s − 2·61-s − 2·67-s + 4·73-s + 2·79-s + 2·97-s + 2·103-s + 4·109-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 3·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{20}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{20}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.799253637\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.799253637\) |
| \(L(1)\) |
|
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 | | \( 1 \) |
| 3 | | \( 1 \) |
| good | 5 | $C_2^3$ | \( 1 - T^{4} + T^{8} \) |
| 7 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 11 | $C_2^3$ | \( 1 - T^{4} + T^{8} \) |
| 13 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{4}( 1 - T + T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - T + T^{2} )^{4} \) |
| 23 | $C_2^3$ | \( 1 - T^{4} + T^{8} \) |
| 29 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 31 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{4}( 1 - T + T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 41 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 43 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{4}( 1 + T + T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 59 | $C_2^3$ | \( 1 - T^{4} + T^{8} \) |
| 61 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{4}( 1 - T + T^{2} )^{2} \) |
| 67 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{4}( 1 - T + T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - T + T^{2} )^{4} \) |
| 79 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{4}( 1 + T + T^{2} )^{2} \) |
| 83 | $C_2^3$ | \( 1 - T^{4} + T^{8} \) |
| 89 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 97 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{4}( 1 + T + T^{2} )^{2} \) |
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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
−6.10978260143317395715461133130, −6.09053865758676589763139872092, −5.66207671587129000373193480172, −5.54818110689922294056004165096, −5.36358208762661813662214256012, −5.10026881245132127360845305760, −5.00054722187726560962981100692, −4.90647692300110461702931091386, −4.85179141651602118790912110506, −4.31760543643979077456664762134, −4.11269110314432836324289300077, −4.02387001413965781910743015464, −3.71373542318687590120962931788, −3.31047700858787553932037146888, −3.29653685652047644422763227432, −3.22581476028159468597205959802, −2.94499156292873590666922487481, −2.44619159362145186826583967988, −2.36038333770136979914717449043, −2.16823808429689552537651228391, −1.99952525415419847654523341150, −1.50130149491383411467772983689, −1.07775195513092428815528055893, −1.00003197452409065119794701104, −0.55354268833888863708761740332,
0.55354268833888863708761740332, 1.00003197452409065119794701104, 1.07775195513092428815528055893, 1.50130149491383411467772983689, 1.99952525415419847654523341150, 2.16823808429689552537651228391, 2.36038333770136979914717449043, 2.44619159362145186826583967988, 2.94499156292873590666922487481, 3.22581476028159468597205959802, 3.29653685652047644422763227432, 3.31047700858787553932037146888, 3.71373542318687590120962931788, 4.02387001413965781910743015464, 4.11269110314432836324289300077, 4.31760543643979077456664762134, 4.85179141651602118790912110506, 4.90647692300110461702931091386, 5.00054722187726560962981100692, 5.10026881245132127360845305760, 5.36358208762661813662214256012, 5.54818110689922294056004165096, 5.66207671587129000373193480172, 6.09053865758676589763139872092, 6.10978260143317395715461133130
