| L(s) = 1 | + 2·3-s + 2·7-s + 2·9-s + 4·21-s − 2·23-s + 2·27-s − 2·43-s − 2·47-s + 2·49-s + 4·63-s − 2·67-s − 4·69-s + 3·81-s − 2·83-s − 4·101-s + 2·103-s + 2·107-s − 2·121-s + 127-s − 4·129-s + 131-s + 137-s + 139-s − 4·141-s + 4·147-s + 149-s + 151-s + ⋯ |
| L(s) = 1 | + 2·3-s + 2·7-s + 2·9-s + 4·21-s − 2·23-s + 2·27-s − 2·43-s − 2·47-s + 2·49-s + 4·63-s − 2·67-s − 4·69-s + 3·81-s − 2·83-s − 4·101-s + 2·103-s + 2·107-s − 2·121-s + 127-s − 4·129-s + 131-s + 137-s + 139-s − 4·141-s + 4·147-s + 149-s + 151-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2560000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2560000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(2.800781532\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.800781532\) |
| \(L(1)\) |
|
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 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 3 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T^{2} ) \) |
| 7 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 + T^{4} \) |
| 17 | $C_2^2$ | \( 1 + T^{4} \) |
| 19 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 23 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 + T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 + T^{4} \) |
| 41 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 43 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 + T^{2} ) \) |
| 47 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 + T^{2} ) \) |
| 53 | $C_2^2$ | \( 1 + T^{4} \) |
| 59 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 61 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 67 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 + T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + T^{4} \) |
| 79 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 83 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 + T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 + T^{4} \) |
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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
−9.702788932087076384253681683392, −9.336137767135761875116565430071, −8.768539091866812237486005072799, −8.424259432951252227075141732133, −8.290612172734234462516375814948, −8.081474969791895606429605224165, −7.49852891830371670949769420491, −7.42271215906803915216995867190, −6.59533025265282169763422979388, −6.35910984698418105891142023115, −5.55617438434589328247878086889, −5.22812367134743977701325713085, −4.50916375256853110461324208476, −4.48116622592691338547202548855, −3.84278008463460074942882744315, −3.27385646867688932235280580080, −2.89458068973806471730627222799, −2.24492217076563886293446112366, −1.64300569239688915817781434294, −1.59584208834128298357390072734,
1.59584208834128298357390072734, 1.64300569239688915817781434294, 2.24492217076563886293446112366, 2.89458068973806471730627222799, 3.27385646867688932235280580080, 3.84278008463460074942882744315, 4.48116622592691338547202548855, 4.50916375256853110461324208476, 5.22812367134743977701325713085, 5.55617438434589328247878086889, 6.35910984698418105891142023115, 6.59533025265282169763422979388, 7.42271215906803915216995867190, 7.49852891830371670949769420491, 8.081474969791895606429605224165, 8.290612172734234462516375814948, 8.424259432951252227075141732133, 8.768539091866812237486005072799, 9.336137767135761875116565430071, 9.702788932087076384253681683392
