| L(s) = 1 | + 2-s − 8-s − 2·11-s − 16-s − 2·22-s − 25-s − 4·43-s − 50-s + 64-s − 2·67-s − 4·86-s + 2·88-s + 2·107-s + 4·113-s + 121-s + 127-s + 128-s + 131-s − 2·134-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 2·169-s + ⋯ |
| L(s) = 1 | + 2-s − 8-s − 2·11-s − 16-s − 2·22-s − 25-s − 4·43-s − 50-s + 64-s − 2·67-s − 4·86-s + 2·88-s + 2·107-s + 4·113-s + 121-s + 127-s + 128-s + 131-s − 2·134-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 2·169-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 12446784 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12446784 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.9401008512\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9401008512\) |
| \(L(1)\) |
|
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 | $C_2$ | \( 1 - T + T^{2} \) |
| 3 | | \( 1 \) |
| 7 | | \( 1 \) |
| good | 5 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 13 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 17 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 19 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 23 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 29 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 31 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 43 | $C_1$ | \( ( 1 + T )^{4} \) |
| 47 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 61 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 73 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 79 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 97 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
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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
−8.972404168941035647368013574691, −8.445083235320813996020907802080, −8.177447599615119292780337015819, −7.87742902625199362523657061016, −7.44901414019852637275647194798, −7.07125722992744299467600493364, −6.59697450785000371193694496398, −6.15554936294227981925344633043, −5.91800724205383763113232228712, −5.28638409699505385508433620772, −5.24902643120829221504967623541, −4.78820787998930189007439371839, −4.46868294677704997994827425708, −3.92666724090627853094105128638, −3.43084779623484295203855753939, −2.93617752672696660285027792109, −2.92177008504583755043945183811, −1.91780899580110526910302843824, −1.84665010616777858824728853159, −0.45089173415561438820596481864,
0.45089173415561438820596481864, 1.84665010616777858824728853159, 1.91780899580110526910302843824, 2.92177008504583755043945183811, 2.93617752672696660285027792109, 3.43084779623484295203855753939, 3.92666724090627853094105128638, 4.46868294677704997994827425708, 4.78820787998930189007439371839, 5.24902643120829221504967623541, 5.28638409699505385508433620772, 5.91800724205383763113232228712, 6.15554936294227981925344633043, 6.59697450785000371193694496398, 7.07125722992744299467600493364, 7.44901414019852637275647194798, 7.87742902625199362523657061016, 8.177447599615119292780337015819, 8.445083235320813996020907802080, 8.972404168941035647368013574691
