| L(s) = 1 | + 2-s − 7-s + 9-s − 11-s − 14-s + 18-s − 22-s − 25-s − 32-s − 3·37-s + 2·43-s − 50-s − 3·53-s − 63-s − 64-s + 5·71-s − 3·74-s + 77-s − 3·79-s + 2·86-s − 99-s − 3·106-s + 3·107-s − 2·113-s − 126-s + 127-s + 131-s + ⋯ |
| L(s) = 1 | + 2-s − 7-s + 9-s − 11-s − 14-s + 18-s − 22-s − 25-s − 32-s − 3·37-s + 2·43-s − 50-s − 3·53-s − 63-s − 64-s + 5·71-s − 3·74-s + 77-s − 3·79-s + 2·86-s − 99-s − 3·106-s + 3·107-s − 2·113-s − 126-s + 127-s + 131-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 7^{4} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 7^{4} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.5000821996\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5000821996\) |
| \(L(1)\) |
|
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 | $C_4$ | \( 1 - T + T^{2} - T^{3} + T^{4} \) |
| 7 | $C_4$ | \( 1 + T + T^{2} + T^{3} + T^{4} \) |
| 11 | $C_4$ | \( 1 + T + T^{2} + T^{3} + T^{4} \) |
| good | 3 | $C_4\times C_2$ | \( 1 - T^{2} + T^{4} - T^{6} + T^{8} \) |
| 5 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 13 | $C_4\times C_2$ | \( 1 - T^{2} + T^{4} - T^{6} + T^{8} \) |
| 17 | $C_4\times C_2$ | \( 1 - T^{2} + T^{4} - T^{6} + T^{8} \) |
| 19 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 23 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 29 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 31 | $C_4\times C_2$ | \( 1 - T^{2} + T^{4} - T^{6} + T^{8} \) |
| 37 | $C_1$$\times$$C_4$ | \( ( 1 + T )^{4}( 1 - T + T^{2} - T^{3} + T^{4} ) \) |
| 41 | $C_4\times C_2$ | \( 1 - T^{2} + T^{4} - T^{6} + T^{8} \) |
| 43 | $C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2} \) |
| 47 | $C_4\times C_2$ | \( 1 - T^{2} + T^{4} - T^{6} + T^{8} \) |
| 53 | $C_1$$\times$$C_4$ | \( ( 1 + T )^{4}( 1 - T + T^{2} - T^{3} + T^{4} ) \) |
| 59 | $C_4\times C_2$ | \( 1 - T^{2} + T^{4} - T^{6} + T^{8} \) |
| 61 | $C_4\times C_2$ | \( 1 - T^{2} + T^{4} - T^{6} + T^{8} \) |
| 67 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 71 | $C_1$$\times$$C_4$ | \( ( 1 - T )^{4}( 1 - T + T^{2} - T^{3} + T^{4} ) \) |
| 73 | $C_4\times C_2$ | \( 1 - T^{2} + T^{4} - T^{6} + T^{8} \) |
| 79 | $C_1$$\times$$C_4$ | \( ( 1 + T )^{4}( 1 - T + T^{2} - T^{3} + T^{4} ) \) |
| 83 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 89 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 97 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
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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
−8.881399509964142578173480154335, −8.404162664876180116596919548400, −8.229789242283892413502559059665, −7.86797693417786416749683553178, −7.74008318749546535126414633261, −7.44695276480689231780791241560, −7.25577357353975236947200284418, −6.77248739965654946590809032076, −6.73663614361854083901690259650, −6.35566491626277483957974503507, −6.30963892285751417809317182030, −5.64883353293780937296635881754, −5.48816545679951374629477683142, −5.22452612829304758347818127061, −5.21194012104362180367471639432, −4.53111078957030650189012729291, −4.43424937145661039226077371555, −4.20732118965755099865304951168, −3.73957420713434814425022933589, −3.45067266594494925634135014028, −3.26922858556763795880648304961, −2.86573205785808218237549130098, −2.20078112360369189262110859727, −2.01470570013651892664210429723, −1.41049860462560735985848609586,
1.41049860462560735985848609586, 2.01470570013651892664210429723, 2.20078112360369189262110859727, 2.86573205785808218237549130098, 3.26922858556763795880648304961, 3.45067266594494925634135014028, 3.73957420713434814425022933589, 4.20732118965755099865304951168, 4.43424937145661039226077371555, 4.53111078957030650189012729291, 5.21194012104362180367471639432, 5.22452612829304758347818127061, 5.48816545679951374629477683142, 5.64883353293780937296635881754, 6.30963892285751417809317182030, 6.35566491626277483957974503507, 6.73663614361854083901690259650, 6.77248739965654946590809032076, 7.25577357353975236947200284418, 7.44695276480689231780791241560, 7.74008318749546535126414633261, 7.86797693417786416749683553178, 8.229789242283892413502559059665, 8.404162664876180116596919548400, 8.881399509964142578173480154335
