| L(s) = 1 | − 4·11-s − 4·23-s − 4·37-s − 4·43-s + 81-s + 4·107-s − 4·113-s + 6·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + ⋯ |
| L(s) = 1 | − 4·11-s − 4·23-s − 4·37-s − 4·43-s + 81-s + 4·107-s − 4·113-s + 6·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 5^{4} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 5^{4} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.08713021569\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.08713021569\) |
| \(L(1)\) |
|
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 | $C_2^2$ | \( 1 + T^{4} \) |
| 7 | | \( 1 \) |
| good | 3 | $C_2^3$ | \( 1 - T^{4} + T^{8} \) |
| 11 | $C_2$ | \( ( 1 + T + T^{2} )^{4} \) |
| 13 | $C_2^3$ | \( 1 - T^{4} + T^{8} \) |
| 17 | $C_2^3$ | \( 1 - T^{4} + T^{8} \) |
| 19 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 23 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{4}( 1 + T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 37 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{4}( 1 + T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 43 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{4}( 1 + T^{2} )^{2} \) |
| 47 | $C_2^3$ | \( 1 - T^{4} + T^{8} \) |
| 53 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 59 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 61 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 67 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 73 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 79 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 89 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 97 | $C_2^3$ | \( 1 - T^{4} + T^{8} \) |
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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
−7.50297468971467792394933511309, −7.05344801088825716690434129927, −6.91406619680705215927694020682, −6.78252651089909879221005121327, −6.68275259552373492169209450092, −6.06925952296824449769060128887, −5.93463645754754882786196146325, −5.79361056681761986120264298205, −5.57027541242876101860383896610, −5.21329053695135169403030741217, −5.16855975458253997647170586219, −4.94594334465607589755504266419, −4.63664439114456741867729062313, −4.57217603771545592910284399114, −3.96789358529827480857973350759, −3.72029848004240896549124337484, −3.51859022295874098333628347250, −3.33107101727954278402600243010, −2.97169001499720657383057040768, −2.67292826717381102709188148376, −2.26843445963207952645417291882, −2.08825176846897764606425559998, −1.73565657595194071449317819409, −1.66660624681013410520308477510, −0.20638402307577713385347462409,
0.20638402307577713385347462409, 1.66660624681013410520308477510, 1.73565657595194071449317819409, 2.08825176846897764606425559998, 2.26843445963207952645417291882, 2.67292826717381102709188148376, 2.97169001499720657383057040768, 3.33107101727954278402600243010, 3.51859022295874098333628347250, 3.72029848004240896549124337484, 3.96789358529827480857973350759, 4.57217603771545592910284399114, 4.63664439114456741867729062313, 4.94594334465607589755504266419, 5.16855975458253997647170586219, 5.21329053695135169403030741217, 5.57027541242876101860383896610, 5.79361056681761986120264298205, 5.93463645754754882786196146325, 6.06925952296824449769060128887, 6.68275259552373492169209450092, 6.78252651089909879221005121327, 6.91406619680705215927694020682, 7.05344801088825716690434129927, 7.50297468971467792394933511309
