| L(s) = 1 | + 6·3-s + 21·9-s − 2·11-s − 6·17-s + 4·19-s + 25-s + 56·27-s − 12·33-s − 2·41-s − 8·43-s + 4·49-s − 36·51-s + 24·57-s − 2·67-s + 6·75-s + 126·81-s + 2·83-s − 4·97-s − 42·99-s + 4·107-s − 2·113-s + 3·121-s − 12·123-s + 127-s − 48·129-s + 131-s + 137-s + ⋯ |
| L(s) = 1 | + 6·3-s + 21·9-s − 2·11-s − 6·17-s + 4·19-s + 25-s + 56·27-s − 12·33-s − 2·41-s − 8·43-s + 4·49-s − 36·51-s + 24·57-s − 2·67-s + 6·75-s + 126·81-s + 2·83-s − 4·97-s − 42·99-s + 4·107-s − 2·113-s + 3·121-s − 12·123-s + 127-s − 48·129-s + 131-s + 137-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 5^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 5^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(12.73992249\) |
| \(L(\frac12)\) |
\(\approx\) |
\(12.73992249\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - T^{2} + T^{4} - T^{6} + T^{8} \) |
| good | 3 | \( ( 1 - T )^{8}( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \) |
| 7 | \( ( 1 - T^{2} + T^{4} )^{4} \) |
| 11 | \( ( 1 + T - T^{3} - T^{4} - T^{5} + T^{7} + T^{8} )^{2} \) |
| 13 | \( 1 + T^{2} - T^{6} - T^{8} - T^{10} + T^{14} + T^{16} \) |
| 17 | \( ( 1 + T )^{8}( 1 - T + T^{2} - T^{3} + T^{4} )^{2} \) |
| 19 | \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{4} \) |
| 23 | \( 1 + T^{2} - T^{6} - T^{8} - T^{10} + T^{14} + T^{16} \) |
| 29 | \( 1 + T^{2} - T^{6} - T^{8} - T^{10} + T^{14} + T^{16} \) |
| 31 | \( 1 + T^{2} - T^{6} - T^{8} - T^{10} + T^{14} + T^{16} \) |
| 37 | \( ( 1 + T^{2} )^{4}( 1 - T^{2} + T^{4} - T^{6} + T^{8} ) \) |
| 41 | \( ( 1 + T - T^{3} - T^{4} - T^{5} + T^{7} + T^{8} )^{2} \) |
| 43 | \( ( 1 + T + T^{2} )^{8} \) |
| 47 | \( 1 + T^{2} - T^{6} - T^{8} - T^{10} + T^{14} + T^{16} \) |
| 53 | \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \) |
| 59 | \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2} \) |
| 61 | \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2} \) |
| 67 | \( ( 1 + T - T^{3} - T^{4} - T^{5} + T^{7} + T^{8} )^{2} \) |
| 71 | \( 1 + T^{2} - T^{6} - T^{8} - T^{10} + T^{14} + T^{16} \) |
| 73 | \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2} \) |
| 79 | \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \) |
| 83 | \( ( 1 - T + T^{3} - T^{4} + T^{5} - T^{7} + T^{8} )^{2} \) |
| 89 | \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2} \) |
| 97 | \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−4.07533061721839503246788145211, −3.78130870919699588966202629136, −3.78026945593509392405392727301, −3.72567002901169173172690439621, −3.70077766951106518706165218665, −3.49302110174771449763827792115, −3.44072190004143317947859809484, −3.40748246258609209332808353975, −3.08708747189423826347610553239, −2.99870716620426868469770204119, −2.86690673448392636947303102821, −2.78407815483222289425974616803, −2.68896384230871911619396135975, −2.59021186282743696511813370249, −2.46049345173612335336974572945, −2.29241112556012279011613663510, −2.22609653196111025837970414060, −2.07990678878757648559283531012, −1.90223674138987521859474340529, −1.63573677018300844177800003656, −1.62156375519937490333112924818, −1.51759610574786977608539274289, −1.34079449661293068961215765629, −1.18415819880667281241119596464, −0.70612389567034749941737344533,
0.70612389567034749941737344533, 1.18415819880667281241119596464, 1.34079449661293068961215765629, 1.51759610574786977608539274289, 1.62156375519937490333112924818, 1.63573677018300844177800003656, 1.90223674138987521859474340529, 2.07990678878757648559283531012, 2.22609653196111025837970414060, 2.29241112556012279011613663510, 2.46049345173612335336974572945, 2.59021186282743696511813370249, 2.68896384230871911619396135975, 2.78407815483222289425974616803, 2.86690673448392636947303102821, 2.99870716620426868469770204119, 3.08708747189423826347610553239, 3.40748246258609209332808353975, 3.44072190004143317947859809484, 3.49302110174771449763827792115, 3.70077766951106518706165218665, 3.72567002901169173172690439621, 3.78026945593509392405392727301, 3.78130870919699588966202629136, 4.07533061721839503246788145211
Plot not available for L-functions of degree greater than 10.
