| L(s) = 1 | − 8·2-s + 3-s + 36·4-s − 5-s − 8·6-s − 120·8-s + 8·10-s + 11-s + 36·12-s − 13-s − 15-s + 330·16-s − 17-s − 36·20-s − 8·22-s + 23-s − 120·24-s + 8·26-s − 29-s + 8·30-s − 792·32-s + 33-s + 8·34-s − 39-s + 120·40-s − 41-s + 36·44-s + ⋯ |
| L(s) = 1 | − 8·2-s + 3-s + 36·4-s − 5-s − 8·6-s − 120·8-s + 8·10-s + 11-s + 36·12-s − 13-s − 15-s + 330·16-s − 17-s − 36·20-s − 8·22-s + 23-s − 120·24-s + 8·26-s − 29-s + 8·30-s − 792·32-s + 33-s + 8·34-s − 39-s + 120·40-s − 41-s + 36·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 701^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 701^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.04733300459\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.04733300459\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( ( 1 + T )^{8} \) |
| 701 | \( ( 1 - T )^{8} \) |
| good | 3 | \( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} - T^{11} + T^{12} - T^{13} + T^{14} - T^{15} + T^{16} \) |
| 5 | \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} + T^{11} + T^{12} + T^{13} + T^{14} + T^{15} + T^{16} \) |
| 7 | \( ( 1 - T )^{8}( 1 + T )^{8} \) |
| 11 | \( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} - T^{11} + T^{12} - T^{13} + T^{14} - T^{15} + T^{16} \) |
| 13 | \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} + T^{11} + T^{12} + T^{13} + T^{14} + T^{15} + T^{16} \) |
| 17 | \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} + T^{11} + T^{12} + T^{13} + T^{14} + T^{15} + T^{16} \) |
| 19 | \( ( 1 - T )^{8}( 1 + T )^{8} \) |
| 23 | \( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} - T^{11} + T^{12} - T^{13} + T^{14} - T^{15} + T^{16} \) |
| 29 | \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} + T^{11} + T^{12} + T^{13} + T^{14} + T^{15} + T^{16} \) |
| 31 | \( ( 1 - T )^{8}( 1 + T )^{8} \) |
| 37 | \( ( 1 - T )^{8}( 1 + T )^{8} \) |
| 41 | \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} + T^{11} + T^{12} + T^{13} + T^{14} + T^{15} + T^{16} \) |
| 43 | \( ( 1 - T )^{8}( 1 + T )^{8} \) |
| 47 | \( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} - T^{11} + T^{12} - T^{13} + T^{14} - T^{15} + T^{16} \) |
| 53 | \( ( 1 - T )^{8}( 1 + T )^{8} \) |
| 59 | \( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} - T^{11} + T^{12} - T^{13} + T^{14} - T^{15} + T^{16} \) |
| 61 | \( ( 1 - T )^{8}( 1 + T )^{8} \) |
| 67 | \( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} - T^{11} + T^{12} - T^{13} + T^{14} - T^{15} + T^{16} \) |
| 71 | \( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} - T^{11} + T^{12} - T^{13} + T^{14} - T^{15} + T^{16} \) |
| 73 | \( ( 1 - T )^{8}( 1 + T )^{8} \) |
| 79 | \( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} - T^{11} + T^{12} - T^{13} + T^{14} - T^{15} + T^{16} \) |
| 83 | \( ( 1 - T )^{8}( 1 + T )^{8} \) |
| 89 | \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} + T^{11} + T^{12} + T^{13} + T^{14} + T^{15} + T^{16} \) |
| 97 | \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} + T^{11} + T^{12} + T^{13} + T^{14} + T^{15} + T^{16} \) |
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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
−3.80642658011263653297078555141, −3.76437926059240789088596879847, −3.30223964035611724733609718710, −3.19787900908226755241798301228, −3.17527248639336012488012904466, −3.04305579025869939290784636491, −2.93091165829713083256841943632, −2.85408114831330708037109695947, −2.81091317571186736137974445574, −2.49555732933272116472955829328, −2.36670539411092366513667299140, −2.28266759207303809638257782514, −2.24302655909573327290725459407, −2.14662660463702519635917196713, −2.11597635926812016971399776759, −2.11476836511370390829149032485, −1.64090611925824925553859100538, −1.59727607229403092339675820149, −1.43320741742030384736609466220, −1.16652205259605001079905540390, −1.06087306204815760432566247763, −1.03976494447053543862312684232, −0.74854703968546146769893733064, −0.53828749899104840074917411452, −0.43801926170625730243950948416,
0.43801926170625730243950948416, 0.53828749899104840074917411452, 0.74854703968546146769893733064, 1.03976494447053543862312684232, 1.06087306204815760432566247763, 1.16652205259605001079905540390, 1.43320741742030384736609466220, 1.59727607229403092339675820149, 1.64090611925824925553859100538, 2.11476836511370390829149032485, 2.11597635926812016971399776759, 2.14662660463702519635917196713, 2.24302655909573327290725459407, 2.28266759207303809638257782514, 2.36670539411092366513667299140, 2.49555732933272116472955829328, 2.81091317571186736137974445574, 2.85408114831330708037109695947, 2.93091165829713083256841943632, 3.04305579025869939290784636491, 3.17527248639336012488012904466, 3.19787900908226755241798301228, 3.30223964035611724733609718710, 3.76437926059240789088596879847, 3.80642658011263653297078555141
Plot not available for L-functions of degree greater than 10.
