| L(s) = 1 | − 2·3-s + 6·7-s + 6·9-s + 12·11-s − 8·13-s + 6·17-s − 12·21-s − 6·23-s − 4·25-s − 12·27-s − 24·33-s + 6·37-s + 16·39-s + 12·41-s + 10·43-s + 2·49-s − 12·51-s + 24·53-s − 24·59-s − 4·61-s + 36·63-s − 54·67-s + 12·69-s − 36·71-s + 8·75-s + 72·77-s − 16·79-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 2.26·7-s + 2·9-s + 3.61·11-s − 2.21·13-s + 1.45·17-s − 2.61·21-s − 1.25·23-s − 4/5·25-s − 2.30·27-s − 4.17·33-s + 0.986·37-s + 2.56·39-s + 1.87·41-s + 1.52·43-s + 2/7·49-s − 1.68·51-s + 3.29·53-s − 3.12·59-s − 0.512·61-s + 4.53·63-s − 6.59·67-s + 1.44·69-s − 4.27·71-s + 0.923·75-s + 8.20·77-s − 1.80·79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 5^{8} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 5^{8} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.575509124\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.575509124\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( ( 1 + T^{2} )^{4} \) |
| 13 | \( 1 + 8 T + 16 T^{2} - 40 T^{3} - 290 T^{4} - 40 p T^{5} + 16 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) |
| good | 3 | \( 1 + 2 T - 2 T^{2} - 4 T^{3} + T^{4} - 4 T^{5} + 10 T^{6} + 26 T^{7} - 20 T^{8} + 26 p T^{9} + 10 p^{2} T^{10} - 4 p^{3} T^{11} + p^{4} T^{12} - 4 p^{5} T^{13} - 2 p^{6} T^{14} + 2 p^{7} T^{15} + p^{8} T^{16} \) |
| 7 | \( 1 - 6 T + 34 T^{2} - 132 T^{3} + 493 T^{4} - 1620 T^{5} + 682 p T^{6} - 13830 T^{7} + 35716 T^{8} - 13830 p T^{9} + 682 p^{3} T^{10} - 1620 p^{3} T^{11} + 493 p^{4} T^{12} - 132 p^{5} T^{13} + 34 p^{6} T^{14} - 6 p^{7} T^{15} + p^{8} T^{16} \) |
| 11 | \( ( 1 - 3 T + 14 T^{2} - 3 p T^{3} + p^{2} T^{4} )^{4} \) |
| 17 | \( 1 - 6 T - 14 T^{2} + 12 p T^{3} - 107 T^{4} - 3744 T^{5} + 11950 T^{6} + 36282 T^{7} - 332828 T^{8} + 36282 p T^{9} + 11950 p^{2} T^{10} - 3744 p^{3} T^{11} - 107 p^{4} T^{12} + 12 p^{6} T^{13} - 14 p^{6} T^{14} - 6 p^{7} T^{15} + p^{8} T^{16} \) |
| 19 | \( 1 + 46 T^{2} + 1057 T^{4} + 15502 T^{6} + 227284 T^{8} + 15502 p^{2} T^{10} + 1057 p^{4} T^{12} + 46 p^{6} T^{14} + p^{8} T^{16} \) |
| 23 | \( 1 + 6 T - 38 T^{2} - 324 T^{3} + 829 T^{4} + 9132 T^{5} - 626 T^{6} - 99546 T^{7} - 202460 T^{8} - 99546 p T^{9} - 626 p^{2} T^{10} + 9132 p^{3} T^{11} + 829 p^{4} T^{12} - 324 p^{5} T^{13} - 38 p^{6} T^{14} + 6 p^{7} T^{15} + p^{8} T^{16} \) |
| 29 | \( 1 - 74 T^{2} - 192 T^{3} + 2905 T^{4} + 9888 T^{5} - 56570 T^{6} - 165216 T^{7} + 1030180 T^{8} - 165216 p T^{9} - 56570 p^{2} T^{10} + 9888 p^{3} T^{11} + 2905 p^{4} T^{12} - 192 p^{5} T^{13} - 74 p^{6} T^{14} + p^{8} T^{16} \) |
| 31 | \( ( 1 - 28 T^{2} + 1926 T^{4} - 28 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 37 | \( 1 - 6 T + 70 T^{2} - 348 T^{3} + 2305 T^{4} - 3024 T^{5} - 84422 T^{6} + 586362 T^{7} - 5624996 T^{8} + 586362 p T^{9} - 84422 p^{2} T^{10} - 3024 p^{3} T^{11} + 2305 p^{4} T^{12} - 348 p^{5} T^{13} + 70 p^{6} T^{14} - 6 p^{7} T^{15} + p^{8} T^{16} \) |
| 41 | \( ( 1 - 6 T + 61 T^{2} - 294 T^{3} + 1212 T^{4} - 294 p T^{5} + 61 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 43 | \( 1 - 10 T - 6 T^{2} - 308 T^{3} + 4301 T^{4} + 7476 T^{5} + 35134 T^{6} - 521194 T^{7} - 4291308 T^{8} - 521194 p T^{9} + 35134 p^{2} T^{10} + 7476 p^{3} T^{11} + 4301 p^{4} T^{12} - 308 p^{5} T^{13} - 6 p^{6} T^{14} - 10 p^{7} T^{15} + p^{8} T^{16} \) |
| 47 | \( ( 1 - 82 T^{2} + p^{2} T^{4} )^{4} \) |
| 53 | \( ( 1 - 12 T + 56 T^{2} + 12 T^{3} - 306 T^{4} + 12 p T^{5} + 56 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 59 | \( 1 + 24 T + 410 T^{2} + 5232 T^{3} + 55429 T^{4} + 515160 T^{5} + 4379474 T^{6} + 35226912 T^{7} + 274579084 T^{8} + 35226912 p T^{9} + 4379474 p^{2} T^{10} + 515160 p^{3} T^{11} + 55429 p^{4} T^{12} + 5232 p^{5} T^{13} + 410 p^{6} T^{14} + 24 p^{7} T^{15} + p^{8} T^{16} \) |
| 61 | \( 1 + 4 T - 126 T^{2} + 392 T^{3} + 10901 T^{4} - 53424 T^{5} - 222326 T^{6} + 2281108 T^{7} - 2679156 T^{8} + 2281108 p T^{9} - 222326 p^{2} T^{10} - 53424 p^{3} T^{11} + 10901 p^{4} T^{12} + 392 p^{5} T^{13} - 126 p^{6} T^{14} + 4 p^{7} T^{15} + p^{8} T^{16} \) |
| 67 | \( 1 + 54 T + 1558 T^{2} + 31644 T^{3} + 504145 T^{4} + 6670188 T^{5} + 75661330 T^{6} + 748893006 T^{7} + 6525069388 T^{8} + 748893006 p T^{9} + 75661330 p^{2} T^{10} + 6670188 p^{3} T^{11} + 504145 p^{4} T^{12} + 31644 p^{5} T^{13} + 1558 p^{6} T^{14} + 54 p^{7} T^{15} + p^{8} T^{16} \) |
| 71 | \( 1 + 36 T + 698 T^{2} + 9576 T^{3} + 99037 T^{4} + 800280 T^{5} + 5013074 T^{6} + 24891156 T^{7} + 146900620 T^{8} + 24891156 p T^{9} + 5013074 p^{2} T^{10} + 800280 p^{3} T^{11} + 99037 p^{4} T^{12} + 9576 p^{5} T^{13} + 698 p^{6} T^{14} + 36 p^{7} T^{15} + p^{8} T^{16} \) |
| 73 | \( 1 - 320 T^{2} + 54748 T^{4} - 6368960 T^{6} + 541047430 T^{8} - 6368960 p^{2} T^{10} + 54748 p^{4} T^{12} - 320 p^{6} T^{14} + p^{8} T^{16} \) |
| 79 | \( ( 1 + 8 T + 136 T^{2} + 392 T^{3} + 8638 T^{4} + 392 p T^{5} + 136 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 83 | \( 1 - 184 T^{2} + 17788 T^{4} - 2042440 T^{6} + 209794726 T^{8} - 2042440 p^{2} T^{10} + 17788 p^{4} T^{12} - 184 p^{6} T^{14} + p^{8} T^{16} \) |
| 89 | \( 1 + 24 T + 530 T^{2} + 8112 T^{3} + 116461 T^{4} + 1452168 T^{5} + 16850810 T^{6} + 177993840 T^{7} + 1745476972 T^{8} + 177993840 p T^{9} + 16850810 p^{2} T^{10} + 1452168 p^{3} T^{11} + 116461 p^{4} T^{12} + 8112 p^{5} T^{13} + 530 p^{6} T^{14} + 24 p^{7} T^{15} + p^{8} T^{16} \) |
| 97 | \( 1 + 30 T + 634 T^{2} + 10020 T^{3} + 129661 T^{4} + 1433376 T^{5} + 14194246 T^{6} + 134625630 T^{7} + 1292575636 T^{8} + 134625630 p T^{9} + 14194246 p^{2} T^{10} + 1433376 p^{3} T^{11} + 129661 p^{4} T^{12} + 10020 p^{5} T^{13} + 634 p^{6} T^{14} + 30 p^{7} T^{15} + p^{8} T^{16} \) |
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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
−5.66556952044906277306580885754, −5.24966203764217923475134394409, −4.97294123988410212046570306359, −4.76084435262839747445338023786, −4.69644018803483543420545100368, −4.61980104263590401138682521249, −4.55419633479545802237819638477, −4.36805520518789901397622507788, −4.22777394802076983601264433189, −4.20116655315744911936204547023, −4.03421503065901668692345532547, −3.81196785478660970369013151617, −3.50528892861905740085357026812, −3.43960957350700788253176807088, −3.00947218793149423274226479325, −2.93506040657268022627593500901, −2.75674761174743290793420948402, −2.45996850235239659036784096039, −2.19855523080941054973023126502, −1.79395502158262866374458092708, −1.61517614365116339326544243150, −1.39986556318128548056087364612, −1.38241620604394333990585995176, −1.35036155603496843858478719400, −0.49500227612639370727210155501,
0.49500227612639370727210155501, 1.35036155603496843858478719400, 1.38241620604394333990585995176, 1.39986556318128548056087364612, 1.61517614365116339326544243150, 1.79395502158262866374458092708, 2.19855523080941054973023126502, 2.45996850235239659036784096039, 2.75674761174743290793420948402, 2.93506040657268022627593500901, 3.00947218793149423274226479325, 3.43960957350700788253176807088, 3.50528892861905740085357026812, 3.81196785478660970369013151617, 4.03421503065901668692345532547, 4.20116655315744911936204547023, 4.22777394802076983601264433189, 4.36805520518789901397622507788, 4.55419633479545802237819638477, 4.61980104263590401138682521249, 4.69644018803483543420545100368, 4.76084435262839747445338023786, 4.97294123988410212046570306359, 5.24966203764217923475134394409, 5.66556952044906277306580885754
Plot not available for L-functions of degree greater than 10.
