| L(s) = 1 | + 14·5-s − 6·7-s + 38·9-s + 26·11-s − 18·17-s − 16·19-s + 12·23-s + 15·25-s − 84·35-s − 62·43-s + 532·45-s + 22·47-s − 167·49-s + 364·55-s + 158·61-s − 228·63-s − 170·73-s − 156·77-s + 667·81-s + 64·83-s − 252·85-s − 224·95-s + 988·99-s − 144·101-s + 168·115-s + 108·119-s − 113·121-s + ⋯ |
| L(s) = 1 | + 14/5·5-s − 6/7·7-s + 38/9·9-s + 2.36·11-s − 1.05·17-s − 0.842·19-s + 0.521·23-s + 3/5·25-s − 2.39·35-s − 1.44·43-s + 11.8·45-s + 0.468·47-s − 3.40·49-s + 6.61·55-s + 2.59·61-s − 3.61·63-s − 2.32·73-s − 2.02·77-s + 8.23·81-s + 0.771·83-s − 2.96·85-s − 2.35·95-s + 9.97·99-s − 1.42·101-s + 1.46·115-s + 0.907·119-s − 0.933·121-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 19^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 19^{8}\right)^{s/2} \, \Gamma_{\C}(s+1)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.6593902286\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6593902286\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 19 | \( 1 + 16 T + 1024 T^{2} + 16784 T^{3} + 25258 p T^{4} + 16784 p^{2} T^{5} + 1024 p^{4} T^{6} + 16 p^{6} T^{7} + p^{8} T^{8} \) |
| good | 3 | \( 1 - 38 T^{2} + 259 p T^{4} - 10870 T^{6} + 112324 T^{8} - 10870 p^{4} T^{10} + 259 p^{9} T^{12} - 38 p^{12} T^{14} + p^{16} T^{16} \) |
| 5 | \( ( 1 - 7 T + 66 T^{2} - 269 T^{3} + 1698 T^{4} - 269 p^{2} T^{5} + 66 p^{4} T^{6} - 7 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 7 | \( ( 1 + 3 T + 97 T^{2} + 662 T^{3} + 5042 T^{4} + 662 p^{2} T^{5} + 97 p^{4} T^{6} + 3 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 11 | \( ( 1 - 13 T + 310 T^{2} - 2459 T^{3} + 39826 T^{4} - 2459 p^{2} T^{5} + 310 p^{4} T^{6} - 13 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 13 | \( 1 - 574 T^{2} + 172793 T^{4} - 35200710 T^{6} + 6184140468 T^{8} - 35200710 p^{4} T^{10} + 172793 p^{8} T^{12} - 574 p^{12} T^{14} + p^{16} T^{16} \) |
| 17 | \( ( 1 + 9 T + 961 T^{2} + 7550 T^{3} + 393230 T^{4} + 7550 p^{2} T^{5} + 961 p^{4} T^{6} + 9 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 23 | \( ( 1 - 6 T + 1617 T^{2} - 10026 T^{3} + 1178940 T^{4} - 10026 p^{2} T^{5} + 1617 p^{4} T^{6} - 6 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 29 | \( 1 - 4230 T^{2} + 9126089 T^{4} - 12816162454 T^{6} + 15119017764 p^{2} T^{8} - 12816162454 p^{4} T^{10} + 9126089 p^{8} T^{12} - 4230 p^{12} T^{14} + p^{16} T^{16} \) |
| 31 | \( 1 - 2456 T^{2} + 5828684 T^{4} - 7563279976 T^{6} + 9147516798310 T^{8} - 7563279976 p^{4} T^{10} + 5828684 p^{8} T^{12} - 2456 p^{12} T^{14} + p^{16} T^{16} \) |
| 37 | \( 1 - 2680 T^{2} + 6548812 T^{4} - 11648081160 T^{6} + 19219307024678 T^{8} - 11648081160 p^{4} T^{10} + 6548812 p^{8} T^{12} - 2680 p^{12} T^{14} + p^{16} T^{16} \) |
| 41 | \( 1 - 152 p T^{2} + 23450188 T^{4} - 58856569000 T^{6} + 113964380393446 T^{8} - 58856569000 p^{4} T^{10} + 23450188 p^{8} T^{12} - 152 p^{13} T^{14} + p^{16} T^{16} \) |
| 43 | \( ( 1 + 31 T + 5414 T^{2} + 164693 T^{3} + 13442026 T^{4} + 164693 p^{2} T^{5} + 5414 p^{4} T^{6} + 31 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 47 | \( ( 1 - 11 T + 2534 T^{2} - 157669 T^{3} + 4049362 T^{4} - 157669 p^{2} T^{5} + 2534 p^{4} T^{6} - 11 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 53 | \( 1 - 17062 T^{2} + 136055849 T^{4} - 669634945014 T^{6} + 2246747683623492 T^{8} - 669634945014 p^{4} T^{10} + 136055849 p^{8} T^{12} - 17062 p^{12} T^{14} + p^{16} T^{16} \) |
| 59 | \( 1 - 7486 T^{2} + 40383577 T^{4} - 117111635910 T^{6} + 421631927790260 T^{8} - 117111635910 p^{4} T^{10} + 40383577 p^{8} T^{12} - 7486 p^{12} T^{14} + p^{16} T^{16} \) |
| 61 | \( ( 1 - 79 T + 7578 T^{2} - 583157 T^{3} + 45737394 T^{4} - 583157 p^{2} T^{5} + 7578 p^{4} T^{6} - 79 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 67 | \( 1 - 20374 T^{2} + 217748153 T^{4} - 1576483164230 T^{6} + 8252062911497188 T^{8} - 1576483164230 p^{4} T^{10} + 217748153 p^{8} T^{12} - 20374 p^{12} T^{14} + p^{16} T^{16} \) |
| 71 | \( 1 - 25632 T^{2} + 320394172 T^{4} - 2622179822560 T^{6} + 15430330602633734 T^{8} - 2622179822560 p^{4} T^{10} + 320394172 p^{8} T^{12} - 25632 p^{12} T^{14} + p^{16} T^{16} \) |
| 73 | \( ( 1 + 85 T + 9205 T^{2} + 1070570 T^{3} + 69877894 T^{4} + 1070570 p^{2} T^{5} + 9205 p^{4} T^{6} + 85 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 79 | \( 1 - 30616 T^{2} + 471334732 T^{4} - 4813265721960 T^{6} + 35213431967679590 T^{8} - 4813265721960 p^{4} T^{10} + 471334732 p^{8} T^{12} - 30616 p^{12} T^{14} + p^{16} T^{16} \) |
| 83 | \( ( 1 - 32 T + 8256 T^{2} - 17248 T^{3} + 18380510 T^{4} - 17248 p^{2} T^{5} + 8256 p^{4} T^{6} - 32 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 89 | \( 1 - 33600 T^{2} + 633167852 T^{4} - 8047414082688 T^{6} + 74241105074276134 T^{8} - 8047414082688 p^{4} T^{10} + 633167852 p^{8} T^{12} - 33600 p^{12} T^{14} + p^{16} T^{16} \) |
| 97 | \( 1 - 17088 T^{2} + 293381372 T^{4} - 2765846651200 T^{6} + 32564321945822214 T^{8} - 2765846651200 p^{4} T^{10} + 293381372 p^{8} T^{12} - 17088 p^{12} T^{14} + p^{16} T^{16} \) |
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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
−3.85082099007628164929697961481, −3.79410101865583299040193846762, −3.78523005152924465846447853702, −3.56936884621608761712814779970, −3.46348079633043982461986594201, −3.38284480523926770418801435756, −3.29757081920693703137985300529, −3.00858838901977980644673901411, −2.68447525591737174051097275196, −2.53368760194414098337303679142, −2.45127091884934280105056731604, −2.40480304500729751711629030700, −2.27232995297995693654506991583, −2.02418170386205082457347930756, −1.94986336236992861028353851071, −1.73001906677043213735925019252, −1.64923463037847795363862716893, −1.42327820629775891484352913849, −1.31690430358357537681731919151, −1.28805684367483648191954125102, −1.22983048059242293525906376186, −1.09088548609037308245604612564, −0.69633376871348169151216139132, −0.12049005880654991553113841988, −0.088411025906869278873778109268,
0.088411025906869278873778109268, 0.12049005880654991553113841988, 0.69633376871348169151216139132, 1.09088548609037308245604612564, 1.22983048059242293525906376186, 1.28805684367483648191954125102, 1.31690430358357537681731919151, 1.42327820629775891484352913849, 1.64923463037847795363862716893, 1.73001906677043213735925019252, 1.94986336236992861028353851071, 2.02418170386205082457347930756, 2.27232995297995693654506991583, 2.40480304500729751711629030700, 2.45127091884934280105056731604, 2.53368760194414098337303679142, 2.68447525591737174051097275196, 3.00858838901977980644673901411, 3.29757081920693703137985300529, 3.38284480523926770418801435756, 3.46348079633043982461986594201, 3.56936884621608761712814779970, 3.78523005152924465846447853702, 3.79410101865583299040193846762, 3.85082099007628164929697961481
Plot not available for L-functions of degree greater than 10.
