| L(s) = 1 | + 4·2-s + 8·4-s + 4·5-s − 4·7-s + 8·8-s + 16·10-s − 16·11-s + 32·13-s − 16·14-s + 16-s − 80·17-s + 32·20-s − 64·22-s − 56·23-s + 128·26-s − 32·28-s + 16·31-s + 16·32-s − 320·34-s − 16·35-s + 128·37-s + 32·40-s + 56·41-s + 8·43-s − 128·44-s − 224·46-s − 128·47-s + ⋯ |
| L(s) = 1 | + 2·2-s + 2·4-s + 4/5·5-s − 4/7·7-s + 8-s + 8/5·10-s − 1.45·11-s + 2.46·13-s − 8/7·14-s + 1/16·16-s − 4.70·17-s + 8/5·20-s − 2.90·22-s − 2.43·23-s + 4.92·26-s − 8/7·28-s + 0.516·31-s + 1/2·32-s − 9.41·34-s − 0.457·35-s + 3.45·37-s + 4/5·40-s + 1.36·41-s + 8/43·43-s − 2.90·44-s − 4.86·46-s − 2.72·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{32} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{32} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+1)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(3.627314894\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.627314894\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 - 4 T + 16 T^{2} + 8 p^{2} T^{3} - 41 p^{2} T^{4} + 8 p^{4} T^{5} + 16 p^{4} T^{6} - 4 p^{6} T^{7} + p^{8} T^{8} \) |
| good | 2 | \( 1 - p^{2} T + p^{3} T^{2} - p^{3} T^{3} - T^{4} - p^{3} T^{5} + 9 p^{3} T^{6} - 51 p^{2} T^{7} + 481 T^{8} - 51 p^{4} T^{9} + 9 p^{7} T^{10} - p^{9} T^{11} - p^{8} T^{12} - p^{13} T^{13} + p^{15} T^{14} - p^{16} T^{15} + p^{16} T^{16} \) |
| 7 | \( 1 + 4 T + 8 T^{2} - 40 p T^{3} - 3502 T^{4} - 2092 p T^{5} + 8640 T^{6} + 884076 T^{7} + 5309203 T^{8} + 884076 p^{2} T^{9} + 8640 p^{4} T^{10} - 2092 p^{7} T^{11} - 3502 p^{8} T^{12} - 40 p^{11} T^{13} + 8 p^{12} T^{14} + 4 p^{14} T^{15} + p^{16} T^{16} \) |
| 11 | \( ( 1 + 8 T - 140 T^{2} - 304 T^{3} + 19231 T^{4} - 304 p^{2} T^{5} - 140 p^{4} T^{6} + 8 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 13 | \( 1 - 32 T + 512 T^{2} + 1856 T^{3} - 178690 T^{4} + 3324896 T^{5} - 13185024 T^{6} - 402955872 T^{7} + 9725217763 T^{8} - 402955872 p^{2} T^{9} - 13185024 p^{4} T^{10} + 3324896 p^{6} T^{11} - 178690 p^{8} T^{12} + 1856 p^{10} T^{13} + 512 p^{12} T^{14} - 32 p^{14} T^{15} + p^{16} T^{16} \) |
| 17 | \( ( 1 + 40 T + 800 T^{2} + 15240 T^{3} + 281858 T^{4} + 15240 p^{2} T^{5} + 800 p^{4} T^{6} + 40 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 19 | \( ( 1 - 940 T^{2} + 450438 T^{4} - 940 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 23 | \( 1 + 56 T + 1568 T^{2} - 14000 T^{3} - 1900162 T^{4} - 62168456 T^{5} - 403979520 T^{6} + 25186881384 T^{7} + 1065945022723 T^{8} + 25186881384 p^{2} T^{9} - 403979520 p^{4} T^{10} - 62168456 p^{6} T^{11} - 1900162 p^{8} T^{12} - 14000 p^{10} T^{13} + 1568 p^{12} T^{14} + 56 p^{14} T^{15} + p^{16} T^{16} \) |
| 29 | \( 1 + 2128 T^{2} + 2362750 T^{4} + 1598281216 T^{6} + 986884245091 T^{8} + 1598281216 p^{4} T^{10} + 2362750 p^{8} T^{12} + 2128 p^{12} T^{14} + p^{16} T^{16} \) |
| 31 | \( ( 1 - 8 T - 1658 T^{2} + 1600 T^{3} + 1980259 T^{4} + 1600 p^{2} T^{5} - 1658 p^{4} T^{6} - 8 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 37 | \( ( 1 - 64 T + 2048 T^{2} - 58176 T^{3} + 1440962 T^{4} - 58176 p^{2} T^{5} + 2048 p^{4} T^{6} - 64 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 41 | \( ( 1 - 28 T - 2558 T^{2} + 560 T^{3} + 7025299 T^{4} + 560 p^{2} T^{5} - 2558 p^{4} T^{6} - 28 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 43 | \( 1 - 8 T + 32 T^{2} + 10256 T^{3} + 516542 T^{4} - 18807592 T^{5} + 186524160 T^{6} - 47893576536 T^{7} - 10622766431261 T^{8} - 47893576536 p^{2} T^{9} + 186524160 p^{4} T^{10} - 18807592 p^{6} T^{11} + 516542 p^{8} T^{12} + 10256 p^{10} T^{13} + 32 p^{12} T^{14} - 8 p^{14} T^{15} + p^{16} T^{16} \) |
| 47 | \( 1 + 128 T + 8192 T^{2} + 35584 T^{3} - 25982818 T^{4} - 1966544768 T^{5} - 38233374720 T^{6} + 2478898271616 T^{7} + 233877878146819 T^{8} + 2478898271616 p^{2} T^{9} - 38233374720 p^{4} T^{10} - 1966544768 p^{6} T^{11} - 25982818 p^{8} T^{12} + 35584 p^{10} T^{13} + 8192 p^{12} T^{14} + 128 p^{14} T^{15} + p^{16} T^{16} \) |
| 53 | \( ( 1 - 56 T + 1568 T^{2} - 155064 T^{3} + 15333122 T^{4} - 155064 p^{2} T^{5} + 1568 p^{4} T^{6} - 56 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 59 | \( 1 - 200 T^{2} + 5686222 T^{4} + 5976188800 T^{6} - 115435309171037 T^{8} + 5976188800 p^{4} T^{10} + 5686222 p^{8} T^{12} - 200 p^{12} T^{14} + p^{16} T^{16} \) |
| 61 | \( ( 1 + 100 T + 2002 T^{2} + 55600 T^{3} + 12912163 T^{4} + 55600 p^{2} T^{5} + 2002 p^{4} T^{6} + 100 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 67 | \( 1 - 200 T + 20000 T^{2} - 223600 T^{3} - 131382658 T^{4} + 15256933400 T^{5} - 398735040000 T^{6} - 49947855269400 T^{7} + 6402331096508323 T^{8} - 49947855269400 p^{2} T^{9} - 398735040000 p^{4} T^{10} + 15256933400 p^{6} T^{11} - 131382658 p^{8} T^{12} - 223600 p^{10} T^{13} + 20000 p^{12} T^{14} - 200 p^{14} T^{15} + p^{16} T^{16} \) |
| 71 | \( ( 1 + 68 T + p^{2} T^{2} )^{8} \) |
| 73 | \( ( 1 - 76 T + 2888 T^{2} + 65436 T^{3} - 36833458 T^{4} + 65436 p^{2} T^{5} + 2888 p^{4} T^{6} - 76 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 79 | \( ( 1 + 11882 T^{2} + 102231843 T^{4} + 11882 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 83 | \( 1 - 16 T + 128 T^{2} + 200608 T^{3} - 81509506 T^{4} + 1845929968 T^{5} + 1020122112 T^{6} - 13246117740336 T^{7} + 4228343627312611 T^{8} - 13246117740336 p^{2} T^{9} + 1020122112 p^{4} T^{10} + 1845929968 p^{6} T^{11} - 81509506 p^{8} T^{12} + 200608 p^{10} T^{13} + 128 p^{12} T^{14} - 16 p^{14} T^{15} + p^{16} T^{16} \) |
| 89 | \( ( 1 - 16060 T^{2} + 188845638 T^{4} - 16060 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 97 | \( 1 - 20 T + 200 T^{2} + 343640 T^{3} - 153987838 T^{4} + 4351821140 T^{5} + 2805369600 T^{6} - 42528219223740 T^{7} + 15206860513102483 T^{8} - 42528219223740 p^{2} T^{9} + 2805369600 p^{4} T^{10} + 4351821140 p^{6} T^{11} - 153987838 p^{8} T^{12} + 343640 p^{10} T^{13} + 200 p^{12} T^{14} - 20 p^{14} T^{15} + p^{16} T^{16} \) |
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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
−4.67219442494162932630083170968, −4.55797579822303830895792828780, −4.49134508332764119373547434289, −4.48597793867068312478616851612, −4.21312700436042516210498467015, −4.12002230919333999436279150144, −3.84034250556260428468875750024, −3.70375841609506865161761868880, −3.49798299850646984600414273768, −3.45115393625661500734754348218, −3.41617810057394756715598203865, −2.99948782466115868452340185067, −2.74771114727062370896431330658, −2.69940558214163811297135266721, −2.63715773862741152731155332174, −2.41238404632099709933292433707, −2.18877332901193193389352014644, −2.07114848661952479089890671422, −1.98247319964299755417936588059, −1.67668036791352579525836818871, −1.36614358578366839335291495645, −1.14158378233546107363186497348, −0.820143388375480665618943817086, −0.36401519979949832026447891143, −0.16385126761001890608876670694,
0.16385126761001890608876670694, 0.36401519979949832026447891143, 0.820143388375480665618943817086, 1.14158378233546107363186497348, 1.36614358578366839335291495645, 1.67668036791352579525836818871, 1.98247319964299755417936588059, 2.07114848661952479089890671422, 2.18877332901193193389352014644, 2.41238404632099709933292433707, 2.63715773862741152731155332174, 2.69940558214163811297135266721, 2.74771114727062370896431330658, 2.99948782466115868452340185067, 3.41617810057394756715598203865, 3.45115393625661500734754348218, 3.49798299850646984600414273768, 3.70375841609506865161761868880, 3.84034250556260428468875750024, 4.12002230919333999436279150144, 4.21312700436042516210498467015, 4.48597793867068312478616851612, 4.49134508332764119373547434289, 4.55797579822303830895792828780, 4.67219442494162932630083170968
Plot not available for L-functions of degree greater than 10.
