| L(s) = 1 | + 4·2-s + 6·4-s + 4·7-s + 2·9-s − 12·11-s + 16·14-s − 15·16-s + 8·18-s − 12·19-s − 48·22-s + 12·23-s + 24·28-s − 4·29-s − 24·32-s + 12·36-s + 16·37-s − 48·38-s − 24·43-s − 72·44-s + 48·46-s − 32·47-s + 30·49-s − 16·58-s + 12·59-s − 16·61-s + 8·63-s − 6·64-s + ⋯ |
| L(s) = 1 | + 2.82·2-s + 3·4-s + 1.51·7-s + 2/3·9-s − 3.61·11-s + 4.27·14-s − 3.75·16-s + 1.88·18-s − 2.75·19-s − 10.2·22-s + 2.50·23-s + 4.53·28-s − 0.742·29-s − 4.24·32-s + 2·36-s + 2.63·37-s − 7.78·38-s − 3.65·43-s − 10.8·44-s + 7.07·46-s − 4.66·47-s + 30/7·49-s − 2.10·58-s + 1.56·59-s − 2.04·61-s + 1.00·63-s − 3/4·64-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 5^{16} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 5^{16} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.07108930430\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.07108930430\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( ( 1 - T + T^{2} )^{4} \) |
| 3 | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 5 | \( 1 \) |
| 13 | \( 1 + 191 T^{4} + p^{4} T^{8} \) |
| good | 7 | \( 1 - 4 T - 2 p T^{2} + 40 T^{3} + 260 T^{4} - 400 T^{5} - 2500 T^{6} + 748 T^{7} + 22075 T^{8} + 748 p T^{9} - 2500 p^{2} T^{10} - 400 p^{3} T^{11} + 260 p^{4} T^{12} + 40 p^{5} T^{13} - 2 p^{7} T^{14} - 4 p^{7} T^{15} + p^{8} T^{16} \) |
| 11 | \( 1 + 12 T + 98 T^{2} + 600 T^{3} + 3105 T^{4} + 14040 T^{5} + 57010 T^{6} + 212028 T^{7} + 727508 T^{8} + 212028 p T^{9} + 57010 p^{2} T^{10} + 14040 p^{3} T^{11} + 3105 p^{4} T^{12} + 600 p^{5} T^{13} + 98 p^{6} T^{14} + 12 p^{7} T^{15} + p^{8} T^{16} \) |
| 17 | \( 1 + 30 T^{2} + 412 T^{4} + 1404 T^{5} + 1188 T^{6} + 52704 T^{7} - 39381 T^{8} + 52704 p T^{9} + 1188 p^{2} T^{10} + 1404 p^{3} T^{11} + 412 p^{4} T^{12} + 30 p^{6} T^{14} + p^{8} T^{16} \) |
| 19 | \( 1 + 12 T + 130 T^{2} + 984 T^{3} + 6940 T^{4} + 40824 T^{5} + 226348 T^{6} + 1102116 T^{7} + 5088619 T^{8} + 1102116 p T^{9} + 226348 p^{2} T^{10} + 40824 p^{3} T^{11} + 6940 p^{4} T^{12} + 984 p^{5} T^{13} + 130 p^{6} T^{14} + 12 p^{7} T^{15} + p^{8} T^{16} \) |
| 23 | \( 1 - 12 T + 104 T^{2} - 672 T^{3} + 3201 T^{4} - 15048 T^{5} + 66616 T^{6} - 300612 T^{7} + 1581440 T^{8} - 300612 p T^{9} + 66616 p^{2} T^{10} - 15048 p^{3} T^{11} + 3201 p^{4} T^{12} - 672 p^{5} T^{13} + 104 p^{6} T^{14} - 12 p^{7} T^{15} + p^{8} T^{16} \) |
| 29 | \( 1 + 4 T - 16 T^{2} + 360 T^{3} + 1898 T^{4} - 3796 T^{5} + 93856 T^{6} + 16140 p T^{7} - 1010141 T^{8} + 16140 p^{2} T^{9} + 93856 p^{2} T^{10} - 3796 p^{3} T^{11} + 1898 p^{4} T^{12} + 360 p^{5} T^{13} - 16 p^{6} T^{14} + 4 p^{7} T^{15} + p^{8} T^{16} \) |
| 31 | \( 1 - 44 T^{2} + 1092 T^{4} - 59836 T^{6} + 2531498 T^{8} - 59836 p^{2} T^{10} + 1092 p^{4} T^{12} - 44 p^{6} T^{14} + p^{8} T^{16} \) |
| 37 | \( 1 - 16 T + 112 T^{2} - 848 T^{3} + 5177 T^{4} - 6616 T^{5} - 69904 T^{6} + 945736 T^{7} - 8640896 T^{8} + 945736 p T^{9} - 69904 p^{2} T^{10} - 6616 p^{3} T^{11} + 5177 p^{4} T^{12} - 848 p^{5} T^{13} + 112 p^{6} T^{14} - 16 p^{7} T^{15} + p^{8} T^{16} \) |
| 41 | \( 1 + 62 T^{2} + 2268 T^{4} + 8820 T^{5} - 51932 T^{6} + 661920 T^{7} - 4392949 T^{8} + 661920 p T^{9} - 51932 p^{2} T^{10} + 8820 p^{3} T^{11} + 2268 p^{4} T^{12} + 62 p^{6} T^{14} + p^{8} T^{16} \) |
| 43 | \( 1 + 24 T + 358 T^{2} + 3984 T^{3} + 36420 T^{4} + 308484 T^{5} + 2479268 T^{6} + 18815016 T^{7} + 130697051 T^{8} + 18815016 p T^{9} + 2479268 p^{2} T^{10} + 308484 p^{3} T^{11} + 36420 p^{4} T^{12} + 3984 p^{5} T^{13} + 358 p^{6} T^{14} + 24 p^{7} T^{15} + p^{8} T^{16} \) |
| 47 | \( ( 1 + 16 T + 208 T^{2} + 1712 T^{3} + 13410 T^{4} + 1712 p T^{5} + 208 p^{2} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 53 | \( 1 - 348 T^{2} + 56308 T^{4} - 5507820 T^{6} + 355337610 T^{8} - 5507820 p^{2} T^{10} + 56308 p^{4} T^{12} - 348 p^{6} T^{14} + p^{8} T^{16} \) |
| 59 | \( 1 - 12 T + 172 T^{2} - 1488 T^{3} + 11014 T^{4} - 14820 T^{5} - 245792 T^{6} + 6441804 T^{7} - 56520173 T^{8} + 6441804 p T^{9} - 245792 p^{2} T^{10} - 14820 p^{3} T^{11} + 11014 p^{4} T^{12} - 1488 p^{5} T^{13} + 172 p^{6} T^{14} - 12 p^{7} T^{15} + p^{8} T^{16} \) |
| 61 | \( 1 + 16 T + 48 T^{2} - 64 T^{3} + 1121 T^{4} - 24000 T^{5} - 352400 T^{6} - 1570640 T^{7} - 8868240 T^{8} - 1570640 p T^{9} - 352400 p^{2} T^{10} - 24000 p^{3} T^{11} + 1121 p^{4} T^{12} - 64 p^{5} T^{13} + 48 p^{6} T^{14} + 16 p^{7} T^{15} + p^{8} T^{16} \) |
| 67 | \( 1 + 24 T + 180 T^{2} + 1008 T^{3} + 15322 T^{4} + 68808 T^{5} - 888624 T^{6} - 7559688 T^{7} - 23098173 T^{8} - 7559688 p T^{9} - 888624 p^{2} T^{10} + 68808 p^{3} T^{11} + 15322 p^{4} T^{12} + 1008 p^{5} T^{13} + 180 p^{6} T^{14} + 24 p^{7} T^{15} + p^{8} T^{16} \) |
| 71 | \( 1 - 60 T + 1880 T^{2} - 40800 T^{3} + 688329 T^{4} - 9588096 T^{5} + 113898952 T^{6} - 1173297084 T^{7} + 10563264272 T^{8} - 1173297084 p T^{9} + 113898952 p^{2} T^{10} - 9588096 p^{3} T^{11} + 688329 p^{4} T^{12} - 40800 p^{5} T^{13} + 1880 p^{6} T^{14} - 60 p^{7} T^{15} + p^{8} T^{16} \) |
| 73 | \( ( 1 - 12 T + 258 T^{2} - 1824 T^{3} + 24779 T^{4} - 1824 p T^{5} + 258 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 79 | \( ( 1 - 4 T + 214 T^{2} - 736 T^{3} + 21952 T^{4} - 736 p T^{5} + 214 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 83 | \( ( 1 - 4 T + 262 T^{2} - 656 T^{3} + 29451 T^{4} - 656 p T^{5} + 262 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 89 | \( 1 + 24 T + 494 T^{2} + 7248 T^{3} + 93564 T^{4} + 1132308 T^{5} + 12981412 T^{6} + 139202424 T^{7} + 1395966731 T^{8} + 139202424 p T^{9} + 12981412 p^{2} T^{10} + 1132308 p^{3} T^{11} + 93564 p^{4} T^{12} + 7248 p^{5} T^{13} + 494 p^{6} T^{14} + 24 p^{7} T^{15} + p^{8} T^{16} \) |
| 97 | \( 1 - 174 T^{2} - 672 T^{3} + 7489 T^{4} + 91056 T^{5} - 577710 T^{6} - 3685584 T^{7} + 134900292 T^{8} - 3685584 p T^{9} - 577710 p^{2} T^{10} + 91056 p^{3} T^{11} + 7489 p^{4} T^{12} - 672 p^{5} T^{13} - 174 p^{6} T^{14} + p^{8} T^{16} \) |
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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
−3.78036990056367113233537519578, −3.71080955485916705948140575743, −3.70985588834321272869696564841, −3.53571956560385495095430034467, −3.51800141134890447993372972252, −3.27104090659391209746382636436, −3.25511793243621752905550488228, −2.97145806558802533348409756119, −2.91976143334587753784511808447, −2.65964565107735102766456039420, −2.62580517181091582533775461405, −2.49342639903648603551198180196, −2.48865381665560229435569301823, −2.31174384641433197032996891411, −2.30746907426642587573542048683, −1.89838215918216042582059927383, −1.75376432649991535748656086351, −1.69744329265963482506117627508, −1.63735962728241484229058542178, −1.41414527747597979536837462619, −0.945493926788141734074948948399, −0.875667863558646217505392187554, −0.70296541671444803453310681754, −0.34874072373444794730460983453, −0.01701668494604432881328023677,
0.01701668494604432881328023677, 0.34874072373444794730460983453, 0.70296541671444803453310681754, 0.875667863558646217505392187554, 0.945493926788141734074948948399, 1.41414527747597979536837462619, 1.63735962728241484229058542178, 1.69744329265963482506117627508, 1.75376432649991535748656086351, 1.89838215918216042582059927383, 2.30746907426642587573542048683, 2.31174384641433197032996891411, 2.48865381665560229435569301823, 2.49342639903648603551198180196, 2.62580517181091582533775461405, 2.65964565107735102766456039420, 2.91976143334587753784511808447, 2.97145806558802533348409756119, 3.25511793243621752905550488228, 3.27104090659391209746382636436, 3.51800141134890447993372972252, 3.53571956560385495095430034467, 3.70985588834321272869696564841, 3.71080955485916705948140575743, 3.78036990056367113233537519578
Plot not available for L-functions of degree greater than 10.
