| L(s) = 1 | + 16·2-s + 136·4-s + 816·8-s − 8·9-s + 3.87e3·16-s − 128·18-s − 8·23-s − 22·25-s + 20·29-s + 1.55e4·32-s − 1.08e3·36-s − 128·46-s − 352·50-s + 320·58-s + 5.42e4·64-s − 48·71-s − 6.52e3·72-s + 36·81-s − 1.08e3·92-s − 2.99e3·100-s + 2.72e3·116-s − 4·121-s + 127-s + 1.70e5·128-s + 131-s + 137-s + 139-s + ⋯ |
| L(s) = 1 | + 11.3·2-s + 68·4-s + 288.·8-s − 8/3·9-s + 969·16-s − 30.1·18-s − 1.66·23-s − 4.39·25-s + 3.71·29-s + 2.74e3·32-s − 181.·36-s − 18.8·46-s − 49.7·50-s + 42.0·58-s + 6.78e3·64-s − 5.69·71-s − 769.·72-s + 4·81-s − 113.·92-s − 299.·100-s + 252.·116-s − 0.363·121-s + 0.0887·127-s + 1.50e4·128-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{16} \cdot 7^{16} \cdot 23^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{16} \cdot 7^{16} \cdot 23^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(13452.62721\) |
| \(L(\frac12)\) |
\(\approx\) |
\(13452.62721\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( ( 1 - T )^{16} \) |
| 3 | \( ( 1 + T^{2} )^{8} \) |
| 7 | \( 1 - 64 T^{4} + 5566 T^{8} - 64 p^{4} T^{12} + p^{8} T^{16} \) |
| 23 | \( ( 1 + 2 T - 18 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{4} \) |
| good | 5 | \( ( 1 + 11 T^{2} + 56 T^{4} + 181 T^{6} + 766 T^{8} + 181 p^{2} T^{10} + 56 p^{4} T^{12} + 11 p^{6} T^{14} + p^{8} T^{16} )^{2} \) |
| 11 | \( ( 1 + 2 T^{2} + 128 T^{4} - 1906 T^{6} + 1950 T^{8} - 1906 p^{2} T^{10} + 128 p^{4} T^{12} + 2 p^{6} T^{14} + p^{8} T^{16} )^{2} \) |
| 13 | \( ( 1 - 55 T^{2} + 1526 T^{4} - 30065 T^{6} + 450706 T^{8} - 30065 p^{2} T^{10} + 1526 p^{4} T^{12} - 55 p^{6} T^{14} + p^{8} T^{16} )^{2} \) |
| 17 | \( ( 1 + 80 T^{2} + 3296 T^{4} + 90960 T^{6} + 1810686 T^{8} + 90960 p^{2} T^{10} + 3296 p^{4} T^{12} + 80 p^{6} T^{14} + p^{8} T^{16} )^{2} \) |
| 19 | \( ( 1 + 62 T^{2} + 2528 T^{4} + 73234 T^{6} + 1573150 T^{8} + 73234 p^{2} T^{10} + 2528 p^{4} T^{12} + 62 p^{6} T^{14} + p^{8} T^{16} )^{2} \) |
| 29 | \( ( 1 - 5 T + 36 T^{2} + 65 T^{3} + 6 T^{4} + 65 p T^{5} + 36 p^{2} T^{6} - 5 p^{3} T^{7} + p^{4} T^{8} )^{4} \) |
| 31 | \( ( 1 - p T^{2} )^{16} \) |
| 37 | \( ( 1 - 195 T^{2} + 17176 T^{4} - 948285 T^{6} + 39151806 T^{8} - 948285 p^{2} T^{10} + 17176 p^{4} T^{12} - 195 p^{6} T^{14} + p^{8} T^{16} )^{2} \) |
| 41 | \( ( 1 - 223 T^{2} + 23918 T^{4} - 1631921 T^{6} + 78518050 T^{8} - 1631921 p^{2} T^{10} + 23918 p^{4} T^{12} - 223 p^{6} T^{14} + p^{8} T^{16} )^{2} \) |
| 43 | \( ( 1 - 5 p T^{2} + 24056 T^{4} - 1733905 T^{6} + 88109326 T^{8} - 1733905 p^{2} T^{10} + 24056 p^{4} T^{12} - 5 p^{7} T^{14} + p^{8} T^{16} )^{2} \) |
| 47 | \( ( 1 - 195 T^{2} + 20926 T^{4} - 1539005 T^{6} + 83012546 T^{8} - 1539005 p^{2} T^{10} + 20926 p^{4} T^{12} - 195 p^{6} T^{14} + p^{8} T^{16} )^{2} \) |
| 53 | \( ( 1 - 150 T^{2} + 18536 T^{4} - 1409370 T^{6} + 89381326 T^{8} - 1409370 p^{2} T^{10} + 18536 p^{4} T^{12} - 150 p^{6} T^{14} + p^{8} T^{16} )^{2} \) |
| 59 | \( ( 1 - 208 T^{2} + 21308 T^{4} - 1748656 T^{6} + 119202150 T^{8} - 1748656 p^{2} T^{10} + 21308 p^{4} T^{12} - 208 p^{6} T^{14} + p^{8} T^{16} )^{2} \) |
| 61 | \( ( 1 + 362 T^{2} + 62888 T^{4} + 6803174 T^{6} + 498621390 T^{8} + 6803174 p^{2} T^{10} + 62888 p^{4} T^{12} + 362 p^{6} T^{14} + p^{8} T^{16} )^{2} \) |
| 67 | \( ( 1 - 130 T^{2} + 17696 T^{4} - 1500590 T^{6} + 118838686 T^{8} - 1500590 p^{2} T^{10} + 17696 p^{4} T^{12} - 130 p^{6} T^{14} + p^{8} T^{16} )^{2} \) |
| 71 | \( ( 1 + 12 T + 208 T^{2} + 1884 T^{3} + 18430 T^{4} + 1884 p T^{5} + 208 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} )^{4} \) |
| 73 | \( ( 1 - 340 T^{2} + 56996 T^{4} - 6277740 T^{6} + 517870326 T^{8} - 6277740 p^{2} T^{10} + 56996 p^{4} T^{12} - 340 p^{6} T^{14} + p^{8} T^{16} )^{2} \) |
| 79 | \( ( 1 - 388 T^{2} + 77128 T^{4} - 10074636 T^{6} + 936162830 T^{8} - 10074636 p^{2} T^{10} + 77128 p^{4} T^{12} - 388 p^{6} T^{14} + p^{8} T^{16} )^{2} \) |
| 83 | \( ( 1 + 470 T^{2} + 102496 T^{4} + 13995450 T^{6} + 1353683486 T^{8} + 13995450 p^{2} T^{10} + 102496 p^{4} T^{12} + 470 p^{6} T^{14} + p^{8} T^{16} )^{2} \) |
| 89 | \( ( 1 + 332 T^{2} + 45848 T^{4} + 3627844 T^{6} + 262456750 T^{8} + 3627844 p^{2} T^{10} + 45848 p^{4} T^{12} + 332 p^{6} T^{14} + p^{8} T^{16} )^{2} \) |
| 97 | \( ( 1 + 65 T^{2} + 5786 T^{4} - 121745 T^{6} - 33558614 T^{8} - 121745 p^{2} T^{10} + 5786 p^{4} T^{12} + 65 p^{6} T^{14} + p^{8} T^{16} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{32} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−2.79494349721897366480981545337, −2.73568492129839724669798394906, −2.65257446559921405231324643388, −2.64301233439825789808078496642, −2.55329248069224456677461320401, −2.31442113253445170359463676884, −2.29288062733223649849704586312, −2.23654992034401569302678734988, −2.17055356994319726608135970294, −2.15572536887119119389730000879, −2.04674014584405730668731531213, −1.97930310972824931392128144116, −1.84554192349600011005050269523, −1.76561149618192031991952249137, −1.70415801616692373578297891522, −1.46269018140867535809920387365, −1.44191531365675699045935681492, −1.40992859418983396070486789904, −1.28782009270965295239021251812, −1.26435548458558100734672988266, −1.15616509516127515989703305788, −0.795847048610270155761882714827, −0.61115702707839104302933696082, −0.45692993508611996510842782456, −0.11896233994203977967644911414,
0.11896233994203977967644911414, 0.45692993508611996510842782456, 0.61115702707839104302933696082, 0.795847048610270155761882714827, 1.15616509516127515989703305788, 1.26435548458558100734672988266, 1.28782009270965295239021251812, 1.40992859418983396070486789904, 1.44191531365675699045935681492, 1.46269018140867535809920387365, 1.70415801616692373578297891522, 1.76561149618192031991952249137, 1.84554192349600011005050269523, 1.97930310972824931392128144116, 2.04674014584405730668731531213, 2.15572536887119119389730000879, 2.17055356994319726608135970294, 2.23654992034401569302678734988, 2.29288062733223649849704586312, 2.31442113253445170359463676884, 2.55329248069224456677461320401, 2.64301233439825789808078496642, 2.65257446559921405231324643388, 2.73568492129839724669798394906, 2.79494349721897366480981545337
Plot not available for L-functions of degree greater than 10.
