| L(s) = 1 | + 16·7-s + 8·13-s − 8·19-s + 108·25-s − 8·31-s + 8·37-s + 40·43-s − 148·49-s + 72·61-s + 16·67-s − 64·79-s + 128·91-s − 24·97-s − 88·103-s − 472·109-s − 44·121-s + 127-s + 131-s − 128·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 684·169-s + ⋯ |
| L(s) = 1 | + 16/7·7-s + 8/13·13-s − 0.421·19-s + 4.31·25-s − 0.258·31-s + 8/37·37-s + 0.930·43-s − 3.02·49-s + 1.18·61-s + 0.238·67-s − 0.810·79-s + 1.40·91-s − 0.247·97-s − 0.854·103-s − 4.33·109-s − 0.363·121-s + 0.00787·127-s + 0.00763·131-s − 0.962·133-s + 0.00729·137-s + 0.00719·139-s + 0.00671·149-s + 0.00662·151-s + 0.00636·157-s + 0.00613·163-s + 0.00598·167-s − 4.04·169-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{16} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{16} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s+1)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(8.906145029\) |
| \(L(\frac12)\) |
\(\approx\) |
\(8.906145029\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 11 | \( ( 1 + p T^{2} )^{4} \) |
| good | 5 | \( 1 - 108 T^{2} + 6032 T^{4} - 229428 T^{6} + 6536814 T^{8} - 229428 p^{4} T^{10} + 6032 p^{8} T^{12} - 108 p^{12} T^{14} + p^{16} T^{16} \) |
| 7 | \( ( 1 - 8 T + 170 T^{2} - 832 T^{3} + 11166 T^{4} - 832 p^{2} T^{5} + 170 p^{4} T^{6} - 8 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 13 | \( ( 1 - 4 T + 366 T^{2} - 1884 T^{3} + 85702 T^{4} - 1884 p^{2} T^{5} + 366 p^{4} T^{6} - 4 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 17 | \( 1 - 1616 T^{2} + 1222012 T^{4} - 581803440 T^{6} + 196216557318 T^{8} - 581803440 p^{4} T^{10} + 1222012 p^{8} T^{12} - 1616 p^{12} T^{14} + p^{16} T^{16} \) |
| 19 | \( ( 1 + 4 T + 964 T^{2} + 7940 T^{3} + 429074 T^{4} + 7940 p^{2} T^{5} + 964 p^{4} T^{6} + 4 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 23 | \( 1 - 332 T^{2} - 86704 T^{4} - 30291348 T^{6} + 129128075822 T^{8} - 30291348 p^{4} T^{10} - 86704 p^{8} T^{12} - 332 p^{12} T^{14} + p^{16} T^{16} \) |
| 29 | \( 1 - 1176 T^{2} + 424220 T^{4} - 662538664 T^{6} + 1144574423942 T^{8} - 662538664 p^{4} T^{10} + 424220 p^{8} T^{12} - 1176 p^{12} T^{14} + p^{16} T^{16} \) |
| 31 | \( ( 1 + 4 T + 2872 T^{2} - 6788 T^{3} + 3579678 T^{4} - 6788 p^{2} T^{5} + 2872 p^{4} T^{6} + 4 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 37 | \( ( 1 - 4 T + 3032 T^{2} - 12764 T^{3} + 5942526 T^{4} - 12764 p^{2} T^{5} + 3032 p^{4} T^{6} - 4 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 41 | \( 1 - 6704 T^{2} + 24872444 T^{4} - 1574523216 p T^{6} + 124787581589894 T^{8} - 1574523216 p^{5} T^{10} + 24872444 p^{8} T^{12} - 6704 p^{12} T^{14} + p^{16} T^{16} \) |
| 43 | \( ( 1 - 20 T + 3140 T^{2} + 53356 T^{3} + 3217970 T^{4} + 53356 p^{2} T^{5} + 3140 p^{4} T^{6} - 20 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 47 | \( 1 - 11228 T^{2} + 57556048 T^{4} - 186994471236 T^{6} + 458903060844078 T^{8} - 186994471236 p^{4} T^{10} + 57556048 p^{8} T^{12} - 11228 p^{12} T^{14} + p^{16} T^{16} \) |
| 53 | \( 1 - 10828 T^{2} + 64244688 T^{4} - 264093123156 T^{6} + 835356584267758 T^{8} - 264093123156 p^{4} T^{10} + 64244688 p^{8} T^{12} - 10828 p^{12} T^{14} + p^{16} T^{16} \) |
| 59 | \( 1 - 18568 T^{2} + 170500188 T^{4} - 1007538767928 T^{6} + 4155443188638598 T^{8} - 1007538767928 p^{4} T^{10} + 170500188 p^{8} T^{12} - 18568 p^{12} T^{14} + p^{16} T^{16} \) |
| 61 | \( ( 1 - 36 T + 5374 T^{2} - 124140 T^{3} + 21784550 T^{4} - 124140 p^{2} T^{5} + 5374 p^{4} T^{6} - 36 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 67 | \( ( 1 - 8 T + 11084 T^{2} + 145096 T^{3} + 58196294 T^{4} + 145096 p^{2} T^{5} + 11084 p^{4} T^{6} - 8 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 71 | \( 1 - 18188 T^{2} + 198821392 T^{4} - 1524834301524 T^{6} + 8869330769068974 T^{8} - 1524834301524 p^{4} T^{10} + 198821392 p^{8} T^{12} - 18188 p^{12} T^{14} + p^{16} T^{16} \) |
| 73 | \( ( 1 + 14444 T^{2} + 123904 T^{3} + 96871718 T^{4} + 123904 p^{2} T^{5} + 14444 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 79 | \( ( 1 + 32 T + 9194 T^{2} + 972888 T^{3} + 51952702 T^{4} + 972888 p^{2} T^{5} + 9194 p^{4} T^{6} + 32 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 83 | \( 1 - 37216 T^{2} + 679673244 T^{4} - 7967837141664 T^{6} + 65114186922062854 T^{8} - 7967837141664 p^{4} T^{10} + 679673244 p^{8} T^{12} - 37216 p^{12} T^{14} + p^{16} T^{16} \) |
| 89 | \( 1 - 45344 T^{2} + 965811460 T^{4} - 12892629471072 T^{6} + 120248488936603014 T^{8} - 12892629471072 p^{4} T^{10} + 965811460 p^{8} T^{12} - 45344 p^{12} T^{14} + p^{16} T^{16} \) |
| 97 | \( ( 1 + 12 T + 33904 T^{2} + 288660 T^{3} + 460866542 T^{4} + 288660 p^{2} T^{5} + 33904 p^{4} T^{6} + 12 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
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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
−3.77982339008043418588864167036, −3.67106972907809094129849794326, −3.45084516796043222930568794332, −3.44271286100943526950445160416, −3.21094606735149100132604040091, −3.17610507686766003895969414189, −2.90264487023400924195647649879, −2.78740582572309498310081923561, −2.77824357587274010881590726118, −2.65399816372106478755088263822, −2.40138794614152441690759816833, −2.38826153543061417109678684677, −2.28899091332625256945544803130, −1.84249800038943176577623078400, −1.79826111138178872351749496228, −1.70852937683611755689103098342, −1.46439726689328518680863948177, −1.38725954452693766780172875704, −1.37774321566302264581358843611, −1.15614559887047763737849237713, −0.973342196678825573052989625362, −0.797859784439019155306936729447, −0.53482729477010998371194193999, −0.35407823441539770892048489511, −0.15362373611467626315986233879,
0.15362373611467626315986233879, 0.35407823441539770892048489511, 0.53482729477010998371194193999, 0.797859784439019155306936729447, 0.973342196678825573052989625362, 1.15614559887047763737849237713, 1.37774321566302264581358843611, 1.38725954452693766780172875704, 1.46439726689328518680863948177, 1.70852937683611755689103098342, 1.79826111138178872351749496228, 1.84249800038943176577623078400, 2.28899091332625256945544803130, 2.38826153543061417109678684677, 2.40138794614152441690759816833, 2.65399816372106478755088263822, 2.77824357587274010881590726118, 2.78740582572309498310081923561, 2.90264487023400924195647649879, 3.17610507686766003895969414189, 3.21094606735149100132604040091, 3.44271286100943526950445160416, 3.45084516796043222930568794332, 3.67106972907809094129849794326, 3.77982339008043418588864167036
Plot not available for L-functions of degree greater than 10.
