| L(s) = 1 | + 8·3-s + 16·9-s + 32·11-s − 16·17-s + 88·19-s + 48·25-s − 8·27-s + 256·33-s − 144·41-s − 224·43-s − 28·49-s − 128·51-s + 704·57-s + 232·59-s − 368·67-s − 272·73-s + 384·75-s + 100·81-s − 424·83-s + 80·89-s + 528·97-s + 512·99-s − 176·107-s − 160·113-s + 216·121-s − 1.15e3·123-s + 127-s + ⋯ |
| L(s) = 1 | + 8/3·3-s + 16/9·9-s + 2.90·11-s − 0.941·17-s + 4.63·19-s + 1.91·25-s − 0.296·27-s + 7.75·33-s − 3.51·41-s − 5.20·43-s − 4/7·49-s − 2.50·51-s + 12.3·57-s + 3.93·59-s − 5.49·67-s − 3.72·73-s + 5.11·75-s + 1.23·81-s − 5.10·83-s + 0.898·89-s + 5.44·97-s + 5.17·99-s − 1.64·107-s − 1.41·113-s + 1.78·121-s − 9.36·123-s + 0.00787·127-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{64} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{64} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(8.840547713\) |
| \(L(\frac12)\) |
\(\approx\) |
\(8.840547713\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( ( 1 + p T^{2} )^{4} \) |
| good | 3 | \( ( 1 - 4 T + 16 T^{2} - 20 p T^{3} + 158 T^{4} - 20 p^{3} T^{5} + 16 p^{4} T^{6} - 4 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 5 | \( 1 - 48 T^{2} + 732 T^{4} + 16432 T^{6} - 1001466 T^{8} + 16432 p^{4} T^{10} + 732 p^{8} T^{12} - 48 p^{12} T^{14} + p^{16} T^{16} \) |
| 11 | \( ( 1 - 16 T + 276 T^{2} - 1840 T^{3} + 27782 T^{4} - 1840 p^{2} T^{5} + 276 p^{4} T^{6} - 16 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 13 | \( 1 - 1136 T^{2} + 596060 T^{4} - 187756944 T^{6} + 38751742342 T^{8} - 187756944 p^{4} T^{10} + 596060 p^{8} T^{12} - 1136 p^{12} T^{14} + p^{16} T^{16} \) |
| 17 | \( ( 1 + 8 T + 428 T^{2} - 1416 T^{3} + 75814 T^{4} - 1416 p^{2} T^{5} + 428 p^{4} T^{6} + 8 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 19 | \( ( 1 - 44 T + 1936 T^{2} - 47828 T^{3} + 1128094 T^{4} - 47828 p^{2} T^{5} + 1936 p^{4} T^{6} - 44 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 23 | \( 1 - 1928 T^{2} + 2366876 T^{4} - 1917103032 T^{6} + 1186521008582 T^{8} - 1917103032 p^{4} T^{10} + 2366876 p^{8} T^{12} - 1928 p^{12} T^{14} + p^{16} T^{16} \) |
| 29 | \( 1 - 5720 T^{2} + 14827292 T^{4} - 22985018856 T^{6} + 23490964042630 T^{8} - 22985018856 p^{4} T^{10} + 14827292 p^{8} T^{12} - 5720 p^{12} T^{14} + p^{16} T^{16} \) |
| 31 | \( 1 - 3624 T^{2} + 5953500 T^{4} - 5905968152 T^{6} + 5382616134 p^{2} T^{8} - 5905968152 p^{4} T^{10} + 5953500 p^{8} T^{12} - 3624 p^{12} T^{14} + p^{16} T^{16} \) |
| 37 | \( 1 + 360 T^{2} + 3032604 T^{4} + 2655359320 T^{6} + 6705554738694 T^{8} + 2655359320 p^{4} T^{10} + 3032604 p^{8} T^{12} + 360 p^{12} T^{14} + p^{16} T^{16} \) |
| 41 | \( ( 1 + 72 T + 4364 T^{2} + 180408 T^{3} + 8880422 T^{4} + 180408 p^{2} T^{5} + 4364 p^{4} T^{6} + 72 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 43 | \( ( 1 + 112 T + 8468 T^{2} + 425040 T^{3} + 19578758 T^{4} + 425040 p^{2} T^{5} + 8468 p^{4} T^{6} + 112 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 47 | \( 1 - 6696 T^{2} + 34041564 T^{4} - 108976927256 T^{6} + 286725887177670 T^{8} - 108976927256 p^{4} T^{10} + 34041564 p^{8} T^{12} - 6696 p^{12} T^{14} + p^{16} T^{16} \) |
| 53 | \( 1 - 11640 T^{2} + 63508956 T^{4} - 235884348872 T^{6} + 714590361466758 T^{8} - 235884348872 p^{4} T^{10} + 63508956 p^{8} T^{12} - 11640 p^{12} T^{14} + p^{16} T^{16} \) |
| 59 | \( ( 1 - 116 T + 17424 T^{2} - 1222412 T^{3} + 96865118 T^{4} - 1222412 p^{2} T^{5} + 17424 p^{4} T^{6} - 116 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 61 | \( 1 - 16240 T^{2} + 130126300 T^{4} - 693430250128 T^{6} + 2856274871188870 T^{8} - 693430250128 p^{4} T^{10} + 130126300 p^{8} T^{12} - 16240 p^{12} T^{14} + p^{16} T^{16} \) |
| 67 | \( ( 1 + 184 T + 25028 T^{2} + 2353896 T^{3} + 180133414 T^{4} + 2353896 p^{2} T^{5} + 25028 p^{4} T^{6} + 184 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 71 | \( 1 - 21000 T^{2} + 247457180 T^{4} - 1991232124728 T^{6} + 11615880585488070 T^{8} - 1991232124728 p^{4} T^{10} + 247457180 p^{8} T^{12} - 21000 p^{12} T^{14} + p^{16} T^{16} \) |
| 73 | \( ( 1 + 136 T + 21660 T^{2} + 1811512 T^{3} + 170528390 T^{4} + 1811512 p^{2} T^{5} + 21660 p^{4} T^{6} + 136 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 79 | \( 1 - 26248 T^{2} + 289540636 T^{4} - 1851543245752 T^{6} + 10296312398061382 T^{8} - 1851543245752 p^{4} T^{10} + 289540636 p^{8} T^{12} - 26248 p^{12} T^{14} + p^{16} T^{16} \) |
| 83 | \( ( 1 + 212 T + 32112 T^{2} + 3651372 T^{3} + 342998878 T^{4} + 3651372 p^{2} T^{5} + 32112 p^{4} T^{6} + 212 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 89 | \( ( 1 - 40 T + 25916 T^{2} - 837912 T^{3} + 285914566 T^{4} - 837912 p^{2} T^{5} + 25916 p^{4} T^{6} - 40 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 97 | \( ( 1 - 264 T + 52364 T^{2} - 7260024 T^{3} + 808802534 T^{4} - 7260024 p^{2} T^{5} + 52364 p^{4} T^{6} - 264 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
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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
−3.49756349165512377120508565206, −3.43176851108153458306312006581, −3.39730028786017277342229358647, −3.23583500997218240039798817529, −3.09297172056718003294421429196, −3.07620286292336150794311298990, −3.04043719151172060203597400714, −3.01437725144366112151198099778, −2.69370365080033507493858363472, −2.67137496213850795347856224534, −2.58263382501433454249003172484, −2.50123796598617634537183652165, −1.98192352513985541161029299005, −1.88237551733600271315460265742, −1.70863459729737546247399340908, −1.67518599991345412715659400034, −1.60180545328914363066668434277, −1.55235299367950704248002676026, −1.41390832086841241997924985594, −1.05520189629176357201456878997, −1.02078915903581091519949678320, −0.78200009438667895537188212725, −0.53289431989420737226166392981, −0.43044475892864852544744528629, −0.085346640442176882852302135767,
0.085346640442176882852302135767, 0.43044475892864852544744528629, 0.53289431989420737226166392981, 0.78200009438667895537188212725, 1.02078915903581091519949678320, 1.05520189629176357201456878997, 1.41390832086841241997924985594, 1.55235299367950704248002676026, 1.60180545328914363066668434277, 1.67518599991345412715659400034, 1.70863459729737546247399340908, 1.88237551733600271315460265742, 1.98192352513985541161029299005, 2.50123796598617634537183652165, 2.58263382501433454249003172484, 2.67137496213850795347856224534, 2.69370365080033507493858363472, 3.01437725144366112151198099778, 3.04043719151172060203597400714, 3.07620286292336150794311298990, 3.09297172056718003294421429196, 3.23583500997218240039798817529, 3.39730028786017277342229358647, 3.43176851108153458306312006581, 3.49756349165512377120508565206
Plot not available for L-functions of degree greater than 10.
