| L(s) = 1 | + 32·13-s + 16·17-s − 48·25-s + 80·29-s − 176·37-s − 144·41-s − 28·49-s − 48·53-s − 192·61-s + 272·73-s + 80·89-s + 528·97-s + 128·101-s + 208·109-s + 160·113-s + 296·121-s − 448·125-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 624·169-s + ⋯ |
| L(s) = 1 | + 2.46·13-s + 0.941·17-s − 1.91·25-s + 2.75·29-s − 4.75·37-s − 3.51·41-s − 4/7·49-s − 0.905·53-s − 3.14·61-s + 3.72·73-s + 0.898·89-s + 5.44·97-s + 1.26·101-s + 1.90·109-s + 1.41·113-s + 2.44·121-s − 3.58·125-s + 0.00787·127-s + 0.00763·131-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 − 3.69·169-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{40} \cdot 3^{16} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{40} \cdot 3^{16} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.677684644\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.677684644\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( ( 1 + p T^{2} )^{4} \) |
| good | 5 | \( ( 1 + 24 T^{2} + 224 T^{3} + 78 T^{4} + 224 p^{2} T^{5} + 24 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 11 | \( 1 - 296 T^{2} + 72860 T^{4} - 12907416 T^{6} + 1704788486 T^{8} - 12907416 p^{4} T^{10} + 72860 p^{8} T^{12} - 296 p^{12} T^{14} + p^{16} T^{16} \) |
| 13 | \( ( 1 - 16 T + 696 T^{2} - 7536 T^{3} + 176398 T^{4} - 7536 p^{2} T^{5} + 696 p^{4} T^{6} - 16 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 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 | \( 1 - 1936 T^{2} + 1795420 T^{4} - 1065665392 T^{6} + 449725089670 T^{8} - 1065665392 p^{4} T^{10} + 1795420 p^{8} T^{12} - 1936 p^{12} T^{14} + p^{16} T^{16} \) |
| 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 | \( ( 1 - 40 T + 3660 T^{2} - 100632 T^{3} + 4741126 T^{4} - 100632 p^{2} T^{5} + 3660 p^{4} T^{6} - 40 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 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 | \( ( 1 + 88 T + 3692 T^{2} + 66024 T^{3} + 510982 T^{4} + 66024 p^{2} T^{5} + 3692 p^{4} T^{6} + 88 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 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 | \( 1 - 4392 T^{2} + 15655580 T^{4} - 32786290584 T^{6} + 70340912004102 T^{8} - 32786290584 p^{4} T^{10} + 15655580 p^{8} T^{12} - 4392 p^{12} T^{14} + p^{16} T^{16} \) |
| 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 | \( ( 1 + 24 T + 6108 T^{2} + 242920 T^{3} + 18930726 T^{4} + 242920 p^{2} T^{5} + 6108 p^{4} T^{6} + 24 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 59 | \( 1 - 21392 T^{2} + 213726428 T^{4} - 1316320178544 T^{6} + 5495654048256518 T^{8} - 1316320178544 p^{4} T^{10} + 213726428 p^{8} T^{12} - 21392 p^{12} T^{14} + p^{16} T^{16} \) |
| 61 | \( ( 1 + 96 T + 12728 T^{2} + 895872 T^{3} + 70065870 T^{4} + 895872 p^{2} T^{5} + 12728 p^{4} T^{6} + 96 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 67 | \( 1 - 16200 T^{2} + 120433884 T^{4} - 596093100152 T^{6} + 2634880977729030 T^{8} - 596093100152 p^{4} T^{10} + 120433884 p^{8} T^{12} - 16200 p^{12} T^{14} + p^{16} T^{16} \) |
| 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 | \( 1 - 19280 T^{2} + 168996572 T^{4} - 1078781742000 T^{6} + 6945567152070790 T^{8} - 1078781742000 p^{4} T^{10} + 168996572 p^{8} T^{12} - 19280 p^{12} T^{14} + p^{16} T^{16} \) |
| 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
−3.60837547942905631281840889431, −3.51623264768618148489714139785, −3.47101079786706958122403280634, −3.25301957724537622915658687561, −3.22529659709668625789641438501, −3.20962907647488606088815228575, −2.91463483658251957255966941653, −2.76937137469501745165422467206, −2.73026124910802363250251780844, −2.59069221273005491449235707433, −2.15578457245629357207234758579, −2.10634140875638387355172344740, −1.95695736983072527012688622839, −1.93052229447422281014159098174, −1.79190029212925728936622272662, −1.68187797557070768250987252299, −1.53835575874196013512451557281, −1.46751387908249705463144284599, −1.21180600293873835474697863464, −0.957836417453114833973034344763, −0.792504659276148726884460527571, −0.70458246481393302480937237528, −0.59862510529094673392568028528, −0.25277075329775112567712701600, −0.085709610164939527483237803853,
0.085709610164939527483237803853, 0.25277075329775112567712701600, 0.59862510529094673392568028528, 0.70458246481393302480937237528, 0.792504659276148726884460527571, 0.957836417453114833973034344763, 1.21180600293873835474697863464, 1.46751387908249705463144284599, 1.53835575874196013512451557281, 1.68187797557070768250987252299, 1.79190029212925728936622272662, 1.93052229447422281014159098174, 1.95695736983072527012688622839, 2.10634140875638387355172344740, 2.15578457245629357207234758579, 2.59069221273005491449235707433, 2.73026124910802363250251780844, 2.76937137469501745165422467206, 2.91463483658251957255966941653, 3.20962907647488606088815228575, 3.22529659709668625789641438501, 3.25301957724537622915658687561, 3.47101079786706958122403280634, 3.51623264768618148489714139785, 3.60837547942905631281840889431
Plot not available for L-functions of degree greater than 10.
