| L(s) = 1 | − 220·4-s − 990·9-s + 2.32e4·16-s − 2.85e4·19-s + 400·25-s + 5.89e3·31-s + 2.17e5·36-s − 5.01e4·49-s + 1.49e6·61-s − 1.91e6·64-s + 6.27e6·76-s − 1.28e6·79-s + 5.20e4·81-s − 8.80e4·100-s − 7.29e6·109-s + 7.40e6·121-s − 1.29e6·124-s + 127-s + 131-s + 137-s + 139-s − 2.30e7·144-s + 149-s + 151-s + 157-s + 163-s + 167-s + ⋯ |
| L(s) = 1 | − 3.43·4-s − 1.35·9-s + 5.67·16-s − 4.15·19-s + 0.0255·25-s + 0.197·31-s + 4.66·36-s − 0.426·49-s + 6.57·61-s − 7.31·64-s + 14.2·76-s − 2.60·79-s + 0.0978·81-s − 0.0879·100-s − 5.63·109-s + 4.18·121-s − 0.680·124-s − 7.70·144-s + 6.36·169-s + 5.64·171-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(7-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+3)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(0.1752378926\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1752378926\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + 110 p^{2} T^{2} + 11458 p^{4} T^{4} + 110 p^{14} T^{6} + p^{24} T^{8} \) |
| 5 | \( 1 - 16 p^{2} T^{2} + 122502 p^{5} T^{4} - 16 p^{14} T^{6} + p^{24} T^{8} \) |
| good | 2 | \( ( 1 + 55 p T^{2} + 51 p^{7} T^{4} + 55 p^{13} T^{6} + p^{24} T^{8} )^{2} \) |
| 7 | \( ( 1 + 25070 T^{2} + 24459864738 T^{4} + 25070 p^{12} T^{6} + p^{24} T^{8} )^{2} \) |
| 11 | \( ( 1 - 3703684 T^{2} + 8599901652006 T^{4} - 3703684 p^{12} T^{6} + p^{24} T^{8} )^{2} \) |
| 13 | \( ( 1 - 15357460 T^{2} + 105038551446918 T^{4} - 15357460 p^{12} T^{6} + p^{24} T^{8} )^{2} \) |
| 17 | \( ( 1 + 85683440 T^{2} + 2999150676836958 T^{4} + 85683440 p^{12} T^{6} + p^{24} T^{8} )^{2} \) |
| 19 | \( ( 1 + 7130 T + 106796298 T^{2} + 7130 p^{6} T^{3} + p^{12} T^{4} )^{4} \) |
| 23 | \( ( 1 + 114181310 T^{2} + 1518600265588578 T^{4} + 114181310 p^{12} T^{6} + p^{24} T^{8} )^{2} \) |
| 29 | \( ( 1 - 1601436244 T^{2} + 1282742916791218566 T^{4} - 1601436244 p^{12} T^{6} + p^{24} T^{8} )^{2} \) |
| 31 | \( ( 1 - 1474 T + 1615069506 T^{2} - 1474 p^{6} T^{3} + p^{12} T^{4} )^{4} \) |
| 37 | \( ( 1 - 9431870260 T^{2} + 35235911676992435718 T^{4} - 9431870260 p^{12} T^{6} + p^{24} T^{8} )^{2} \) |
| 41 | \( ( 1 - 15901141504 T^{2} + \)\(10\!\cdots\!66\)\( T^{4} - 15901141504 p^{12} T^{6} + p^{24} T^{8} )^{2} \) |
| 43 | \( ( 1 - 23595056770 T^{2} + \)\(21\!\cdots\!58\)\( T^{4} - 23595056770 p^{12} T^{6} + p^{24} T^{8} )^{2} \) |
| 47 | \( ( 1 + 17629463390 T^{2} + \)\(20\!\cdots\!98\)\( T^{4} + 17629463390 p^{12} T^{6} + p^{24} T^{8} )^{2} \) |
| 53 | \( ( 1 + 61278994640 T^{2} + \)\(18\!\cdots\!98\)\( T^{4} + 61278994640 p^{12} T^{6} + p^{24} T^{8} )^{2} \) |
| 59 | \( ( 1 + 32569142396 T^{2} + \)\(19\!\cdots\!66\)\( T^{4} + 32569142396 p^{12} T^{6} + p^{24} T^{8} )^{2} \) |
| 61 | \( ( 1 - 373354 T + 122780975826 T^{2} - 373354 p^{6} T^{3} + p^{12} T^{4} )^{4} \) |
| 67 | \( ( 1 - 211571710690 T^{2} + \)\(25\!\cdots\!98\)\( T^{4} - 211571710690 p^{12} T^{6} + p^{24} T^{8} )^{2} \) |
| 71 | \( ( 1 - 325988720644 T^{2} + \)\(54\!\cdots\!66\)\( T^{4} - 325988720644 p^{12} T^{6} + p^{24} T^{8} )^{2} \) |
| 73 | \( ( 1 - 156199077220 T^{2} + \)\(36\!\cdots\!18\)\( T^{4} - 156199077220 p^{12} T^{6} + p^{24} T^{8} )^{2} \) |
| 79 | \( ( 1 + 321590 T + 154585780338 T^{2} + 321590 p^{6} T^{3} + p^{12} T^{4} )^{4} \) |
| 83 | \( ( 1 + 1121979728990 T^{2} + \)\(52\!\cdots\!58\)\( T^{4} + 1121979728990 p^{12} T^{6} + p^{24} T^{8} )^{2} \) |
| 89 | \( ( 1 - 1489268090884 T^{2} + \)\(10\!\cdots\!06\)\( T^{4} - 1489268090884 p^{12} T^{6} + p^{24} T^{8} )^{2} \) |
| 97 | \( ( 1 - 798419303140 T^{2} + \)\(19\!\cdots\!38\)\( T^{4} - 798419303140 p^{12} T^{6} + p^{24} T^{8} )^{2} \) |
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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
−8.564441561128573738316790986821, −8.348258755528911077342083124263, −8.138576322406231321037058296845, −8.011763349663693446892611499749, −7.61754281234328233536766956929, −7.23561812431648800926815858812, −6.74642369724622901290048741803, −6.69605048480631403764181322199, −6.36795943408830265416563294029, −6.19753363680240623033358725510, −5.55527682887432286834119884247, −5.48925040432443040983883793623, −5.34798167442422544572047424093, −5.03191574788431188971714958880, −4.43772078034011480427173897634, −4.32448210734120036902524704134, −4.13729530574246336999612599256, −4.09508038044328891778579596216, −3.50111429418126372341179967114, −3.03713602588289620115049922738, −2.33012149940294578393041038118, −2.21846057663547597741491166243, −1.20915478528197878840399843299, −0.39688891356929880813847369461, −0.23268793191604961560974224557,
0.23268793191604961560974224557, 0.39688891356929880813847369461, 1.20915478528197878840399843299, 2.21846057663547597741491166243, 2.33012149940294578393041038118, 3.03713602588289620115049922738, 3.50111429418126372341179967114, 4.09508038044328891778579596216, 4.13729530574246336999612599256, 4.32448210734120036902524704134, 4.43772078034011480427173897634, 5.03191574788431188971714958880, 5.34798167442422544572047424093, 5.48925040432443040983883793623, 5.55527682887432286834119884247, 6.19753363680240623033358725510, 6.36795943408830265416563294029, 6.69605048480631403764181322199, 6.74642369724622901290048741803, 7.23561812431648800926815858812, 7.61754281234328233536766956929, 8.011763349663693446892611499749, 8.138576322406231321037058296845, 8.348258755528911077342083124263, 8.564441561128573738316790986821
Plot not available for L-functions of degree greater than 10.
