| L(s) = 1 | + 32·4-s − 28·9-s + 640·16-s − 816·19-s + 420·25-s + 6.22e3·31-s − 896·36-s − 1.01e3·49-s − 368·61-s + 1.02e4·64-s − 2.61e4·76-s − 1.58e3·79-s − 8.05e3·81-s + 1.34e4·100-s + 6.18e4·109-s + 8.14e4·121-s + 1.99e5·124-s + 127-s + 131-s + 137-s + 139-s − 1.79e4·144-s + 149-s + 151-s + 157-s + 163-s + 167-s + ⋯ |
| L(s) = 1 | + 2·4-s − 0.345·9-s + 5/2·16-s − 2.26·19-s + 0.671·25-s + 6.47·31-s − 0.691·36-s − 0.423·49-s − 0.0988·61-s + 5/2·64-s − 4.52·76-s − 0.253·79-s − 1.22·81-s + 1.34·100-s + 5.20·109-s + 5.56·121-s + 12.9·124-s + 6.20e−5·127-s + 5.82e−5·131-s + 5.32e−5·137-s + 5.17e−5·139-s − 0.864·144-s + 4.50e−5·149-s + 4.38e−5·151-s + 4.05e−5·157-s + 3.76e−5·163-s + 3.58e−5·167-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(5-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(7.925385113\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.925385113\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( ( 1 - p^{3} T^{2} )^{4} \) |
| 3 | \( 1 + 28 T^{2} + 982 p^{2} T^{4} + 28 p^{8} T^{6} + p^{16} T^{8} \) |
| 5 | \( 1 - 84 p T^{2} - 7306 p^{2} T^{4} - 84 p^{9} T^{6} + p^{16} T^{8} \) |
| good | 7 | \( ( 1 + 508 T^{2} + 10948998 T^{4} + 508 p^{8} T^{6} + p^{16} T^{8} )^{2} \) |
| 11 | \( ( 1 - 40740 T^{2} + 794262662 T^{4} - 40740 p^{8} T^{6} + p^{16} T^{8} )^{2} \) |
| 13 | \( ( 1 - 35044 T^{2} + 1711681926 T^{4} - 35044 p^{8} T^{6} + p^{16} T^{8} )^{2} \) |
| 17 | \( ( 1 + 16860 p T^{2} + 34376012102 T^{4} + 16860 p^{9} T^{6} + p^{16} T^{8} )^{2} \) |
| 19 | \( ( 1 + 204 T + 109766 T^{2} + 204 p^{4} T^{3} + p^{8} T^{4} )^{4} \) |
| 23 | \( ( 1 + 792540 T^{2} + 306972915782 T^{4} + 792540 p^{8} T^{6} + p^{16} T^{8} )^{2} \) |
| 29 | \( ( 1 - 1312804 T^{2} + 895663883526 T^{4} - 1312804 p^{8} T^{6} + p^{16} T^{8} )^{2} \) |
| 31 | \( ( 1 - 1556 T + 1444326 T^{2} - 1556 p^{4} T^{3} + p^{8} T^{4} )^{4} \) |
| 37 | \( ( 1 - 6645092 T^{2} + 18064078866438 T^{4} - 6645092 p^{8} T^{6} + p^{16} T^{8} )^{2} \) |
| 41 | \( ( 1 - 6058500 T^{2} + 24422716198022 T^{4} - 6058500 p^{8} T^{6} + p^{16} T^{8} )^{2} \) |
| 43 | \( ( 1 - 10721636 T^{2} + 50104506873606 T^{4} - 10721636 p^{8} T^{6} + p^{16} T^{8} )^{2} \) |
| 47 | \( ( 1 + 19311324 T^{2} + 140845284781766 T^{4} + 19311324 p^{8} T^{6} + p^{16} T^{8} )^{2} \) |
| 53 | \( ( 1 + 10656220 T^{2} + 152855460971142 T^{4} + 10656220 p^{8} T^{6} + p^{16} T^{8} )^{2} \) |
| 59 | \( ( 1 - 39696420 T^{2} + 686958456899462 T^{4} - 39696420 p^{8} T^{6} + p^{16} T^{8} )^{2} \) |
| 61 | \( ( 1 + 92 T + 27048678 T^{2} + 92 p^{4} T^{3} + p^{8} T^{4} )^{4} \) |
| 67 | \( ( 1 - 34970852 T^{2} + 1020422014572678 T^{4} - 34970852 p^{8} T^{6} + p^{16} T^{8} )^{2} \) |
| 71 | \( ( 1 - 28972804 T^{2} + 535921971765126 T^{4} - 28972804 p^{8} T^{6} + p^{16} T^{8} )^{2} \) |
| 73 | \( ( 1 - 64633604 T^{2} + 2064039894495366 T^{4} - 64633604 p^{8} T^{6} + p^{16} T^{8} )^{2} \) |
| 79 | \( ( 1 + 396 T + 72133286 T^{2} + 396 p^{4} T^{3} + p^{8} T^{4} )^{4} \) |
| 83 | \( ( 1 + 31579548 T^{2} - 1477571129795962 T^{4} + 31579548 p^{8} T^{6} + p^{16} T^{8} )^{2} \) |
| 89 | \( ( 1 - 2001060 p T^{2} + 15595341664136582 T^{4} - 2001060 p^{9} T^{6} + p^{16} T^{8} )^{2} \) |
| 97 | \( ( 1 - 310922756 T^{2} + 39804579392094726 T^{4} - 310922756 p^{8} T^{6} + p^{16} T^{8} )^{2} \) |
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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
−7.66741013155420515124983726927, −7.39032339145474042935191622092, −6.87130628101178597967360856936, −6.84517880762444794365584747240, −6.82705233719735879399133145854, −6.41166449542069221985958195491, −6.39909365308044840945163402524, −6.22066320495021197542518157878, −5.95681936101786174564812017740, −5.79729726184132117122295204756, −5.40879112992021783630228440569, −5.21185770014145342411047912372, −4.61613753075980424302399774519, −4.43571734466293053575798129781, −4.42526563934007657718520437631, −4.38160754434020446228324916996, −3.58703962204067489955390828117, −3.08421961901600987018584192312, −2.96850992478208530649046326503, −2.96655466093066505176993197367, −2.22462287005078090795107834185, −2.18769600581901732672312134057, −1.71451413459010018207236108337, −0.843032529509418743279664683210, −0.74152960769971989576400207990,
0.74152960769971989576400207990, 0.843032529509418743279664683210, 1.71451413459010018207236108337, 2.18769600581901732672312134057, 2.22462287005078090795107834185, 2.96655466093066505176993197367, 2.96850992478208530649046326503, 3.08421961901600987018584192312, 3.58703962204067489955390828117, 4.38160754434020446228324916996, 4.42526563934007657718520437631, 4.43571734466293053575798129781, 4.61613753075980424302399774519, 5.21185770014145342411047912372, 5.40879112992021783630228440569, 5.79729726184132117122295204756, 5.95681936101786174564812017740, 6.22066320495021197542518157878, 6.39909365308044840945163402524, 6.41166449542069221985958195491, 6.82705233719735879399133145854, 6.84517880762444794365584747240, 6.87130628101178597967360856936, 7.39032339145474042935191622092, 7.66741013155420515124983726927
Plot not available for L-functions of degree greater than 10.
