| L(s) = 1 | + 4·4-s + 2·11-s + 6·16-s − 4·19-s + 2·29-s − 8·31-s + 10·41-s + 8·44-s + 31·49-s + 34·59-s − 26·61-s − 64-s + 16·71-s − 16·76-s + 14·79-s − 18·89-s + 44·109-s + 8·116-s − 35·121-s − 32·124-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + ⋯ |
| L(s) = 1 | + 2·4-s + 0.603·11-s + 3/2·16-s − 0.917·19-s + 0.371·29-s − 1.43·31-s + 1.56·41-s + 1.20·44-s + 31/7·49-s + 4.42·59-s − 3.32·61-s − 1/8·64-s + 1.89·71-s − 1.83·76-s + 1.57·79-s − 1.90·89-s + 4.21·109-s + 0.742·116-s − 3.18·121-s − 2.87·124-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{32} \cdot 5^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{32} \cdot 5^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6682292974\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6682292974\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 2 | \( 1 - p^{2} T^{2} + 5 p T^{4} - 15 T^{6} + 21 T^{8} - 15 p^{2} T^{10} + 5 p^{5} T^{12} - p^{8} T^{14} + p^{8} T^{16} \) |
| 7 | \( 1 - 31 T^{2} + 496 T^{4} - 5345 T^{6} + 42767 T^{8} - 5345 p^{2} T^{10} + 496 p^{4} T^{12} - 31 p^{6} T^{14} + p^{8} T^{16} \) |
| 11 | \( ( 1 - T + 19 T^{2} - 74 T^{3} + 167 T^{4} - 74 p T^{5} + 19 p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 13 | \( 1 - 40 T^{2} + 874 T^{4} - 13315 T^{6} + 180865 T^{8} - 13315 p^{2} T^{10} + 874 p^{4} T^{12} - 40 p^{6} T^{14} + p^{8} T^{16} \) |
| 17 | \( 1 - 55 T^{2} + 2044 T^{4} - 50325 T^{6} + 994815 T^{8} - 50325 p^{2} T^{10} + 2044 p^{4} T^{12} - 55 p^{6} T^{14} + p^{8} T^{16} \) |
| 19 | \( ( 1 + 2 T + 49 T^{2} + 34 T^{3} + 1115 T^{4} + 34 p T^{5} + 49 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 23 | \( 1 - 73 T^{2} + 2527 T^{4} - 51820 T^{6} + 985903 T^{8} - 51820 p^{2} T^{10} + 2527 p^{4} T^{12} - 73 p^{6} T^{14} + p^{8} T^{16} \) |
| 29 | \( ( 1 - T + 76 T^{2} + 55 T^{3} + 2597 T^{4} + 55 p T^{5} + 76 p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 31 | \( ( 1 + 4 T + 82 T^{2} + 345 T^{3} + 3405 T^{4} + 345 p T^{5} + 82 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 37 | \( 1 - 97 T^{2} + 3667 T^{4} - 33124 T^{6} - 1092101 T^{8} - 33124 p^{2} T^{10} + 3667 p^{4} T^{12} - 97 p^{6} T^{14} + p^{8} T^{16} \) |
| 41 | \( ( 1 - 5 T + 139 T^{2} - 454 T^{3} + 7829 T^{4} - 454 p T^{5} + 139 p^{2} T^{6} - 5 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 43 | \( 1 - 148 T^{2} + 13882 T^{4} - 918160 T^{6} + 44775523 T^{8} - 918160 p^{2} T^{10} + 13882 p^{4} T^{12} - 148 p^{6} T^{14} + p^{8} T^{16} \) |
| 47 | \( 1 - 190 T^{2} + 17287 T^{4} - 1047300 T^{6} + 51910773 T^{8} - 1047300 p^{2} T^{10} + 17287 p^{4} T^{12} - 190 p^{6} T^{14} + p^{8} T^{16} \) |
| 53 | \( 1 - 196 T^{2} + 19762 T^{4} - 1503447 T^{6} + 91143129 T^{8} - 1503447 p^{2} T^{10} + 19762 p^{4} T^{12} - 196 p^{6} T^{14} + p^{8} T^{16} \) |
| 59 | \( ( 1 - 17 T + 238 T^{2} - 2095 T^{3} + 18809 T^{4} - 2095 p T^{5} + 238 p^{2} T^{6} - 17 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 61 | \( ( 1 + 13 T + 241 T^{2} + 2288 T^{3} + 21959 T^{4} + 2288 p T^{5} + 241 p^{2} T^{6} + 13 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 67 | \( 1 - 319 T^{2} + 50440 T^{4} - 5369705 T^{6} + 418333751 T^{8} - 5369705 p^{2} T^{10} + 50440 p^{4} T^{12} - 319 p^{6} T^{14} + p^{8} T^{16} \) |
| 71 | \( ( 1 - 8 T + 244 T^{2} - 1441 T^{3} + 24947 T^{4} - 1441 p T^{5} + 244 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 73 | \( 1 - 388 T^{2} + 71842 T^{4} - 8513491 T^{6} + 722371129 T^{8} - 8513491 p^{2} T^{10} + 71842 p^{4} T^{12} - 388 p^{6} T^{14} + p^{8} T^{16} \) |
| 79 | \( ( 1 - 7 T + 283 T^{2} - 1590 T^{3} + 32439 T^{4} - 1590 p T^{5} + 283 p^{2} T^{6} - 7 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 83 | \( 1 - 340 T^{2} + 59242 T^{4} - 7263475 T^{6} + 683930533 T^{8} - 7263475 p^{2} T^{10} + 59242 p^{4} T^{12} - 340 p^{6} T^{14} + p^{8} T^{16} \) |
| 89 | \( ( 1 + 9 T + 257 T^{2} + 1998 T^{3} + 31929 T^{4} + 1998 p T^{5} + 257 p^{2} T^{6} + 9 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 97 | \( 1 - 577 T^{2} + 158407 T^{4} - 27023440 T^{6} + 3138718723 T^{8} - 27023440 p^{2} T^{10} + 158407 p^{4} T^{12} - 577 p^{6} T^{14} + p^{8} T^{16} \) |
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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
−3.96440540055782343535008519928, −3.71280667756604151921581475353, −3.70937545588801546141846898972, −3.35914670082575413046274042638, −3.24511146376718122466312847478, −3.13845017574912623029005258546, −3.05776874434996838763539315916, −3.02418969554951450189862058689, −2.98152424946178168147724628185, −2.52487105003074796595784306545, −2.40424900581142766609834776623, −2.34180808569065060823998864637, −2.18785395626822785004816997589, −2.18538521619153562359770667852, −2.11031957597352366927071960112, −2.08524687645438160320750296584, −2.03697679636929376988281790137, −1.44667075724397311053468892669, −1.22926269716302564167703473118, −1.19249968965481063077588709557, −1.18383348747072990193945973933, −1.09289610562692701412847447362, −0.65625092717695542506818124164, −0.51340984455350772406430294947, −0.05008132385277651810879759338,
0.05008132385277651810879759338, 0.51340984455350772406430294947, 0.65625092717695542506818124164, 1.09289610562692701412847447362, 1.18383348747072990193945973933, 1.19249968965481063077588709557, 1.22926269716302564167703473118, 1.44667075724397311053468892669, 2.03697679636929376988281790137, 2.08524687645438160320750296584, 2.11031957597352366927071960112, 2.18538521619153562359770667852, 2.18785395626822785004816997589, 2.34180808569065060823998864637, 2.40424900581142766609834776623, 2.52487105003074796595784306545, 2.98152424946178168147724628185, 3.02418969554951450189862058689, 3.05776874434996838763539315916, 3.13845017574912623029005258546, 3.24511146376718122466312847478, 3.35914670082575413046274042638, 3.70937545588801546141846898972, 3.71280667756604151921581475353, 3.96440540055782343535008519928
Plot not available for L-functions of degree greater than 10.
