| L(s) = 1 | + 2·4-s − 8·7-s + 4·13-s + 3·16-s + 8·19-s + 8·25-s − 16·28-s − 56·31-s − 8·37-s + 4·43-s + 4·49-s + 8·52-s − 20·61-s − 6·64-s + 28·67-s − 20·73-s + 16·76-s + 28·79-s − 32·91-s − 32·97-s + 16·100-s − 8·103-s + 16·109-s − 24·112-s − 16·121-s − 112·124-s + 127-s + ⋯ |
| L(s) = 1 | + 4-s − 3.02·7-s + 1.10·13-s + 3/4·16-s + 1.83·19-s + 8/5·25-s − 3.02·28-s − 10.0·31-s − 1.31·37-s + 0.609·43-s + 4/7·49-s + 1.10·52-s − 2.56·61-s − 3/4·64-s + 3.42·67-s − 2.34·73-s + 1.83·76-s + 3.15·79-s − 3.35·91-s − 3.24·97-s + 8/5·100-s − 0.788·103-s + 1.53·109-s − 2.26·112-s − 1.45·121-s − 10.0·124-s + 0.0887·127-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{16} \cdot 19^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{16} \cdot 19^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3696056641\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3696056641\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 19 | \( ( 1 - 2 T + p T^{2} )^{4} \) |
| good | 2 | \( 1 - p T^{2} + T^{4} + 5 p T^{6} - 23 T^{8} + 5 p^{3} T^{10} + p^{4} T^{12} - p^{7} T^{14} + p^{8} T^{16} \) |
| 5 | \( 1 - 8 T^{2} + 22 T^{4} + 64 T^{6} - 461 T^{8} + 64 p^{2} T^{10} + 22 p^{4} T^{12} - 8 p^{6} T^{14} + p^{8} T^{16} \) |
| 7 | \( ( 1 + 2 T + 9 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{4} \) |
| 11 | \( ( 1 + 8 T^{2} + 42 T^{4} + 8 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 13 | \( ( 1 - T - 12 T^{2} - p T^{3} + p^{2} T^{4} )^{4} \) |
| 17 | \( 1 - 20 T^{2} + 106 T^{4} + 5680 T^{6} - 112685 T^{8} + 5680 p^{2} T^{10} + 106 p^{4} T^{12} - 20 p^{6} T^{14} + p^{8} T^{16} \) |
| 23 | \( 1 - 56 T^{2} + 1318 T^{4} - 42560 T^{6} + 1366339 T^{8} - 42560 p^{2} T^{10} + 1318 p^{4} T^{12} - 56 p^{6} T^{14} + p^{8} T^{16} \) |
| 29 | \( 1 + 28 T^{2} - 710 T^{4} - 5264 T^{6} + 865411 T^{8} - 5264 p^{2} T^{10} - 710 p^{4} T^{12} + 28 p^{6} T^{14} + p^{8} T^{16} \) |
| 31 | \( ( 1 + 14 T + 105 T^{2} + 14 p T^{3} + p^{2} T^{4} )^{4} \) |
| 37 | \( ( 1 + 2 T + 51 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{4} \) |
| 41 | \( 1 - 68 T^{2} + 1642 T^{4} + 25840 T^{6} - 1706381 T^{8} + 25840 p^{2} T^{10} + 1642 p^{4} T^{12} - 68 p^{6} T^{14} + p^{8} T^{16} \) |
| 43 | \( ( 1 - 2 T - 29 T^{2} + 106 T^{3} - 932 T^{4} + 106 p T^{5} - 29 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 47 | \( 1 - 92 T^{2} + 3466 T^{4} - 53360 T^{6} + 1403347 T^{8} - 53360 p^{2} T^{10} + 3466 p^{4} T^{12} - 92 p^{6} T^{14} + p^{8} T^{16} \) |
| 53 | \( 1 - 200 T^{2} + 24406 T^{4} - 1995200 T^{6} + 122922355 T^{8} - 1995200 p^{2} T^{10} + 24406 p^{4} T^{12} - 200 p^{6} T^{14} + p^{8} T^{16} \) |
| 59 | \( 1 - 104 T^{2} + 1174 T^{4} - 278720 T^{6} + 43200307 T^{8} - 278720 p^{2} T^{10} + 1174 p^{4} T^{12} - 104 p^{6} T^{14} + p^{8} T^{16} \) |
| 61 | \( ( 1 + 5 T - 36 T^{2} + 5 p T^{3} + p^{2} T^{4} )^{4} \) |
| 67 | \( ( 1 - 14 T + p T^{2} )^{4}( 1 + 14 T + 129 T^{2} + 14 p T^{3} + p^{2} T^{4} )^{2} \) |
| 71 | \( 1 - 140 T^{2} + 5002 T^{4} - 632240 T^{6} + 88886323 T^{8} - 632240 p^{2} T^{10} + 5002 p^{4} T^{12} - 140 p^{6} T^{14} + p^{8} T^{16} \) |
| 73 | \( ( 1 + 5 T - 48 T^{2} + 5 p T^{3} + p^{2} T^{4} )^{4} \) |
| 79 | \( ( 1 - 14 T + 43 T^{2} + 70 T^{3} + 1684 T^{4} + 70 p T^{5} + 43 p^{2} T^{6} - 14 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 83 | \( ( 1 + 188 T^{2} + 19158 T^{4} + 188 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 89 | \( 1 - 56 T^{2} + 1510 T^{4} + 796096 T^{6} - 84938621 T^{8} + 796096 p^{2} T^{10} + 1510 p^{4} T^{12} - 56 p^{6} T^{14} + p^{8} T^{16} \) |
| 97 | \( ( 1 + 16 T + 22 T^{2} + 640 T^{3} + 20515 T^{4} + 640 p T^{5} + 22 p^{2} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
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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
−6.02146487384392664345548551044, −5.69932283889306553982380002765, −5.40587237188096489899156098875, −5.37991315035471063801366666066, −5.32359514577515486715068028075, −5.24330761978999811517949769637, −5.17303939310479416255578736696, −4.96408746434452255443489704236, −4.45028810441098974392498080797, −4.15500534839769717008787518503, −4.03079140635165643282337794595, −4.01353707060061123016859263813, −3.59951872493571199716439445806, −3.52702508145343431196749731985, −3.33908773496856145759819457692, −3.27209989116301653810870929278, −3.20446852748645676549388394871, −3.13451600143324871986974396707, −2.56640660276347950610243771566, −2.53012790942622937604874157457, −1.87374757507543874498583370423, −1.85505864880742908792514435663, −1.55527538572307297230851856321, −1.45612952167493132635072227588, −0.25852575978823346909573396353,
0.25852575978823346909573396353, 1.45612952167493132635072227588, 1.55527538572307297230851856321, 1.85505864880742908792514435663, 1.87374757507543874498583370423, 2.53012790942622937604874157457, 2.56640660276347950610243771566, 3.13451600143324871986974396707, 3.20446852748645676549388394871, 3.27209989116301653810870929278, 3.33908773496856145759819457692, 3.52702508145343431196749731985, 3.59951872493571199716439445806, 4.01353707060061123016859263813, 4.03079140635165643282337794595, 4.15500534839769717008787518503, 4.45028810441098974392498080797, 4.96408746434452255443489704236, 5.17303939310479416255578736696, 5.24330761978999811517949769637, 5.32359514577515486715068028075, 5.37991315035471063801366666066, 5.40587237188096489899156098875, 5.69932283889306553982380002765, 6.02146487384392664345548551044
Plot not available for L-functions of degree greater than 10.
