| L(s) = 1 | − 3·2-s + 4·4-s − 6·5-s − 3·8-s − 3·9-s + 18·10-s − 2·13-s + 16-s + 9·18-s − 24·20-s + 5·25-s + 6·26-s + 6·29-s − 6·32-s − 12·36-s − 8·37-s + 18·40-s + 24·41-s + 18·45-s − 19·49-s − 15·50-s − 8·52-s − 18·58-s − 2·61-s + 23·64-s + 12·65-s + 9·72-s + ⋯ |
| L(s) = 1 | − 2.12·2-s + 2·4-s − 2.68·5-s − 1.06·8-s − 9-s + 5.69·10-s − 0.554·13-s + 1/4·16-s + 2.12·18-s − 5.36·20-s + 25-s + 1.17·26-s + 1.11·29-s − 1.06·32-s − 2·36-s − 1.31·37-s + 2.84·40-s + 3.74·41-s + 2.68·45-s − 2.71·49-s − 2.12·50-s − 1.10·52-s − 2.36·58-s − 0.256·61-s + 23/8·64-s + 1.48·65-s + 1.06·72-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.02827969161\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.02827969161\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 3 T + 5 T^{2} + 3 p T^{3} + 3 p T^{4} + 3 p^{2} T^{5} + 5 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} \) |
| 3 | \( 1 + p T^{2} + 4 p T^{4} + p^{3} T^{6} + p^{4} T^{8} \) |
| good | 5 | \( ( 1 + 3 T + 11 T^{2} + 24 T^{3} + 54 T^{4} + 24 p T^{5} + 11 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 7 | \( 1 + 19 T^{2} + 181 T^{4} + 1558 T^{6} + 12310 T^{8} + 1558 p^{2} T^{10} + 181 p^{4} T^{12} + 19 p^{6} T^{14} + p^{8} T^{16} \) |
| 11 | \( 1 - 32 T^{2} + 559 T^{4} - 7136 T^{6} + 77680 T^{8} - 7136 p^{2} T^{10} + 559 p^{4} T^{12} - 32 p^{6} T^{14} + p^{8} T^{16} \) |
| 13 | \( ( 1 + T - 17 T^{2} - 8 T^{3} + 142 T^{4} - 8 p T^{5} - 17 p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 17 | \( ( 1 - 61 T^{2} + 1500 T^{4} - 61 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 19 | \( ( 1 - 49 T^{2} + 1248 T^{4} - 49 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 23 | \( 1 - 77 T^{2} + 3397 T^{4} - 113498 T^{6} + 2994742 T^{8} - 113498 p^{2} T^{10} + 3397 p^{4} T^{12} - 77 p^{6} T^{14} + p^{8} T^{16} \) |
| 29 | \( ( 1 - 3 T + 59 T^{2} - 168 T^{3} + 2382 T^{4} - 168 p T^{5} + 59 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 31 | \( 1 + 55 T^{2} + 1345 T^{4} - 13310 T^{6} - 1008146 T^{8} - 13310 p^{2} T^{10} + 1345 p^{4} T^{12} + 55 p^{6} T^{14} + p^{8} T^{16} \) |
| 37 | \( ( 1 + 2 T + 42 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{4} \) |
| 41 | \( ( 1 - 12 T + 131 T^{2} - 996 T^{3} + 7176 T^{4} - 996 p T^{5} + 131 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 43 | \( 1 + 64 T^{2} - 593 T^{4} + 63424 T^{6} + 10994416 T^{8} + 63424 p^{2} T^{10} - 593 p^{4} T^{12} + 64 p^{6} T^{14} + p^{8} T^{16} \) |
| 47 | \( 1 - 53 T^{2} + 2053 T^{4} + 194086 T^{6} - 10513226 T^{8} + 194086 p^{2} T^{10} + 2053 p^{4} T^{12} - 53 p^{6} T^{14} + p^{8} T^{16} \) |
| 53 | \( ( 1 - 136 T^{2} + 9054 T^{4} - 136 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 59 | \( 1 - 56 T^{2} - 617 T^{4} + 179704 T^{6} - 8948768 T^{8} + 179704 p^{2} T^{10} - 617 p^{4} T^{12} - 56 p^{6} T^{14} + p^{8} T^{16} \) |
| 61 | \( ( 1 + T - 113 T^{2} - 8 T^{3} + 9214 T^{4} - 8 p T^{5} - 113 p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 67 | \( 1 + 160 T^{2} + 10255 T^{4} + 1018720 T^{6} + 100399504 T^{8} + 1018720 p^{2} T^{10} + 10255 p^{4} T^{12} + 160 p^{6} T^{14} + p^{8} T^{16} \) |
| 71 | \( ( 1 + 140 T^{2} + 10230 T^{4} + 140 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 73 | \( ( 1 - T + 138 T^{2} - p T^{3} + p^{2} T^{4} )^{4} \) |
| 79 | \( 1 + 115 T^{2} - 2555 T^{4} + 379270 T^{6} + 127521094 T^{8} + 379270 p^{2} T^{10} - 2555 p^{4} T^{12} + 115 p^{6} T^{14} + p^{8} T^{16} \) |
| 83 | \( 1 - 221 T^{2} + 22861 T^{4} - 2696642 T^{6} + 291036430 T^{8} - 2696642 p^{2} T^{10} + 22861 p^{4} T^{12} - 221 p^{6} T^{14} + p^{8} T^{16} \) |
| 89 | \( ( 1 - 184 T^{2} + 21006 T^{4} - 184 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 97 | \( ( 1 - 2 T - 59 T^{2} + 262 T^{3} - 5828 T^{4} + 262 p T^{5} - 59 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
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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
−8.306669182248338593189779514203, −7.975639787693418735192102406061, −7.927550193705076381336876952662, −7.81937093685503095898891737418, −7.53554169894284660944570720392, −7.43637056954322732575378759682, −7.39524432508755506327575201218, −7.25147504858081772664063154981, −6.83566204816395454826821293633, −6.49847748207673495252183723233, −6.34259575125884772498092378583, −6.02210048266719320367213666561, −6.00136149408027008257477790444, −5.76614090183951599052206518532, −5.18938050780882079827077089902, −5.11295463965147680559141361569, −4.66978679232252476157993146347, −4.62632618248830518501790350783, −4.28240964674254658949224646414, −3.85338976274152635267506907552, −3.66052421143855954487685300223, −3.35888230345847000471252667227, −3.21460605818579881970173812500, −2.44171528296503042666414328994, −2.12226860007967066366062149714,
2.12226860007967066366062149714, 2.44171528296503042666414328994, 3.21460605818579881970173812500, 3.35888230345847000471252667227, 3.66052421143855954487685300223, 3.85338976274152635267506907552, 4.28240964674254658949224646414, 4.62632618248830518501790350783, 4.66978679232252476157993146347, 5.11295463965147680559141361569, 5.18938050780882079827077089902, 5.76614090183951599052206518532, 6.00136149408027008257477790444, 6.02210048266719320367213666561, 6.34259575125884772498092378583, 6.49847748207673495252183723233, 6.83566204816395454826821293633, 7.25147504858081772664063154981, 7.39524432508755506327575201218, 7.43637056954322732575378759682, 7.53554169894284660944570720392, 7.81937093685503095898891737418, 7.927550193705076381336876952662, 7.975639787693418735192102406061, 8.306669182248338593189779514203
Plot not available for L-functions of degree greater than 10.
