| L(s) = 1 | − 2·4-s + 12·5-s + 12·11-s − 48·17-s + 42·19-s − 24·20-s − 24·23-s + 58·25-s − 102·31-s + 22·37-s + 28·43-s − 24·44-s − 132·47-s − 120·53-s + 144·55-s − 24·59-s + 72·61-s + 8·64-s + 110·67-s + 96·68-s − 312·71-s + 66·73-s − 84·76-s − 10·79-s − 576·85-s − 72·89-s + 48·92-s + ⋯ |
| L(s) = 1 | − 1/2·4-s + 12/5·5-s + 1.09·11-s − 2.82·17-s + 2.21·19-s − 6/5·20-s − 1.04·23-s + 2.31·25-s − 3.29·31-s + 0.594·37-s + 0.651·43-s − 0.545·44-s − 2.80·47-s − 2.26·53-s + 2.61·55-s − 0.406·59-s + 1.18·61-s + 1/8·64-s + 1.64·67-s + 1.41·68-s − 4.39·71-s + 0.904·73-s − 1.10·76-s − 0.126·79-s − 6.77·85-s − 0.808·89-s + 0.521·92-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.02482266960\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.02482266960\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 3 | | \( 1 \) |
| 7 | | \( 1 \) |
| good | 5 | $D_4\times C_2$ | \( 1 - 12 T + 86 T^{2} - 456 T^{3} + 2019 T^{4} - 456 p^{2} T^{5} + 86 p^{4} T^{6} - 12 p^{6} T^{7} + p^{8} T^{8} \) |
| 11 | $C_2^2$ | \( ( 1 - 6 T - 85 T^{2} - 6 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 13 | $D_4\times C_2$ | \( 1 + 98 T^{2} + 54915 T^{4} + 98 p^{4} T^{6} + p^{8} T^{8} \) |
| 17 | $D_4\times C_2$ | \( 1 + 48 T + 1514 T^{2} + 35808 T^{3} + 694947 T^{4} + 35808 p^{2} T^{5} + 1514 p^{4} T^{6} + 48 p^{6} T^{7} + p^{8} T^{8} \) |
| 19 | $D_4\times C_2$ | \( 1 - 42 T + 1433 T^{2} - 35490 T^{3} + 795972 T^{4} - 35490 p^{2} T^{5} + 1433 p^{4} T^{6} - 42 p^{6} T^{7} + p^{8} T^{8} \) |
| 23 | $D_4\times C_2$ | \( 1 + 24 T + 22 T^{2} - 12096 T^{3} - 277629 T^{4} - 12096 p^{2} T^{5} + 22 p^{4} T^{6} + 24 p^{6} T^{7} + p^{8} T^{8} \) |
| 29 | $C_2^2$ | \( ( 1 + 530 T^{2} + p^{4} T^{4} )^{2} \) |
| 31 | $D_4\times C_2$ | \( 1 + 102 T + 6041 T^{2} + 8466 p T^{3} + 9396 p^{2} T^{4} + 8466 p^{3} T^{5} + 6041 p^{4} T^{6} + 102 p^{6} T^{7} + p^{8} T^{8} \) |
| 37 | $D_4\times C_2$ | \( 1 - 22 T - 2087 T^{2} + 3674 T^{3} + 4073284 T^{4} + 3674 p^{2} T^{5} - 2087 p^{4} T^{6} - 22 p^{6} T^{7} + p^{8} T^{8} \) |
| 41 | $D_4\times C_2$ | \( 1 - 2476 T^{2} + 6405414 T^{4} - 2476 p^{4} T^{6} + p^{8} T^{8} \) |
| 43 | $D_{4}$ | \( ( 1 - 14 T + 3675 T^{2} - 14 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 47 | $D_4\times C_2$ | \( 1 + 132 T + 11654 T^{2} + 771672 T^{3} + 42125907 T^{4} + 771672 p^{2} T^{5} + 11654 p^{4} T^{6} + 132 p^{6} T^{7} + p^{8} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 + 120 T + 110 p T^{2} + 354240 T^{3} + 25104819 T^{4} + 354240 p^{2} T^{5} + 110 p^{5} T^{6} + 120 p^{6} T^{7} + p^{8} T^{8} \) |
| 59 | $D_4\times C_2$ | \( 1 + 24 T + 6026 T^{2} + 140016 T^{3} + 22586547 T^{4} + 140016 p^{2} T^{5} + 6026 p^{4} T^{6} + 24 p^{6} T^{7} + p^{8} T^{8} \) |
| 61 | $D_4\times C_2$ | \( 1 - 72 T + 9218 T^{2} - 539280 T^{3} + 48684147 T^{4} - 539280 p^{2} T^{5} + 9218 p^{4} T^{6} - 72 p^{6} T^{7} + p^{8} T^{8} \) |
| 67 | $D_4\times C_2$ | \( 1 - 110 T + 3625 T^{2} + 55330 T^{3} - 2642396 T^{4} + 55330 p^{2} T^{5} + 3625 p^{4} T^{6} - 110 p^{6} T^{7} + p^{8} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 + 156 T + 178 p T^{2} + 156 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 - 66 T + 2873 T^{2} - 93786 T^{3} - 18641292 T^{4} - 93786 p^{2} T^{5} + 2873 p^{4} T^{6} - 66 p^{6} T^{7} + p^{8} T^{8} \) |
| 79 | $D_4\times C_2$ | \( 1 + 10 T - 3695 T^{2} - 86870 T^{3} - 25172156 T^{4} - 86870 p^{2} T^{5} - 3695 p^{4} T^{6} + 10 p^{6} T^{7} + p^{8} T^{8} \) |
| 83 | $D_4\times C_2$ | \( 1 - 116 p T^{2} + 107694438 T^{4} - 116 p^{5} T^{6} + p^{8} T^{8} \) |
| 89 | $C_2^2$ | \( ( 1 + 36 T + 8353 T^{2} + 36 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 97 | $D_4\times C_2$ | \( 1 - 36580 T^{2} + 511416774 T^{4} - 36580 p^{4} T^{6} + p^{8} T^{8} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.09982448015459729271057825384, −6.93475676479372237037378056802, −6.50646523298619175836973026823, −6.16522410797379262053808125077, −6.11293664785474202302029136569, −6.09254104810557329419197520074, −5.67420337430965103043673963074, −5.45220163431749512709517449049, −5.36049545636351860652254495697, −4.94296737508998229011540241242, −4.64321621520816492070001369899, −4.61541498474185050625115214476, −4.30919067929931474074369187245, −3.93609189687439819153110084559, −3.50979265591099789635513051321, −3.50313052271167810548663863676, −3.14249044424638942532684762080, −2.72173509695793278100634883192, −2.25332645593227090224806523461, −2.24961268277246578405710696529, −1.66822693466611157427724257612, −1.58078845379521973355521371910, −1.53271228902145000960551113863, −0.76606488849368643904385243848, −0.02201118099561768348875527335,
0.02201118099561768348875527335, 0.76606488849368643904385243848, 1.53271228902145000960551113863, 1.58078845379521973355521371910, 1.66822693466611157427724257612, 2.24961268277246578405710696529, 2.25332645593227090224806523461, 2.72173509695793278100634883192, 3.14249044424638942532684762080, 3.50313052271167810548663863676, 3.50979265591099789635513051321, 3.93609189687439819153110084559, 4.30919067929931474074369187245, 4.61541498474185050625115214476, 4.64321621520816492070001369899, 4.94296737508998229011540241242, 5.36049545636351860652254495697, 5.45220163431749512709517449049, 5.67420337430965103043673963074, 6.09254104810557329419197520074, 6.11293664785474202302029136569, 6.16522410797379262053808125077, 6.50646523298619175836973026823, 6.93475676479372237037378056802, 7.09982448015459729271057825384
