| L(s) = 1 | + 2·2-s + 2·4-s + 8·5-s − 6·7-s + 4·8-s + 16·10-s + 10·11-s − 10·13-s − 12·14-s + 8·16-s + 16·17-s − 6·19-s + 16·20-s + 20·22-s − 6·23-s + 44·25-s − 20·26-s − 12·28-s + 6·29-s − 8·31-s + 8·32-s + 32·34-s − 48·35-s + 12·37-s − 12·38-s + 32·40-s − 16·43-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 4-s + 3.57·5-s − 2.26·7-s + 1.41·8-s + 5.05·10-s + 3.01·11-s − 2.77·13-s − 3.20·14-s + 2·16-s + 3.88·17-s − 1.37·19-s + 3.57·20-s + 4.26·22-s − 1.25·23-s + 44/5·25-s − 3.92·26-s − 2.26·28-s + 1.11·29-s − 1.43·31-s + 1.41·32-s + 5.48·34-s − 8.11·35-s + 1.97·37-s − 1.94·38-s + 5.05·40-s − 2.43·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{12}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(11.96981923\) |
| \(L(\frac12)\) |
\(\approx\) |
\(11.96981923\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2^2$ | \( 1 - p T + p T^{2} - p^{2} T^{3} + p^{2} T^{4} \) |
| 3 | | \( 1 \) |
| good | 5 | $D_4\times C_2$ | \( 1 - 8 T + 4 p T^{2} + 4 T^{3} - 89 T^{4} + 4 p T^{5} + 4 p^{3} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 7 | $D_4\times C_2$ | \( 1 + 6 T + 4 p T^{2} + 96 T^{3} + 291 T^{4} + 96 p T^{5} + 4 p^{3} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 11 | $D_4\times C_2$ | \( 1 - 10 T + 41 T^{2} - 82 T^{3} + 136 T^{4} - 82 p T^{5} + 41 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8} \) |
| 13 | $C_2$$\times$$C_2^2$ | \( ( 1 + 5 T + p T^{2} )^{2}( 1 + 23 T^{2} + p^{2} T^{4} ) \) |
| 17 | $C_2^2$ | \( ( 1 - 8 T + 47 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 19 | $D_4\times C_2$ | \( 1 + 6 T + 18 T^{2} + 132 T^{3} + 959 T^{4} + 132 p T^{5} + 18 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 23 | $D_4\times C_2$ | \( 1 + 6 T + 52 T^{2} + 240 T^{3} + 1347 T^{4} + 240 p T^{5} + 52 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 29 | $C_2^3$ | \( 1 - 6 T + 18 T^{2} - 36 T^{3} - 457 T^{4} - 36 p T^{5} + 18 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 31 | $D_4\times C_2$ | \( 1 + 8 T - 2 T^{2} + 32 T^{3} + 1411 T^{4} + 32 p T^{5} - 2 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 37 | $D_4\times C_2$ | \( 1 - 12 T + 72 T^{2} - 588 T^{3} + 4658 T^{4} - 588 p T^{5} + 72 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 41 | $C_2^3$ | \( 1 + 73 T^{2} + 3648 T^{4} + 73 p^{2} T^{6} + p^{4} T^{8} \) |
| 43 | $D_4\times C_2$ | \( 1 + 16 T + 65 T^{2} - 624 T^{3} - 8092 T^{4} - 624 p T^{5} + 65 p^{2} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 - 2 T - 16 T^{2} + 148 T^{3} - 1997 T^{4} + 148 p T^{5} - 16 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 - 16 T + 128 T^{2} - 976 T^{3} + 7378 T^{4} - 976 p T^{5} + 128 p^{2} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8} \) |
| 59 | $D_4\times C_2$ | \( 1 + 6 T + 45 T^{2} - 594 T^{3} - 3376 T^{4} - 594 p T^{5} + 45 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $D_4\times C_2$ | \( 1 + 12 T + 180 T^{2} + 1596 T^{3} + 15143 T^{4} + 1596 p T^{5} + 180 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 67 | $D_4\times C_2$ | \( 1 - 16 T + 113 T^{2} - 384 T^{3} - 172 T^{4} - 384 p T^{5} + 113 p^{2} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $D_4\times C_2$ | \( 1 - 156 T^{2} + 13094 T^{4} - 156 p^{2} T^{6} + p^{4} T^{8} \) |
| 73 | $D_4\times C_2$ | \( 1 - 158 T^{2} + 16131 T^{4} - 158 p^{2} T^{6} + p^{4} T^{8} \) |
| 79 | $C_2^2$ | \( ( 1 + 12 T + 65 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 83 | $C_2^3$ | \( 1 - 2 T + 2 T^{2} + 328 T^{3} - 7217 T^{4} + 328 p T^{5} + 2 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 89 | $C_2^2$ | \( ( 1 - 174 T^{2} + p^{2} T^{4} )^{2} \) |
| 97 | $D_4\times C_2$ | \( 1 + 20 T + 109 T^{2} + 20 p T^{3} + 376 p T^{4} + 20 p^{2} T^{5} + 109 p^{2} T^{6} + 20 p^{3} T^{7} + p^{4} T^{8} \) |
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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
−8.116300242509711434264172004038, −7.52466291892398585419045296132, −7.35190037863788649384359848082, −7.18026616249031108967719108142, −6.96470345629069036199717576603, −6.57115600352004861048004696935, −6.45105419250351055909150362279, −6.20240246848151539746791729411, −6.18220274904153142482764237727, −5.72327659372728644057569859643, −5.70472587627970730142466317578, −5.20553773019301408903148037016, −5.15588010148660714451344099554, −4.92376946712246516549545537689, −4.44786218821172797067270103508, −4.06649813398219428227637708107, −3.96631389884197970252020820052, −3.35787598637842151191333601545, −3.12089302398365724040842611025, −3.04285522996453146049582391337, −2.50280141676677704915967407565, −2.24934641777783856426148877324, −1.74033945150501020624702714326, −1.32569669481994300730337353410, −1.14740151092014100777846884239,
1.14740151092014100777846884239, 1.32569669481994300730337353410, 1.74033945150501020624702714326, 2.24934641777783856426148877324, 2.50280141676677704915967407565, 3.04285522996453146049582391337, 3.12089302398365724040842611025, 3.35787598637842151191333601545, 3.96631389884197970252020820052, 4.06649813398219428227637708107, 4.44786218821172797067270103508, 4.92376946712246516549545537689, 5.15588010148660714451344099554, 5.20553773019301408903148037016, 5.70472587627970730142466317578, 5.72327659372728644057569859643, 6.18220274904153142482764237727, 6.20240246848151539746791729411, 6.45105419250351055909150362279, 6.57115600352004861048004696935, 6.96470345629069036199717576603, 7.18026616249031108967719108142, 7.35190037863788649384359848082, 7.52466291892398585419045296132, 8.116300242509711434264172004038
