| L(s) = 1 | − 3-s + 9·5-s − 44·7-s − 10·9-s − 79·11-s − 11·13-s − 9·15-s + 82·17-s − 57·19-s + 44·21-s − 103·23-s − 252·25-s + 55·27-s − 93·29-s + 116·31-s + 79·33-s − 396·35-s − 466·37-s + 11·39-s − 188·41-s + 11·43-s − 90·45-s − 163·47-s + 488·49-s − 82·51-s + 197·53-s − 711·55-s + ⋯ |
| L(s) = 1 | − 0.192·3-s + 0.804·5-s − 2.37·7-s − 0.370·9-s − 2.16·11-s − 0.234·13-s − 0.154·15-s + 1.16·17-s − 0.688·19-s + 0.457·21-s − 0.933·23-s − 2.01·25-s + 0.392·27-s − 0.595·29-s + 0.672·31-s + 0.416·33-s − 1.91·35-s − 2.07·37-s + 0.0451·39-s − 0.716·41-s + 0.0390·43-s − 0.298·45-s − 0.505·47-s + 1.42·49-s − 0.225·51-s + 0.510·53-s − 1.74·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 28094464 ^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 28094464 ^{s/2} \, \Gamma_{\C}(s+3/2)^{3} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
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 | | \( 1 \) |
| 19 | $C_1$ | \( ( 1 + p T )^{3} \) |
| good | 3 | $S_4\times C_2$ | \( 1 + T + 11 T^{2} - 34 T^{3} + 11 p^{3} T^{4} + p^{6} T^{5} + p^{9} T^{6} \) |
| 5 | $S_4\times C_2$ | \( 1 - 9 T + 333 T^{2} - 2034 T^{3} + 333 p^{3} T^{4} - 9 p^{6} T^{5} + p^{9} T^{6} \) |
| 7 | $S_4\times C_2$ | \( 1 + 44 T + 1448 T^{2} + 31328 T^{3} + 1448 p^{3} T^{4} + 44 p^{6} T^{5} + p^{9} T^{6} \) |
| 11 | $S_4\times C_2$ | \( 1 + 79 T + 3327 T^{2} + 101410 T^{3} + 3327 p^{3} T^{4} + 79 p^{6} T^{5} + p^{9} T^{6} \) |
| 13 | $S_4\times C_2$ | \( 1 + 11 T + 157 T^{2} - 153482 T^{3} + 157 p^{3} T^{4} + 11 p^{6} T^{5} + p^{9} T^{6} \) |
| 17 | $S_4\times C_2$ | \( 1 - 82 T + 1674 T^{2} + 216350 T^{3} + 1674 p^{3} T^{4} - 82 p^{6} T^{5} + p^{9} T^{6} \) |
| 23 | $S_4\times C_2$ | \( 1 + 103 T + 33165 T^{2} + 2271010 T^{3} + 33165 p^{3} T^{4} + 103 p^{6} T^{5} + p^{9} T^{6} \) |
| 29 | $S_4\times C_2$ | \( 1 + 93 T + 41103 T^{2} + 2284086 T^{3} + 41103 p^{3} T^{4} + 93 p^{6} T^{5} + p^{9} T^{6} \) |
| 31 | $S_4\times C_2$ | \( 1 - 116 T + 60733 T^{2} - 7995928 T^{3} + 60733 p^{3} T^{4} - 116 p^{6} T^{5} + p^{9} T^{6} \) |
| 37 | $S_4\times C_2$ | \( 1 + 466 T + 150899 T^{2} + 32064268 T^{3} + 150899 p^{3} T^{4} + 466 p^{6} T^{5} + p^{9} T^{6} \) |
| 41 | $S_4\times C_2$ | \( 1 + 188 T + 180207 T^{2} + 20873144 T^{3} + 180207 p^{3} T^{4} + 188 p^{6} T^{5} + p^{9} T^{6} \) |
| 43 | $S_4\times C_2$ | \( 1 - 11 T + 193225 T^{2} - 4108402 T^{3} + 193225 p^{3} T^{4} - 11 p^{6} T^{5} + p^{9} T^{6} \) |
| 47 | $S_4\times C_2$ | \( 1 + 163 T + 164805 T^{2} + 37697050 T^{3} + 164805 p^{3} T^{4} + 163 p^{6} T^{5} + p^{9} T^{6} \) |
| 53 | $S_4\times C_2$ | \( 1 - 197 T + 304365 T^{2} - 70223642 T^{3} + 304365 p^{3} T^{4} - 197 p^{6} T^{5} + p^{9} T^{6} \) |
| 59 | $S_4\times C_2$ | \( 1 + 1381 T + 1171863 T^{2} + 620122726 T^{3} + 1171863 p^{3} T^{4} + 1381 p^{6} T^{5} + p^{9} T^{6} \) |
| 61 | $S_4\times C_2$ | \( 1 + 405 T + 457275 T^{2} + 183007486 T^{3} + 457275 p^{3} T^{4} + 405 p^{6} T^{5} + p^{9} T^{6} \) |
| 67 | $S_4\times C_2$ | \( 1 + 943 T + 985825 T^{2} + 566069114 T^{3} + 985825 p^{3} T^{4} + 943 p^{6} T^{5} + p^{9} T^{6} \) |
| 71 | $S_4\times C_2$ | \( 1 + 1052 T + 553065 T^{2} + 231251816 T^{3} + 553065 p^{3} T^{4} + 1052 p^{6} T^{5} + p^{9} T^{6} \) |
| 73 | $S_4\times C_2$ | \( 1 + 580 T + 48224 T^{2} - 275268242 T^{3} + 48224 p^{3} T^{4} + 580 p^{6} T^{5} + p^{9} T^{6} \) |
| 79 | $S_4\times C_2$ | \( 1 - 1402 T + 1906685 T^{2} - 1400621260 T^{3} + 1906685 p^{3} T^{4} - 1402 p^{6} T^{5} + p^{9} T^{6} \) |
| 83 | $S_4\times C_2$ | \( 1 + 1802 T + 2632785 T^{2} + 2152705340 T^{3} + 2632785 p^{3} T^{4} + 1802 p^{6} T^{5} + p^{9} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 - 1966 T + 3070563 T^{2} - 2819136892 T^{3} + 3070563 p^{3} T^{4} - 1966 p^{6} T^{5} + p^{9} T^{6} \) |
| 97 | $S_4\times C_2$ | \( 1 - 48 T + 2319663 T^{2} - 169626944 T^{3} + 2319663 p^{3} T^{4} - 48 p^{6} T^{5} + p^{9} T^{6} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{6} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.23188407839113984594833520891, −10.04295770834128005997523984575, −10.03130725551444965291870903561, −9.910841325545429073187763028541, −9.264272061005136409950220502785, −8.958875059598570505249450950689, −8.803471050523879066402585402871, −8.202695716701966556295265194936, −7.74494674235714830524074031520, −7.61485190049755229484346652539, −7.45848194320415807351128371701, −6.66668436210589048693041205297, −6.51563126275869104434099225099, −5.94610612784240030886051617776, −5.90605902570103505142967017215, −5.77551866768351498754625685761, −5.02128693551510706994655719396, −4.90706415950642757083701870678, −4.25051876102451649887893109195, −3.59750427184239555121059889331, −3.21789497601185194258130988577, −3.09683038630498494217981867255, −2.52199208728174713755348052591, −1.98516159304502441257232986522, −1.50575812084467461422837399803, 0, 0, 0,
1.50575812084467461422837399803, 1.98516159304502441257232986522, 2.52199208728174713755348052591, 3.09683038630498494217981867255, 3.21789497601185194258130988577, 3.59750427184239555121059889331, 4.25051876102451649887893109195, 4.90706415950642757083701870678, 5.02128693551510706994655719396, 5.77551866768351498754625685761, 5.90605902570103505142967017215, 5.94610612784240030886051617776, 6.51563126275869104434099225099, 6.66668436210589048693041205297, 7.45848194320415807351128371701, 7.61485190049755229484346652539, 7.74494674235714830524074031520, 8.202695716701966556295265194936, 8.803471050523879066402585402871, 8.958875059598570505249450950689, 9.264272061005136409950220502785, 9.910841325545429073187763028541, 10.03130725551444965291870903561, 10.04295770834128005997523984575, 10.23188407839113984594833520891
