| L(s) = 1 | − 81·3-s + 414·5-s − 1.02e3·7-s + 4.37e3·9-s − 9.55e3·11-s + 954·13-s − 3.35e4·15-s − 30·17-s − 3.95e4·19-s + 8.33e4·21-s + 3.11e4·23-s + 3.83e4·25-s − 1.96e5·27-s + 3.88e4·29-s − 1.47e5·31-s + 7.73e5·33-s − 4.26e5·35-s − 2.17e5·37-s − 7.72e4·39-s + 1.74e5·41-s − 1.93e5·43-s + 1.81e6·45-s − 6.75e5·47-s + 7.05e5·49-s + 2.43e3·51-s + 4.76e5·53-s − 3.95e6·55-s + ⋯ |
| L(s) = 1 | − 1.73·3-s + 1.48·5-s − 1.13·7-s + 2·9-s − 2.16·11-s + 0.120·13-s − 2.56·15-s − 0.00148·17-s − 1.32·19-s + 1.96·21-s + 0.534·23-s + 0.490·25-s − 1.92·27-s + 0.296·29-s − 0.891·31-s + 3.74·33-s − 1.67·35-s − 0.706·37-s − 0.208·39-s + 0.395·41-s − 0.371·43-s + 2.96·45-s − 0.949·47-s + 6/7·49-s + 0.00256·51-s + 0.440·53-s − 3.20·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 37933056 ^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 37933056 ^{s/2} \, \Gamma_{\C}(s+7/2)^{3} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(0.2070384579\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2070384579\) |
| \(L(\frac{9}{2})\) |
|
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 | | \( 1 \) |
| 3 | $C_1$ | \( ( 1 + p^{3} T )^{3} \) |
| 7 | $C_1$ | \( ( 1 + p^{3} T )^{3} \) |
| good | 5 | $S_4\times C_2$ | \( 1 - 414 T + 26619 p T^{2} - 2062828 p^{2} T^{3} + 26619 p^{8} T^{4} - 414 p^{14} T^{5} + p^{21} T^{6} \) |
| 11 | $S_4\times C_2$ | \( 1 + 9552 T + 53201373 T^{2} + 210809216496 T^{3} + 53201373 p^{7} T^{4} + 9552 p^{14} T^{5} + p^{21} T^{6} \) |
| 13 | $S_4\times C_2$ | \( 1 - 954 T + 151979403 T^{2} - 186272630172 T^{3} + 151979403 p^{7} T^{4} - 954 p^{14} T^{5} + p^{21} T^{6} \) |
| 17 | $S_4\times C_2$ | \( 1 + 30 T + 33381315 p T^{2} + 5949569367292 T^{3} + 33381315 p^{8} T^{4} + 30 p^{14} T^{5} + p^{21} T^{6} \) |
| 19 | $S_4\times C_2$ | \( 1 + 39588 T + 2320729185 T^{2} + 69765841269592 T^{3} + 2320729185 p^{7} T^{4} + 39588 p^{14} T^{5} + p^{21} T^{6} \) |
| 23 | $S_4\times C_2$ | \( 1 - 31164 T + 1660714713 T^{2} - 304772953066664 T^{3} + 1660714713 p^{7} T^{4} - 31164 p^{14} T^{5} + p^{21} T^{6} \) |
| 29 | $S_4\times C_2$ | \( 1 - 38898 T + 41882247723 T^{2} - 924072889230988 T^{3} + 41882247723 p^{7} T^{4} - 38898 p^{14} T^{5} + p^{21} T^{6} \) |
| 31 | $S_4\times C_2$ | \( 1 + 147888 T + 25627003101 T^{2} + 8970382834919584 T^{3} + 25627003101 p^{7} T^{4} + 147888 p^{14} T^{5} + p^{21} T^{6} \) |
| 37 | $S_4\times C_2$ | \( 1 + 217662 T + 126303786147 T^{2} + 19866972889716564 T^{3} + 126303786147 p^{7} T^{4} + 217662 p^{14} T^{5} + p^{21} T^{6} \) |
| 41 | $S_4\times C_2$ | \( 1 - 174594 T + 485889051435 T^{2} - 70469419124187908 T^{3} + 485889051435 p^{7} T^{4} - 174594 p^{14} T^{5} + p^{21} T^{6} \) |
| 43 | $S_4\times C_2$ | \( 1 + 193644 T + 570626861337 T^{2} + 44073549956088776 T^{3} + 570626861337 p^{7} T^{4} + 193644 p^{14} T^{5} + p^{21} T^{6} \) |
| 47 | $S_4\times C_2$ | \( 1 + 675864 T + 585601789053 T^{2} + 781083008402314192 T^{3} + 585601789053 p^{7} T^{4} + 675864 p^{14} T^{5} + p^{21} T^{6} \) |
| 53 | $S_4\times C_2$ | \( 1 - 476898 T + 3401372161779 T^{2} - 1063374513494419756 T^{3} + 3401372161779 p^{7} T^{4} - 476898 p^{14} T^{5} + p^{21} T^{6} \) |
| 59 | $S_4\times C_2$ | \( 1 + 2107092 T + 7276917709113 T^{2} + 10399345927224705912 T^{3} + 7276917709113 p^{7} T^{4} + 2107092 p^{14} T^{5} + p^{21} T^{6} \) |
| 61 | $S_4\times C_2$ | \( 1 - 190962 T + 6314464452555 T^{2} + 456977593456466420 T^{3} + 6314464452555 p^{7} T^{4} - 190962 p^{14} T^{5} + p^{21} T^{6} \) |
| 67 | $S_4\times C_2$ | \( 1 + 3016380 T + 20229385811169 T^{2} + 36868533152067165480 T^{3} + 20229385811169 p^{7} T^{4} + 3016380 p^{14} T^{5} + p^{21} T^{6} \) |
| 71 | $S_4\times C_2$ | \( 1 + 1285404 T - 1752758319927 T^{2} - 43608279802182862872 T^{3} - 1752758319927 p^{7} T^{4} + 1285404 p^{14} T^{5} + p^{21} T^{6} \) |
| 73 | $S_4\times C_2$ | \( 1 - 6735462 T + 40073872707975 T^{2} - \)\(13\!\cdots\!28\)\( T^{3} + 40073872707975 p^{7} T^{4} - 6735462 p^{14} T^{5} + p^{21} T^{6} \) |
| 79 | $S_4\times C_2$ | \( 1 + 2985240 T + 45288161742237 T^{2} + 79021727041693878736 T^{3} + 45288161742237 p^{7} T^{4} + 2985240 p^{14} T^{5} + p^{21} T^{6} \) |
| 83 | $S_4\times C_2$ | \( 1 + 6168276 T + 59763210259425 T^{2} + \)\(19\!\cdots\!96\)\( T^{3} + 59763210259425 p^{7} T^{4} + 6168276 p^{14} T^{5} + p^{21} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 - 3937938 T + 101375192977947 T^{2} - \)\(21\!\cdots\!04\)\( T^{3} + 101375192977947 p^{7} T^{4} - 3937938 p^{14} T^{5} + p^{21} T^{6} \) |
| 97 | $S_4\times C_2$ | \( 1 - 22816878 T + 377303379784095 T^{2} - \)\(39\!\cdots\!96\)\( T^{3} + 377303379784095 p^{7} T^{4} - 22816878 p^{14} T^{5} + p^{21} T^{6} \) |
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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
−9.496491645144743147219284042189, −8.706083297973213594609240572727, −8.517767271601366323215105571742, −8.348759821600685541369149630289, −7.46901840644612629714460791797, −7.41178346005550270484932758676, −7.24773506059823295026919864056, −6.58039168855900041685262071287, −6.31059776850055532889679968990, −6.21745165003542469160860452711, −5.78616980170300917442065112872, −5.49068938501542911812191847137, −5.37012557860473082253792231378, −4.84711831638402716734745414675, −4.47887718632515388624633134384, −4.34573640052810726232366865943, −3.35295162466964147638019079534, −3.29996497286178970621682476811, −2.84566622697191363700913986915, −2.07030478306206908368266401434, −1.98624980153578435370486332576, −1.77124334357988546592380505505, −0.72440968287035180131518292606, −0.71710893644325363377700778304, −0.099843359451878512151902721204,
0.099843359451878512151902721204, 0.71710893644325363377700778304, 0.72440968287035180131518292606, 1.77124334357988546592380505505, 1.98624980153578435370486332576, 2.07030478306206908368266401434, 2.84566622697191363700913986915, 3.29996497286178970621682476811, 3.35295162466964147638019079534, 4.34573640052810726232366865943, 4.47887718632515388624633134384, 4.84711831638402716734745414675, 5.37012557860473082253792231378, 5.49068938501542911812191847137, 5.78616980170300917442065112872, 6.21745165003542469160860452711, 6.31059776850055532889679968990, 6.58039168855900041685262071287, 7.24773506059823295026919864056, 7.41178346005550270484932758676, 7.46901840644612629714460791797, 8.348759821600685541369149630289, 8.517767271601366323215105571742, 8.706083297973213594609240572727, 9.496491645144743147219284042189
