| L(s) = 1 | + 4·5-s − 6·7-s − 3·9-s − 8·11-s + 14·13-s − 16·17-s + 6·19-s + 6·23-s − 4·25-s + 6·29-s + 8·31-s − 24·35-s + 12·37-s − 2·43-s − 12·45-s + 2·47-s + 8·49-s − 16·53-s − 32·55-s − 12·59-s + 24·61-s + 18·63-s + 56·65-s + 14·67-s + 48·77-s + 24·79-s + 2·83-s + ⋯ |
| L(s) = 1 | + 1.78·5-s − 2.26·7-s − 9-s − 2.41·11-s + 3.88·13-s − 3.88·17-s + 1.37·19-s + 1.25·23-s − 4/5·25-s + 1.11·29-s + 1.43·31-s − 4.05·35-s + 1.97·37-s − 0.304·43-s − 1.78·45-s + 0.291·47-s + 8/7·49-s − 2.19·53-s − 4.31·55-s − 1.56·59-s + 3.07·61-s + 2.26·63-s + 6.94·65-s + 1.71·67-s + 5.47·77-s + 2.70·79-s + 0.219·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.683255830\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.683255830\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| good | 5 | $D_4\times C_2$ | \( 1 - 4 T + 4 p T^{2} - 52 T^{3} + 151 T^{4} - 52 p T^{5} + 4 p^{3} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 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 | $D_4\times C_2$ | \( 1 + 8 T + 41 T^{2} + 152 T^{3} + 532 T^{4} + 152 p T^{5} + 41 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 13 | $C_2$$\times$$C_2^2$ | \( ( 1 - 7 T + p T^{2} )^{2}( 1 - T^{2} + p^{2} T^{4} ) \) |
| 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 | $D_4\times C_2$ | \( 1 + 2 T + 65 T^{2} + 426 T^{3} + 2744 T^{4} + 426 p T^{5} + 65 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 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 | $D_4\times C_2$ | \( 1 + 12 T + 45 T^{2} - 828 T^{3} - 9748 T^{4} - 828 p T^{5} + 45 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $D_4\times C_2$ | \( 1 - 24 T + 180 T^{2} + 300 T^{3} - 11209 T^{4} + 300 p T^{5} + 180 p^{2} T^{6} - 24 p^{3} T^{7} + p^{4} T^{8} \) |
| 67 | $D_4\times C_2$ | \( 1 - 14 T + 113 T^{2} - 726 T^{3} + 3848 T^{4} - 726 p T^{5} + 113 p^{2} T^{6} - 14 p^{3} T^{7} + p^{4} T^{8} \) |
| 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
−8.001089553094951181872760568797, −7.60241232225775011300711658489, −6.90700348871671951527275815094, −6.79201497109745060066968480096, −6.67603951010784548686633086492, −6.53442058697239782247019023900, −6.26950661889468743987492335439, −6.02083944492527300538624081356, −5.88060325453294493863199956584, −5.75477533926051155333888436969, −5.49537223598196121522978518526, −4.99844520234877892242828333485, −4.88034051281882439125679380690, −4.55651518398104424197820009672, −4.15782150827215961161591029308, −3.74179055227124784312286030819, −3.62963650697424264783565712871, −3.14999180499124618960374093457, −2.89993379860249389600465731415, −2.75544305700835246477791237629, −2.44121396100694338404737443707, −2.04753882268207566636267285774, −1.71877354122157718157105671499, −0.946072903074264022935386094781, −0.44716466961339725046195303942,
0.44716466961339725046195303942, 0.946072903074264022935386094781, 1.71877354122157718157105671499, 2.04753882268207566636267285774, 2.44121396100694338404737443707, 2.75544305700835246477791237629, 2.89993379860249389600465731415, 3.14999180499124618960374093457, 3.62963650697424264783565712871, 3.74179055227124784312286030819, 4.15782150827215961161591029308, 4.55651518398104424197820009672, 4.88034051281882439125679380690, 4.99844520234877892242828333485, 5.49537223598196121522978518526, 5.75477533926051155333888436969, 5.88060325453294493863199956584, 6.02083944492527300538624081356, 6.26950661889468743987492335439, 6.53442058697239782247019023900, 6.67603951010784548686633086492, 6.79201497109745060066968480096, 6.90700348871671951527275815094, 7.60241232225775011300711658489, 8.001089553094951181872760568797
