| L(s) = 1 | − 29.3i·2-s − 608·4-s + 823. i·5-s + 1.96e3·7-s + 1.03e4i·8-s + 2.41e4·10-s + 1.25e4i·11-s − 4.55e4·13-s − 5.78e4i·14-s + 1.48e5·16-s + 5.96e4i·17-s + 1.52e5·19-s − 5.00e5i·20-s + 3.69e5·22-s + 1.31e5i·23-s + ⋯ |
| L(s) = 1 | − 1.83i·2-s − 2.37·4-s + 1.31i·5-s + 0.819·7-s + 2.52i·8-s + 2.41·10-s + 0.859i·11-s − 1.59·13-s − 1.50i·14-s + 2.26·16-s + 0.713i·17-s + 1.16·19-s − 3.12i·20-s + 1.57·22-s + 0.469i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(1.10612\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.10612\) |
| \(L(5)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| good | 2 | \( 1 + 29.3iT - 256T^{2} \) |
| 5 | \( 1 - 823. iT - 3.90e5T^{2} \) |
| 7 | \( 1 - 1.96e3T + 5.76e6T^{2} \) |
| 11 | \( 1 - 1.25e4iT - 2.14e8T^{2} \) |
| 13 | \( 1 + 4.55e4T + 8.15e8T^{2} \) |
| 17 | \( 1 - 5.96e4iT - 6.97e9T^{2} \) |
| 19 | \( 1 - 1.52e5T + 1.69e10T^{2} \) |
| 23 | \( 1 - 1.31e5iT - 7.83e10T^{2} \) |
| 29 | \( 1 - 5.88e5iT - 5.00e11T^{2} \) |
| 31 | \( 1 + 1.64e5T + 8.52e11T^{2} \) |
| 37 | \( 1 + 6.63e5T + 3.51e12T^{2} \) |
| 41 | \( 1 - 9.38e5iT - 7.98e12T^{2} \) |
| 43 | \( 1 - 5.75e5T + 1.16e13T^{2} \) |
| 47 | \( 1 - 9.23e6iT - 2.38e13T^{2} \) |
| 53 | \( 1 + 1.03e7iT - 6.22e13T^{2} \) |
| 59 | \( 1 - 5.03e6iT - 1.46e14T^{2} \) |
| 61 | \( 1 + 1.92e7T + 1.91e14T^{2} \) |
| 67 | \( 1 + 5.98e5T + 4.06e14T^{2} \) |
| 71 | \( 1 + 2.92e7iT - 6.45e14T^{2} \) |
| 73 | \( 1 - 1.28e7T + 8.06e14T^{2} \) |
| 79 | \( 1 + 2.35e7T + 1.51e15T^{2} \) |
| 83 | \( 1 + 3.34e7iT - 2.25e15T^{2} \) |
| 89 | \( 1 - 2.82e7iT - 3.93e15T^{2} \) |
| 97 | \( 1 - 1.36e8T + 7.83e15T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−14.85796048591503592714676740965, −14.14734310171420113978962528196, −12.54063389911506352363865170605, −11.52604338076039343415075765899, −10.51158709903341614490408552913, −9.560412922687546050177047019007, −7.52063485810322983908469228758, −4.80754363752921529066023694918, −3.08754576563721815408088786126, −1.80398497337955035900751147504,
0.52040295694008958693358612029, 4.68732513555273015926947567644, 5.46432275902876930682036652135, 7.38257311318120953641060315951, 8.424502580573556382771707379568, 9.496711958628782159689328433733, 12.06162512665889783105698828169, 13.52445252718662654061100376604, 14.44884294482851358939031487818, 15.71400430337188836957098893216
