| L(s) = 1 | − 7.92i·2-s + 193.·4-s + 799. i·5-s − 4.07e3·7-s − 3.55e3i·8-s + 6.33e3·10-s + 2.49e4i·11-s − 1.35e4·13-s + 3.22e4i·14-s + 2.12e4·16-s + 1.00e5i·17-s + 1.84e4·19-s + 1.54e5i·20-s + 1.97e5·22-s − 1.93e4i·23-s + ⋯ |
| L(s) = 1 | − 0.495i·2-s + 0.754·4-s + 1.27i·5-s − 1.69·7-s − 0.868i·8-s + 0.633·10-s + 1.70i·11-s − 0.475·13-s + 0.839i·14-s + 0.324·16-s + 1.20i·17-s + 0.141·19-s + 0.965i·20-s + 0.844·22-s − 0.0693i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(0.910414 + 0.910414i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.910414 + 0.910414i\) |
| \(L(5)\) |
|
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 | \( 1 + 7.92iT - 256T^{2} \) |
| 5 | \( 1 - 799. iT - 3.90e5T^{2} \) |
| 7 | \( 1 + 4.07e3T + 5.76e6T^{2} \) |
| 11 | \( 1 - 2.49e4iT - 2.14e8T^{2} \) |
| 13 | \( 1 + 1.35e4T + 8.15e8T^{2} \) |
| 17 | \( 1 - 1.00e5iT - 6.97e9T^{2} \) |
| 19 | \( 1 - 1.84e4T + 1.69e10T^{2} \) |
| 23 | \( 1 + 1.93e4iT - 7.83e10T^{2} \) |
| 29 | \( 1 - 1.47e5iT - 5.00e11T^{2} \) |
| 31 | \( 1 - 5.37e5T + 8.52e11T^{2} \) |
| 37 | \( 1 + 5.39e5T + 3.51e12T^{2} \) |
| 41 | \( 1 - 1.55e5iT - 7.98e12T^{2} \) |
| 43 | \( 1 + 5.25e6T + 1.16e13T^{2} \) |
| 47 | \( 1 + 6.53e6iT - 2.38e13T^{2} \) |
| 53 | \( 1 - 6.49e6iT - 6.22e13T^{2} \) |
| 59 | \( 1 - 1.01e7iT - 1.46e14T^{2} \) |
| 61 | \( 1 - 1.22e7T + 1.91e14T^{2} \) |
| 67 | \( 1 - 3.31e7T + 4.06e14T^{2} \) |
| 71 | \( 1 + 2.04e7iT - 6.45e14T^{2} \) |
| 73 | \( 1 - 1.03e7T + 8.06e14T^{2} \) |
| 79 | \( 1 + 2.57e7T + 1.51e15T^{2} \) |
| 83 | \( 1 + 2.24e7iT - 2.25e15T^{2} \) |
| 89 | \( 1 + 7.34e6iT - 3.93e15T^{2} \) |
| 97 | \( 1 + 1.55e7T + 7.83e15T^{2} \) |
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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
−15.57038127623494284336410199660, −14.86689615251513833842648560230, −12.97176662638751143059952565962, −12.03996669448772500361077522582, −10.39231898923632552960627442282, −9.884686608877118018433402167695, −7.14689420751090607999344774267, −6.46636129956545820453459482664, −3.50888980550534771018977753200, −2.27782006791329821099902405150,
0.54380496581751825762763512830, 3.06107427372173169401568124509, 5.44745219318776670244609837793, 6.68890569235072867745797694521, 8.396280138056249006489336408612, 9.688482052814087073861334306572, 11.50501654581518552568050453048, 12.72817817518091052957708776414, 13.86083854890463191766173981154, 15.80732808600673294204719113518
