| L(s) = 1 | + 1.58i·2-s + 253.·4-s − 619. i·5-s − 2.36e3·7-s + 807. i·8-s + 982.·10-s + 4.58e3i·11-s − 2.19e4·13-s − 3.75e3i·14-s + 6.36e4·16-s − 9.32e4i·17-s − 2.32e4·19-s − 1.57e5i·20-s − 7.27e3·22-s + 4.75e5i·23-s + ⋯ |
| L(s) = 1 | + 0.0990i·2-s + 0.990·4-s − 0.991i·5-s − 0.985·7-s + 0.197i·8-s + 0.0982·10-s + 0.313i·11-s − 0.768·13-s − 0.0976i·14-s + 0.970·16-s − 1.11i·17-s − 0.178·19-s − 0.982i·20-s − 0.0310·22-s + 1.70i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 243 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 243 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(1.242544171\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.242544171\) |
| \(L(5)\) |
|
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 | \( 1 - 1.58iT - 256T^{2} \) |
| 5 | \( 1 + 619. iT - 3.90e5T^{2} \) |
| 7 | \( 1 + 2.36e3T + 5.76e6T^{2} \) |
| 11 | \( 1 - 4.58e3iT - 2.14e8T^{2} \) |
| 13 | \( 1 + 2.19e4T + 8.15e8T^{2} \) |
| 17 | \( 1 + 9.32e4iT - 6.97e9T^{2} \) |
| 19 | \( 1 + 2.32e4T + 1.69e10T^{2} \) |
| 23 | \( 1 - 4.75e5iT - 7.83e10T^{2} \) |
| 29 | \( 1 + 2.73e5iT - 5.00e11T^{2} \) |
| 31 | \( 1 + 1.69e6T + 8.52e11T^{2} \) |
| 37 | \( 1 - 3.17e5T + 3.51e12T^{2} \) |
| 41 | \( 1 - 3.05e6iT - 7.98e12T^{2} \) |
| 43 | \( 1 - 2.19e6T + 1.16e13T^{2} \) |
| 47 | \( 1 - 1.37e6iT - 2.38e13T^{2} \) |
| 53 | \( 1 - 7.48e6iT - 6.22e13T^{2} \) |
| 59 | \( 1 - 1.19e7iT - 1.46e14T^{2} \) |
| 61 | \( 1 - 2.75e6T + 1.91e14T^{2} \) |
| 67 | \( 1 - 3.39e7T + 4.06e14T^{2} \) |
| 71 | \( 1 + 4.59e7iT - 6.45e14T^{2} \) |
| 73 | \( 1 + 1.26e7T + 8.06e14T^{2} \) |
| 79 | \( 1 - 5.77e6T + 1.51e15T^{2} \) |
| 83 | \( 1 - 8.17e6iT - 2.25e15T^{2} \) |
| 89 | \( 1 - 3.98e7iT - 3.93e15T^{2} \) |
| 97 | \( 1 - 7.32e6T + 7.83e15T^{2} \) |
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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
−11.01136930243413009021810851315, −9.727070430834745831876339656850, −9.225888416933050905830409944691, −7.73159690400259489786626542103, −7.05723147725680078449145068680, −5.88367504653806412528233287242, −4.93986250764028581046574187381, −3.45704390783042498554282302512, −2.32663355219925193379016147992, −1.07429806812831163445729073675,
0.26252442185126875329350610196, 2.01401616897304973178226707024, 2.88740528650765810724602295921, 3.80173552866905243659164311145, 5.66125946340607312085602350190, 6.64769708958372467844823134763, 7.08797059037098821340595321117, 8.400011078200365016845030731600, 9.767966281987531128904884372537, 10.59717027528035472580422191642
