| L(s) = 1 | − 29.3i·2-s − 602.·4-s + 223. i·5-s − 2.11e3·7-s + 1.01e4i·8-s + 6.55e3·10-s + 1.30e4i·11-s − 2.10e4·13-s + 6.20e4i·14-s + 1.43e5·16-s + 1.38e5i·17-s − 1.48e5·19-s − 1.34e5i·20-s + 3.83e5·22-s + 1.52e5i·23-s + ⋯ |
| L(s) = 1 | − 1.83i·2-s − 2.35·4-s + 0.357i·5-s − 0.882·7-s + 2.48i·8-s + 0.655·10-s + 0.893i·11-s − 0.735·13-s + 1.61i·14-s + 2.18·16-s + 1.66i·17-s − 1.13·19-s − 0.842i·20-s + 1.63·22-s + 0.544i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 243 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 243 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(0.5136241714\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5136241714\) |
| \(L(5)\) |
|
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 | \( 1 - 223. iT - 3.90e5T^{2} \) |
| 7 | \( 1 + 2.11e3T + 5.76e6T^{2} \) |
| 11 | \( 1 - 1.30e4iT - 2.14e8T^{2} \) |
| 13 | \( 1 + 2.10e4T + 8.15e8T^{2} \) |
| 17 | \( 1 - 1.38e5iT - 6.97e9T^{2} \) |
| 19 | \( 1 + 1.48e5T + 1.69e10T^{2} \) |
| 23 | \( 1 - 1.52e5iT - 7.83e10T^{2} \) |
| 29 | \( 1 + 5.42e5iT - 5.00e11T^{2} \) |
| 31 | \( 1 + 1.57e6T + 8.52e11T^{2} \) |
| 37 | \( 1 - 1.70e6T + 3.51e12T^{2} \) |
| 41 | \( 1 + 3.80e6iT - 7.98e12T^{2} \) |
| 43 | \( 1 - 4.06e6T + 1.16e13T^{2} \) |
| 47 | \( 1 + 6.46e6iT - 2.38e13T^{2} \) |
| 53 | \( 1 - 1.15e7iT - 6.22e13T^{2} \) |
| 59 | \( 1 - 1.01e7iT - 1.46e14T^{2} \) |
| 61 | \( 1 + 2.44e7T + 1.91e14T^{2} \) |
| 67 | \( 1 - 1.50e7T + 4.06e14T^{2} \) |
| 71 | \( 1 + 1.55e7iT - 6.45e14T^{2} \) |
| 73 | \( 1 + 2.13e7T + 8.06e14T^{2} \) |
| 79 | \( 1 + 2.22e6T + 1.51e15T^{2} \) |
| 83 | \( 1 + 2.99e5iT - 2.25e15T^{2} \) |
| 89 | \( 1 + 9.07e7iT - 3.93e15T^{2} \) |
| 97 | \( 1 + 9.94e7T + 7.83e15T^{2} \) |
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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
−10.41911009473288150636412396903, −9.579957971658242676926391688423, −8.776934946806743413403407486612, −7.34076187410887839423345290205, −5.92666972597365037628662764186, −4.44512529520290391386054873371, −3.66113407954295330720219606092, −2.51690064761583132248903569637, −1.71428753969102409024907326990, −0.19060324980473691771938014432,
0.60527161069554808815708781362, 2.97929122703140646262207566773, 4.43160388347534802386789816774, 5.30720670729292682035611739525, 6.32654603146207965898855037767, 7.05219031328408516750666850526, 8.048692680158497049163416906896, 9.070912361669219529646842323121, 9.554876062182305773651034836238, 11.02231258966376813379159389796
