| L(s) = 1 | − 4.69i·2-s + 233.·4-s − 365. i·5-s − 3.78e3·7-s − 2.30e3i·8-s − 1.71e3·10-s − 1.30e4i·11-s + 2.13e4·13-s + 1.77e4i·14-s + 4.90e4·16-s + 1.48e5i·17-s + 1.92e5·19-s − 8.55e4i·20-s − 6.12e4·22-s + 5.16e5i·23-s + ⋯ |
| L(s) = 1 | − 0.293i·2-s + 0.913·4-s − 0.585i·5-s − 1.57·7-s − 0.561i·8-s − 0.171·10-s − 0.891i·11-s + 0.748·13-s + 0.462i·14-s + 0.749·16-s + 1.77i·17-s + 1.47·19-s − 0.534i·20-s − 0.261·22-s + 1.84i·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
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(2.378230806\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.378230806\) |
| \(L(5)\) |
|
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 | \( 1 + 4.69iT - 256T^{2} \) |
| 5 | \( 1 + 365. iT - 3.90e5T^{2} \) |
| 7 | \( 1 + 3.78e3T + 5.76e6T^{2} \) |
| 11 | \( 1 + 1.30e4iT - 2.14e8T^{2} \) |
| 13 | \( 1 - 2.13e4T + 8.15e8T^{2} \) |
| 17 | \( 1 - 1.48e5iT - 6.97e9T^{2} \) |
| 19 | \( 1 - 1.92e5T + 1.69e10T^{2} \) |
| 23 | \( 1 - 5.16e5iT - 7.83e10T^{2} \) |
| 29 | \( 1 + 8.98e5iT - 5.00e11T^{2} \) |
| 31 | \( 1 - 5.57e5T + 8.52e11T^{2} \) |
| 37 | \( 1 - 3.28e5T + 3.51e12T^{2} \) |
| 41 | \( 1 + 4.45e6iT - 7.98e12T^{2} \) |
| 43 | \( 1 + 2.33e6T + 1.16e13T^{2} \) |
| 47 | \( 1 + 4.92e6iT - 2.38e13T^{2} \) |
| 53 | \( 1 + 1.52e6iT - 6.22e13T^{2} \) |
| 59 | \( 1 + 5.92e6iT - 1.46e14T^{2} \) |
| 61 | \( 1 + 1.85e7T + 1.91e14T^{2} \) |
| 67 | \( 1 + 2.92e7T + 4.06e14T^{2} \) |
| 71 | \( 1 - 6.92e6iT - 6.45e14T^{2} \) |
| 73 | \( 1 - 3.58e7T + 8.06e14T^{2} \) |
| 79 | \( 1 - 1.99e5T + 1.51e15T^{2} \) |
| 83 | \( 1 + 1.96e7iT - 2.25e15T^{2} \) |
| 89 | \( 1 + 1.04e8iT - 3.93e15T^{2} \) |
| 97 | \( 1 - 3.06e7T + 7.83e15T^{2} \) |
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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
−10.41395638835249691076954539520, −9.624712306014727031156274003653, −8.568902091108516646924935655771, −7.40119904460606863194614447432, −6.22426900992982886113221963665, −5.72338190067909685793571482281, −3.66551704707614882849519710132, −3.19950523995442721445135639600, −1.61747017623887916870353235432, −0.58559407946864221578226356593,
0.997635649236857076185991349762, 2.75608143693618737124328137789, 3.10813249476630465303828933157, 4.91542154798202377052332448148, 6.33407909651441216947355891255, 6.78538224387040803261777955727, 7.60254054746709161756209592089, 9.129609541418014389813198420735, 9.995350065176521122928051129807, 10.83377568712352053544982616331
