| L(s) = 1 | + (−0.670 + 1.88i)2-s + (−3.10 − 2.52i)4-s + (−3.98 + 1.45i)5-s + (−2.54 + 0.448i)7-s + (6.83 − 4.15i)8-s + (−0.0621 − 8.47i)10-s + (0.818 − 2.24i)11-s + (−3.62 − 3.04i)13-s + (0.860 − 5.09i)14-s + (3.23 + 15.6i)16-s + (8.77 − 15.2i)17-s + (13.3 − 7.72i)19-s + (16.0 + 5.56i)20-s + (3.69 + 3.05i)22-s + (13.3 + 2.35i)23-s + ⋯ |
| L(s) = 1 | + (−0.335 + 0.942i)2-s + (−0.775 − 0.631i)4-s + (−0.796 + 0.290i)5-s + (−0.363 + 0.0641i)7-s + (0.854 − 0.518i)8-s + (−0.00621 − 0.847i)10-s + (0.0744 − 0.204i)11-s + (−0.279 − 0.234i)13-s + (0.0614 − 0.364i)14-s + (0.202 + 0.979i)16-s + (0.516 − 0.894i)17-s + (0.704 − 0.406i)19-s + (0.801 + 0.278i)20-s + (0.167 + 0.138i)22-s + (0.580 + 0.102i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0298i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.999 - 0.0298i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.898990 + 0.0134378i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.898990 + 0.0134378i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.670 - 1.88i)T \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + (3.98 - 1.45i)T + (19.1 - 16.0i)T^{2} \) |
| 7 | \( 1 + (2.54 - 0.448i)T + (46.0 - 16.7i)T^{2} \) |
| 11 | \( 1 + (-0.818 + 2.24i)T + (-92.6 - 77.7i)T^{2} \) |
| 13 | \( 1 + (3.62 + 3.04i)T + (29.3 + 166. i)T^{2} \) |
| 17 | \( 1 + (-8.77 + 15.2i)T + (-144.5 - 250. i)T^{2} \) |
| 19 | \( 1 + (-13.3 + 7.72i)T + (180.5 - 312. i)T^{2} \) |
| 23 | \( 1 + (-13.3 - 2.35i)T + (497. + 180. i)T^{2} \) |
| 29 | \( 1 + (-33.3 + 27.9i)T + (146. - 828. i)T^{2} \) |
| 31 | \( 1 + (-46.8 - 8.25i)T + (903. + 328. i)T^{2} \) |
| 37 | \( 1 + (-23.0 + 39.8i)T + (-684.5 - 1.18e3i)T^{2} \) |
| 41 | \( 1 + (59.1 + 49.5i)T + (291. + 1.65e3i)T^{2} \) |
| 43 | \( 1 + (11.2 - 30.8i)T + (-1.41e3 - 1.18e3i)T^{2} \) |
| 47 | \( 1 + (-36.6 + 6.45i)T + (2.07e3 - 755. i)T^{2} \) |
| 53 | \( 1 - 35.4T + 2.80e3T^{2} \) |
| 59 | \( 1 + (-31.4 - 86.5i)T + (-2.66e3 + 2.23e3i)T^{2} \) |
| 61 | \( 1 + (17.4 + 98.7i)T + (-3.49e3 + 1.27e3i)T^{2} \) |
| 67 | \( 1 + (-18.1 + 21.6i)T + (-779. - 4.42e3i)T^{2} \) |
| 71 | \( 1 + (-40.2 - 23.2i)T + (2.52e3 + 4.36e3i)T^{2} \) |
| 73 | \( 1 + (18.0 + 31.2i)T + (-2.66e3 + 4.61e3i)T^{2} \) |
| 79 | \( 1 + (58.8 + 70.1i)T + (-1.08e3 + 6.14e3i)T^{2} \) |
| 83 | \( 1 + (50.7 + 60.4i)T + (-1.19e3 + 6.78e3i)T^{2} \) |
| 89 | \( 1 + (-31.6 - 54.8i)T + (-3.96e3 + 6.85e3i)T^{2} \) |
| 97 | \( 1 + (74.9 + 27.2i)T + (7.20e3 + 6.04e3i)T^{2} \) |
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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
−11.39163654004382722178712602361, −10.21277658598701245472207908723, −9.448120694724835201711443756214, −8.375973850400182771759349073647, −7.51348729921802521843044330601, −6.77401645572929356918029615889, −5.58212391236306726516241725816, −4.48557158770510101427002493122, −3.10830402667146348263684165850, −0.59580122848872440930543015076,
1.15015243687363339395763211586, 2.93289574480625630671350275144, 4.00226971309840135083999930161, 5.02835125320235706058861086451, 6.71172637933533259401291955171, 7.967665525516500501101560490720, 8.537354712773609057290816506691, 9.795563266516240324382099987591, 10.33332123679015395283218461078, 11.62457368578904891321406344520
