| L(s) = 1 | + (0.909 + 1.08i)2-s + (−0.347 + 1.96i)4-s + (1.54 + 4.23i)5-s + (0.223 + 1.26i)7-s + (−2.44 + 1.41i)8-s + (−3.18 + 5.51i)10-s + (−2.33 + 6.41i)11-s + (8.48 + 7.11i)13-s + (−1.17 + 1.39i)14-s + (−3.75 − 1.36i)16-s + (−24.1 − 13.9i)17-s + (14.5 + 25.1i)19-s + (−8.87 + 1.56i)20-s + (−9.07 + 3.30i)22-s + (29.2 + 5.15i)23-s + ⋯ |
| L(s) = 1 | + (0.454 + 0.541i)2-s + (−0.0868 + 0.492i)4-s + (0.308 + 0.846i)5-s + (0.0319 + 0.181i)7-s + (−0.306 + 0.176i)8-s + (−0.318 + 0.551i)10-s + (−0.212 + 0.583i)11-s + (0.652 + 0.547i)13-s + (−0.0836 + 0.0997i)14-s + (−0.234 − 0.0855i)16-s + (−1.42 − 0.821i)17-s + (0.764 + 1.32i)19-s + (−0.443 + 0.0782i)20-s + (−0.412 + 0.150i)22-s + (1.27 + 0.223i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.200 - 0.979i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.200 - 0.979i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.15733 + 1.41812i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.15733 + 1.41812i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.909 - 1.08i)T \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + (-1.54 - 4.23i)T + (-19.1 + 16.0i)T^{2} \) |
| 7 | \( 1 + (-0.223 - 1.26i)T + (-46.0 + 16.7i)T^{2} \) |
| 11 | \( 1 + (2.33 - 6.41i)T + (-92.6 - 77.7i)T^{2} \) |
| 13 | \( 1 + (-8.48 - 7.11i)T + (29.3 + 166. i)T^{2} \) |
| 17 | \( 1 + (24.1 + 13.9i)T + (144.5 + 250. i)T^{2} \) |
| 19 | \( 1 + (-14.5 - 25.1i)T + (-180.5 + 312. i)T^{2} \) |
| 23 | \( 1 + (-29.2 - 5.15i)T + (497. + 180. i)T^{2} \) |
| 29 | \( 1 + (9.87 + 11.7i)T + (-146. + 828. i)T^{2} \) |
| 31 | \( 1 + (-8.09 + 45.8i)T + (-903. - 328. i)T^{2} \) |
| 37 | \( 1 + (6.62 - 11.4i)T + (-684.5 - 1.18e3i)T^{2} \) |
| 41 | \( 1 + (-27.2 + 32.4i)T + (-291. - 1.65e3i)T^{2} \) |
| 43 | \( 1 + (48.0 + 17.4i)T + (1.41e3 + 1.18e3i)T^{2} \) |
| 47 | \( 1 + (-57.3 + 10.1i)T + (2.07e3 - 755. i)T^{2} \) |
| 53 | \( 1 + 72.9iT - 2.80e3T^{2} \) |
| 59 | \( 1 + (-10.4 - 28.6i)T + (-2.66e3 + 2.23e3i)T^{2} \) |
| 61 | \( 1 + (-2.62 - 14.9i)T + (-3.49e3 + 1.27e3i)T^{2} \) |
| 67 | \( 1 + (-86.2 - 72.4i)T + (779. + 4.42e3i)T^{2} \) |
| 71 | \( 1 + (7.53 + 4.34i)T + (2.52e3 + 4.36e3i)T^{2} \) |
| 73 | \( 1 + (52.2 + 90.5i)T + (-2.66e3 + 4.61e3i)T^{2} \) |
| 79 | \( 1 + (46.1 - 38.7i)T + (1.08e3 - 6.14e3i)T^{2} \) |
| 83 | \( 1 + (-17.6 - 21.0i)T + (-1.19e3 + 6.78e3i)T^{2} \) |
| 89 | \( 1 + (106. - 61.5i)T + (3.96e3 - 6.85e3i)T^{2} \) |
| 97 | \( 1 + (-40.5 - 14.7i)T + (7.20e3 + 6.04e3i)T^{2} \) |
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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
−13.22414067386678560269593055221, −11.92000127407037011539432075630, −11.05908749083994920520028082401, −9.841957336216179970543054932008, −8.745166466203171879404283247150, −7.36821741667391670689601950825, −6.58944997600149482971832744802, −5.41394986725472946803124645364, −3.98926552261320731905928676041, −2.43920673579725196039087652153,
1.10024339983651028230053538124, 3.00166340660497258575892998672, 4.55810758219926272619914488747, 5.52959106100290831005020113723, 6.86018193431710553252124748461, 8.577460419305248154858899996956, 9.181400233926807321589741182016, 10.69600724052190189006843937866, 11.19994370726809807685967553990, 12.64547004553376474576250982011
