| L(s) = 1 | + (1.72 − 0.0916i)3-s + (−1.33 + 1.12i)5-s + (2.31 − 0.841i)7-s + (2.98 − 0.317i)9-s + (0.960 + 0.806i)11-s + (−0.789 − 4.47i)13-s + (−2.21 + 2.06i)15-s + (3.32 + 5.75i)17-s + (0.124 − 0.215i)19-s + (3.91 − 1.66i)21-s + (0.791 + 0.287i)23-s + (−0.339 + 1.92i)25-s + (5.13 − 0.821i)27-s + (−0.0889 + 0.504i)29-s + (−0.770 − 0.280i)31-s + ⋯ |
| L(s) = 1 | + (0.998 − 0.0529i)3-s + (−0.598 + 0.501i)5-s + (0.873 − 0.317i)7-s + (0.994 − 0.105i)9-s + (0.289 + 0.243i)11-s + (−0.219 − 1.24i)13-s + (−0.570 + 0.532i)15-s + (0.806 + 1.39i)17-s + (0.0285 − 0.0495i)19-s + (0.855 − 0.363i)21-s + (0.164 + 0.0600i)23-s + (−0.0678 + 0.384i)25-s + (0.987 − 0.158i)27-s + (−0.0165 + 0.0937i)29-s + (−0.138 − 0.0503i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.994 - 0.100i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.994 - 0.100i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.94991 + 0.0982345i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.94991 + 0.0982345i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (-1.72 + 0.0916i)T \) |
| good | 5 | \( 1 + (1.33 - 1.12i)T + (0.868 - 4.92i)T^{2} \) |
| 7 | \( 1 + (-2.31 + 0.841i)T + (5.36 - 4.49i)T^{2} \) |
| 11 | \( 1 + (-0.960 - 0.806i)T + (1.91 + 10.8i)T^{2} \) |
| 13 | \( 1 + (0.789 + 4.47i)T + (-12.2 + 4.44i)T^{2} \) |
| 17 | \( 1 + (-3.32 - 5.75i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-0.124 + 0.215i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-0.791 - 0.287i)T + (17.6 + 14.7i)T^{2} \) |
| 29 | \( 1 + (0.0889 - 0.504i)T + (-27.2 - 9.91i)T^{2} \) |
| 31 | \( 1 + (0.770 + 0.280i)T + (23.7 + 19.9i)T^{2} \) |
| 37 | \( 1 + (1.30 + 2.25i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (1.41 + 8.02i)T + (-38.5 + 14.0i)T^{2} \) |
| 43 | \( 1 + (3.31 + 2.78i)T + (7.46 + 42.3i)T^{2} \) |
| 47 | \( 1 + (4.98 - 1.81i)T + (36.0 - 30.2i)T^{2} \) |
| 53 | \( 1 + 10.4T + 53T^{2} \) |
| 59 | \( 1 + (-2.30 + 1.93i)T + (10.2 - 58.1i)T^{2} \) |
| 61 | \( 1 + (2.70 - 0.986i)T + (46.7 - 39.2i)T^{2} \) |
| 67 | \( 1 + (1.75 + 9.93i)T + (-62.9 + 22.9i)T^{2} \) |
| 71 | \( 1 + (-0.0447 - 0.0774i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (-2.66 + 4.60i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (0.829 - 4.70i)T + (-74.2 - 27.0i)T^{2} \) |
| 83 | \( 1 + (1.39 - 7.91i)T + (-77.9 - 28.3i)T^{2} \) |
| 89 | \( 1 + (3.35 - 5.80i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-4.20 - 3.52i)T + (16.8 + 95.5i)T^{2} \) |
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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
−10.93794203594350669466113455090, −10.37755571150865199042863166862, −9.303270258765146453372557911598, −8.016254458350267300036272582220, −7.895027697547777406196073103532, −6.79003341325370978859956222286, −5.29192991325195808594680263486, −3.99111717843475740879926204415, −3.18687972796385590946623696280, −1.62414877588048692117009808849,
1.55225038835295249907134847453, 2.99934454109806908446165053933, 4.30806003194132277793147047206, 5.01057996143284601262276357132, 6.70788530508507303023905078571, 7.71802520363666977933936104364, 8.390257039474713246292222580307, 9.200652564690171790464053015554, 9.961901472710852498782794949899, 11.46628649835015854475505865824
