| L(s) = 1 | + (−0.0686 − 0.188i)5-s + (−1.47 − 8.38i)7-s + (−2.58 + 7.09i)11-s + (−12.5 − 10.5i)13-s + (−5.21 − 3.01i)17-s + (−0.189 − 0.328i)19-s + (−27.6 − 4.88i)23-s + (19.1 − 16.0i)25-s + (−26.6 − 31.7i)29-s + (2.35 − 13.3i)31-s + (−1.48 + 0.854i)35-s + (−2.26 + 3.91i)37-s + (49.3 − 58.7i)41-s + (1.63 + 0.596i)43-s + (−75.3 + 13.2i)47-s + ⋯ |
| L(s) = 1 | + (−0.0137 − 0.0377i)5-s + (−0.211 − 1.19i)7-s + (−0.234 + 0.645i)11-s + (−0.966 − 0.810i)13-s + (−0.306 − 0.177i)17-s + (−0.00999 − 0.0173i)19-s + (−1.20 − 0.212i)23-s + (0.764 − 0.641i)25-s + (−0.918 − 1.09i)29-s + (0.0760 − 0.431i)31-s + (−0.0423 + 0.0244i)35-s + (−0.0611 + 0.105i)37-s + (1.20 − 1.43i)41-s + (0.0381 + 0.0138i)43-s + (−1.60 + 0.282i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.623 + 0.781i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.623 + 0.781i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.383232 - 0.796251i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.383232 - 0.796251i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + (0.0686 + 0.188i)T + (-19.1 + 16.0i)T^{2} \) |
| 7 | \( 1 + (1.47 + 8.38i)T + (-46.0 + 16.7i)T^{2} \) |
| 11 | \( 1 + (2.58 - 7.09i)T + (-92.6 - 77.7i)T^{2} \) |
| 13 | \( 1 + (12.5 + 10.5i)T + (29.3 + 166. i)T^{2} \) |
| 17 | \( 1 + (5.21 + 3.01i)T + (144.5 + 250. i)T^{2} \) |
| 19 | \( 1 + (0.189 + 0.328i)T + (-180.5 + 312. i)T^{2} \) |
| 23 | \( 1 + (27.6 + 4.88i)T + (497. + 180. i)T^{2} \) |
| 29 | \( 1 + (26.6 + 31.7i)T + (-146. + 828. i)T^{2} \) |
| 31 | \( 1 + (-2.35 + 13.3i)T + (-903. - 328. i)T^{2} \) |
| 37 | \( 1 + (2.26 - 3.91i)T + (-684.5 - 1.18e3i)T^{2} \) |
| 41 | \( 1 + (-49.3 + 58.7i)T + (-291. - 1.65e3i)T^{2} \) |
| 43 | \( 1 + (-1.63 - 0.596i)T + (1.41e3 + 1.18e3i)T^{2} \) |
| 47 | \( 1 + (75.3 - 13.2i)T + (2.07e3 - 755. i)T^{2} \) |
| 53 | \( 1 - 85.8iT - 2.80e3T^{2} \) |
| 59 | \( 1 + (6.23 + 17.1i)T + (-2.66e3 + 2.23e3i)T^{2} \) |
| 61 | \( 1 + (6.51 + 36.9i)T + (-3.49e3 + 1.27e3i)T^{2} \) |
| 67 | \( 1 + (53.9 + 45.2i)T + (779. + 4.42e3i)T^{2} \) |
| 71 | \( 1 + (-38.9 - 22.4i)T + (2.52e3 + 4.36e3i)T^{2} \) |
| 73 | \( 1 + (-51.2 - 88.7i)T + (-2.66e3 + 4.61e3i)T^{2} \) |
| 79 | \( 1 + (-64.7 + 54.2i)T + (1.08e3 - 6.14e3i)T^{2} \) |
| 83 | \( 1 + (-44.2 - 52.6i)T + (-1.19e3 + 6.78e3i)T^{2} \) |
| 89 | \( 1 + (-119. + 68.8i)T + (3.96e3 - 6.85e3i)T^{2} \) |
| 97 | \( 1 + (-112. - 41.0i)T + (7.20e3 + 6.04e3i)T^{2} \) |
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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
−10.86491145403470741162241482608, −10.17032262767511254188748857990, −9.423254077492167110625440076864, −7.928350085196485363433662503608, −7.38087595755517077492514984360, −6.23465128850659640069605961951, −4.87462342811258647117875166579, −3.91223529225584193545816309354, −2.37178649808907630735839618126, −0.39344067946566041431200835979,
2.02518023284983352700007433475, 3.27603208628516577982648166921, 4.82541043710018383494165652327, 5.81464562515130004320603453777, 6.82687645960635190291871869177, 8.039564115349397750464358132663, 9.006662634445300959188531579024, 9.683891744388738566943858493905, 10.92701796420033078879018789225, 11.76414029704774274954314564175
