| L(s) = 1 | + (−1.02 + 2.82i)3-s + (1.26 − 3.47i)5-s + (0.0728 − 0.412i)7-s + (−6.91 − 5.75i)9-s + (1.69 + 4.64i)11-s + (3.65 − 3.06i)13-s + (8.52 + 7.12i)15-s + (20.4 − 11.8i)17-s + (13.5 − 23.4i)19-s + (1.09 + 0.626i)21-s + (−20.7 + 3.66i)23-s + (8.64 + 7.25i)25-s + (23.3 − 13.6i)27-s + (−2.12 + 2.53i)29-s + (−3.01 − 17.0i)31-s + ⋯ |
| L(s) = 1 | + (−0.340 + 0.940i)3-s + (0.253 − 0.695i)5-s + (0.0104 − 0.0589i)7-s + (−0.768 − 0.639i)9-s + (0.153 + 0.422i)11-s + (0.281 − 0.236i)13-s + (0.568 + 0.474i)15-s + (1.20 − 0.696i)17-s + (0.712 − 1.23i)19-s + (0.0519 + 0.0298i)21-s + (−0.902 + 0.159i)23-s + (0.345 + 0.290i)25-s + (0.862 − 0.505i)27-s + (−0.0732 + 0.0873i)29-s + (−0.0972 − 0.551i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 + 0.0521i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.998 + 0.0521i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.59269 - 0.0415304i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.59269 - 0.0415304i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (1.02 - 2.82i)T \) |
| good | 5 | \( 1 + (-1.26 + 3.47i)T + (-19.1 - 16.0i)T^{2} \) |
| 7 | \( 1 + (-0.0728 + 0.412i)T + (-46.0 - 16.7i)T^{2} \) |
| 11 | \( 1 + (-1.69 - 4.64i)T + (-92.6 + 77.7i)T^{2} \) |
| 13 | \( 1 + (-3.65 + 3.06i)T + (29.3 - 166. i)T^{2} \) |
| 17 | \( 1 + (-20.4 + 11.8i)T + (144.5 - 250. i)T^{2} \) |
| 19 | \( 1 + (-13.5 + 23.4i)T + (-180.5 - 312. i)T^{2} \) |
| 23 | \( 1 + (20.7 - 3.66i)T + (497. - 180. i)T^{2} \) |
| 29 | \( 1 + (2.12 - 2.53i)T + (-146. - 828. i)T^{2} \) |
| 31 | \( 1 + (3.01 + 17.0i)T + (-903. + 328. i)T^{2} \) |
| 37 | \( 1 + (-24.9 - 43.2i)T + (-684.5 + 1.18e3i)T^{2} \) |
| 41 | \( 1 + (-26.0 - 31.0i)T + (-291. + 1.65e3i)T^{2} \) |
| 43 | \( 1 + (-61.2 + 22.2i)T + (1.41e3 - 1.18e3i)T^{2} \) |
| 47 | \( 1 + (-26.9 - 4.75i)T + (2.07e3 + 755. i)T^{2} \) |
| 53 | \( 1 + 59.1iT - 2.80e3T^{2} \) |
| 59 | \( 1 + (-16.8 + 46.1i)T + (-2.66e3 - 2.23e3i)T^{2} \) |
| 61 | \( 1 + (-11.9 + 67.9i)T + (-3.49e3 - 1.27e3i)T^{2} \) |
| 67 | \( 1 + (56.4 - 47.3i)T + (779. - 4.42e3i)T^{2} \) |
| 71 | \( 1 + (88.6 - 51.1i)T + (2.52e3 - 4.36e3i)T^{2} \) |
| 73 | \( 1 + (3.81 - 6.60i)T + (-2.66e3 - 4.61e3i)T^{2} \) |
| 79 | \( 1 + (-99.5 - 83.5i)T + (1.08e3 + 6.14e3i)T^{2} \) |
| 83 | \( 1 + (-44.1 + 52.5i)T + (-1.19e3 - 6.78e3i)T^{2} \) |
| 89 | \( 1 + (137. + 79.5i)T + (3.96e3 + 6.85e3i)T^{2} \) |
| 97 | \( 1 + (8.07 - 2.93i)T + (7.20e3 - 6.04e3i)T^{2} \) |
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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
−10.94650843156194217686742336005, −9.761731958602317640175121953033, −9.465419880160325746111919285980, −8.384934032685379332042603794610, −7.23664456730743057834453954812, −5.87432644966996833947843307507, −5.13377073398214513205122834822, −4.20344226077818612207901420109, −2.91281887456939228687673557750, −0.875444586765681459232732709027,
1.19465764700308537793864064892, 2.56581950619015762503960066985, 3.86647316594616206201037839841, 5.75612702321066182705862951723, 6.02419703663235569704168578905, 7.30465646809378214399005196333, 7.941319430427811855163399564571, 9.052576852418822383262974917618, 10.34543545450353975038806659990, 10.86808099793607642102020295410
