| 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
−10.86808099793607642102020295410, −10.34543545450353975038806659990, −9.052576852418822383262974917618, −7.941319430427811855163399564571, −7.30465646809378214399005196333, −6.02419703663235569704168578905, −5.75612702321066182705862951723, −3.86647316594616206201037839841, −2.56581950619015762503960066985, −1.19465764700308537793864064892,
0.875444586765681459232732709027, 2.91281887456939228687673557750, 4.20344226077818612207901420109, 5.13377073398214513205122834822, 5.87432644966996833947843307507, 7.23664456730743057834453954812, 8.384934032685379332042603794610, 9.465419880160325746111919285980, 9.761731958602317640175121953033, 10.94650843156194217686742336005
