| L(s) = 1 | + (−2.91 + 0.945i)2-s + (1.65 − 2.49i)3-s + (4.34 − 3.15i)4-s + (6.31 + 2.05i)5-s + (−2.46 + 8.84i)6-s + (2.47 − 1.80i)7-s + (−2.45 + 3.37i)8-s + (−3.49 − 8.29i)9-s − 20.3·10-s + (−10.9 − 1.01i)11-s + (−0.678 − 16.0i)12-s + (5.01 + 15.4i)13-s + (−5.50 + 7.58i)14-s + (15.6 − 12.3i)15-s + (−2.68 + 8.25i)16-s + (−0.766 − 0.248i)17-s + ⋯ |
| L(s) = 1 | + (−1.45 + 0.472i)2-s + (0.553 − 0.833i)3-s + (1.08 − 0.788i)4-s + (1.26 + 0.410i)5-s + (−0.411 + 1.47i)6-s + (0.353 − 0.257i)7-s + (−0.306 + 0.422i)8-s + (−0.388 − 0.921i)9-s − 2.03·10-s + (−0.995 − 0.0924i)11-s + (−0.0565 − 1.34i)12-s + (0.386 + 1.18i)13-s + (−0.393 + 0.541i)14-s + (1.04 − 0.825i)15-s + (−0.167 + 0.515i)16-s + (−0.0450 − 0.0146i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.995 + 0.0960i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.995 + 0.0960i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.708028 - 0.0340898i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.708028 - 0.0340898i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-1.65 + 2.49i)T \) |
| 11 | \( 1 + (10.9 + 1.01i)T \) |
| good | 2 | \( 1 + (2.91 - 0.945i)T + (3.23 - 2.35i)T^{2} \) |
| 5 | \( 1 + (-6.31 - 2.05i)T + (20.2 + 14.6i)T^{2} \) |
| 7 | \( 1 + (-2.47 + 1.80i)T + (15.1 - 46.6i)T^{2} \) |
| 13 | \( 1 + (-5.01 - 15.4i)T + (-136. + 99.3i)T^{2} \) |
| 17 | \( 1 + (0.766 + 0.248i)T + (233. + 169. i)T^{2} \) |
| 19 | \( 1 + (16.7 + 12.1i)T + (111. + 343. i)T^{2} \) |
| 23 | \( 1 - 27.3iT - 529T^{2} \) |
| 29 | \( 1 + (-2.22 - 3.06i)T + (-259. + 799. i)T^{2} \) |
| 31 | \( 1 + (6.42 + 19.7i)T + (-777. + 564. i)T^{2} \) |
| 37 | \( 1 + (31.1 - 22.6i)T + (423. - 1.30e3i)T^{2} \) |
| 41 | \( 1 + (7.86 - 10.8i)T + (-519. - 1.59e3i)T^{2} \) |
| 43 | \( 1 - 43.4T + 1.84e3T^{2} \) |
| 47 | \( 1 + (-11.6 + 16.0i)T + (-682. - 2.10e3i)T^{2} \) |
| 53 | \( 1 + (-16.8 + 5.46i)T + (2.27e3 - 1.65e3i)T^{2} \) |
| 59 | \( 1 + (25.5 + 35.2i)T + (-1.07e3 + 3.31e3i)T^{2} \) |
| 61 | \( 1 + (-3.29 + 10.1i)T + (-3.01e3 - 2.18e3i)T^{2} \) |
| 67 | \( 1 - 72.2T + 4.48e3T^{2} \) |
| 71 | \( 1 + (-2.44 - 0.794i)T + (4.07e3 + 2.96e3i)T^{2} \) |
| 73 | \( 1 + (-36.7 + 26.7i)T + (1.64e3 - 5.06e3i)T^{2} \) |
| 79 | \( 1 + (30.3 + 93.4i)T + (-5.04e3 + 3.66e3i)T^{2} \) |
| 83 | \( 1 + (30.3 + 9.85i)T + (5.57e3 + 4.04e3i)T^{2} \) |
| 89 | \( 1 - 18.5iT - 7.92e3T^{2} \) |
| 97 | \( 1 + (19.5 + 60.2i)T + (-7.61e3 + 5.53e3i)T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−17.05374708250955228039502034076, −15.49025025953051299906806639909, −14.05082115735995627096917930437, −13.19714301968440324342943642554, −11.06734289449833975949823636559, −9.740915177146072822255516820260, −8.706442874186940975449938001149, −7.38554747539762058044716530994, −6.25477249529844030319829444772, −1.93702728513667002522623300336,
2.33579253706577828701355707008, 5.36594415819512618964682143260, 8.110075002517620245538863551803, 8.933994234419569624739383124062, 10.21194834854105395326661820027, 10.66180925866846258097571503155, 12.81701969265400677211575233689, 14.22241321981993730416610747758, 15.68636705207099501720245643913, 16.84411212399931448815402511514
