| L(s) = 1 | + (−0.869 + 2.58i)2-s + (2.78 + 1.12i)3-s + (−2.71 − 2.06i)4-s + (3.05 + 1.21i)5-s + (−5.32 + 6.19i)6-s + (9.38 − 4.34i)7-s + (−1.32 + 0.896i)8-s + (6.46 + 6.26i)9-s + (−5.79 + 6.82i)10-s + (1.49 + 0.415i)11-s + (−5.22 − 8.80i)12-s + (0.591 − 1.11i)13-s + (3.04 + 28.0i)14-s + (7.12 + 6.82i)15-s + (−4.81 − 17.3i)16-s + (−1.27 + 2.75i)17-s + ⋯ |
| L(s) = 1 | + (−0.434 + 1.29i)2-s + (0.926 + 0.375i)3-s + (−0.679 − 0.516i)4-s + (0.611 + 0.243i)5-s + (−0.887 + 1.03i)6-s + (1.34 − 0.620i)7-s + (−0.165 + 0.112i)8-s + (0.718 + 0.696i)9-s + (−0.579 + 0.682i)10-s + (0.135 + 0.0377i)11-s + (−0.435 − 0.733i)12-s + (0.0454 − 0.0857i)13-s + (0.217 + 2.00i)14-s + (0.475 + 0.455i)15-s + (−0.301 − 1.08i)16-s + (−0.0750 + 0.162i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.474 - 0.880i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.474 - 0.880i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.980736 + 1.64247i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.980736 + 1.64247i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-2.78 - 1.12i)T \) |
| 59 | \( 1 + (-58.9 + 3.16i)T \) |
| good | 2 | \( 1 + (0.869 - 2.58i)T + (-3.18 - 2.42i)T^{2} \) |
| 5 | \( 1 + (-3.05 - 1.21i)T + (18.1 + 17.1i)T^{2} \) |
| 7 | \( 1 + (-9.38 + 4.34i)T + (31.7 - 37.3i)T^{2} \) |
| 11 | \( 1 + (-1.49 - 0.415i)T + (103. + 62.3i)T^{2} \) |
| 13 | \( 1 + (-0.591 + 1.11i)T + (-94.8 - 139. i)T^{2} \) |
| 17 | \( 1 + (1.27 - 2.75i)T + (-187. - 220. i)T^{2} \) |
| 19 | \( 1 + (8.93 - 1.96i)T + (327. - 151. i)T^{2} \) |
| 23 | \( 1 + (11.6 + 1.90i)T + (501. + 168. i)T^{2} \) |
| 29 | \( 1 + (15.6 + 46.3i)T + (-669. + 508. i)T^{2} \) |
| 31 | \( 1 + (28.5 + 6.28i)T + (872. + 403. i)T^{2} \) |
| 37 | \( 1 + (20.1 - 29.7i)T + (-506. - 1.27e3i)T^{2} \) |
| 41 | \( 1 + (47.6 - 7.81i)T + (1.59e3 - 536. i)T^{2} \) |
| 43 | \( 1 + (-13.6 - 49.1i)T + (-1.58e3 + 953. i)T^{2} \) |
| 47 | \( 1 + (-40.7 + 16.2i)T + (1.60e3 - 1.51e3i)T^{2} \) |
| 53 | \( 1 + (-55.3 + 47.0i)T + (454. - 2.77e3i)T^{2} \) |
| 61 | \( 1 + (-49.7 - 16.7i)T + (2.96e3 + 2.25e3i)T^{2} \) |
| 67 | \( 1 + (39.0 + 57.6i)T + (-1.66e3 + 4.17e3i)T^{2} \) |
| 71 | \( 1 + (-28.4 + 11.3i)T + (3.65e3 - 3.46e3i)T^{2} \) |
| 73 | \( 1 + (-61.5 + 6.69i)T + (5.20e3 - 1.14e3i)T^{2} \) |
| 79 | \( 1 + (-7.60 + 4.57i)T + (2.92e3 - 5.51e3i)T^{2} \) |
| 83 | \( 1 + (138. - 7.52i)T + (6.84e3 - 744. i)T^{2} \) |
| 89 | \( 1 + (-35.6 - 105. i)T + (-6.30e3 + 4.79e3i)T^{2} \) |
| 97 | \( 1 + (2.06 + 0.224i)T + (9.18e3 + 2.02e3i)T^{2} \) |
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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
−13.38591438440832966924438309727, −11.64177624045632156569746160613, −10.42899191555254734491686544987, −9.518067422711254446982551711411, −8.341036608946346506141787529865, −7.87645327276005209605376239097, −6.76557810126516824605976203635, −5.42745631055810908613244723688, −4.13821634943258923243822483448, −2.11022591382332873814372018334,
1.54560666507130969634019955160, 2.26478345225060129872092894571, 3.81896595572844398430871299252, 5.50407794760448836872784125304, 7.20791126019904930348116856083, 8.675855273984213171629256812400, 8.958457227510986811626374331124, 10.14701961811693040031518952449, 11.15725509307544013564518060646, 12.10412010816239168302205648158
