| L(s) = 1 | + (−0.322 + 0.955i)2-s + (−0.589 − 2.94i)3-s + (2.37 + 1.80i)4-s + (5.46 + 2.17i)5-s + (3.00 + 0.383i)6-s + (3.90 − 1.80i)7-s + (−5.82 + 3.95i)8-s + (−8.30 + 3.46i)9-s + (−3.84 + 4.52i)10-s + (0.609 + 0.169i)11-s + (3.90 − 8.04i)12-s + (5.83 − 11.0i)13-s + (0.469 + 4.31i)14-s + (3.18 − 17.3i)15-s + (1.29 + 4.64i)16-s + (−10.1 + 22.0i)17-s + ⋯ |
| L(s) = 1 | + (−0.161 + 0.477i)2-s + (−0.196 − 0.980i)3-s + (0.593 + 0.451i)4-s + (1.09 + 0.435i)5-s + (0.500 + 0.0639i)6-s + (0.558 − 0.258i)7-s + (−0.728 + 0.494i)8-s + (−0.922 + 0.385i)9-s + (−0.384 + 0.452i)10-s + (0.0553 + 0.0153i)11-s + (0.325 − 0.670i)12-s + (0.448 − 0.846i)13-s + (0.0335 + 0.308i)14-s + (0.212 − 1.15i)15-s + (0.0806 + 0.290i)16-s + (−0.599 + 1.29i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.943 - 0.331i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.943 - 0.331i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.76854 + 0.301594i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.76854 + 0.301594i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (0.589 + 2.94i)T \) |
| 59 | \( 1 + (45.6 + 37.3i)T \) |
| good | 2 | \( 1 + (0.322 - 0.955i)T + (-3.18 - 2.42i)T^{2} \) |
| 5 | \( 1 + (-5.46 - 2.17i)T + (18.1 + 17.1i)T^{2} \) |
| 7 | \( 1 + (-3.90 + 1.80i)T + (31.7 - 37.3i)T^{2} \) |
| 11 | \( 1 + (-0.609 - 0.169i)T + (103. + 62.3i)T^{2} \) |
| 13 | \( 1 + (-5.83 + 11.0i)T + (-94.8 - 139. i)T^{2} \) |
| 17 | \( 1 + (10.1 - 22.0i)T + (-187. - 220. i)T^{2} \) |
| 19 | \( 1 + (-35.3 + 7.77i)T + (327. - 151. i)T^{2} \) |
| 23 | \( 1 + (-27.2 - 4.46i)T + (501. + 168. i)T^{2} \) |
| 29 | \( 1 + (8.57 + 25.4i)T + (-669. + 508. i)T^{2} \) |
| 31 | \( 1 + (47.8 + 10.5i)T + (872. + 403. i)T^{2} \) |
| 37 | \( 1 + (-25.6 + 37.7i)T + (-506. - 1.27e3i)T^{2} \) |
| 41 | \( 1 + (20.9 - 3.43i)T + (1.59e3 - 536. i)T^{2} \) |
| 43 | \( 1 + (-12.4 - 44.6i)T + (-1.58e3 + 953. i)T^{2} \) |
| 47 | \( 1 + (44.4 - 17.7i)T + (1.60e3 - 1.51e3i)T^{2} \) |
| 53 | \( 1 + (-36.6 + 31.1i)T + (454. - 2.77e3i)T^{2} \) |
| 61 | \( 1 + (98.0 + 33.0i)T + (2.96e3 + 2.25e3i)T^{2} \) |
| 67 | \( 1 + (-1.30 - 1.92i)T + (-1.66e3 + 4.17e3i)T^{2} \) |
| 71 | \( 1 + (62.3 - 24.8i)T + (3.65e3 - 3.46e3i)T^{2} \) |
| 73 | \( 1 + (-4.55 + 0.495i)T + (5.20e3 - 1.14e3i)T^{2} \) |
| 79 | \( 1 + (27.6 - 16.6i)T + (2.92e3 - 5.51e3i)T^{2} \) |
| 83 | \( 1 + (-4.60 + 0.249i)T + (6.84e3 - 744. i)T^{2} \) |
| 89 | \( 1 + (-2.90 - 8.63i)T + (-6.30e3 + 4.79e3i)T^{2} \) |
| 97 | \( 1 + (122. + 13.3i)T + (9.18e3 + 2.02e3i)T^{2} \) |
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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
−12.73681889013209628803063567586, −11.35880213399998682380693800649, −10.88462752139545130569164662660, −9.308559673760529294482193231975, −8.044447953149298619300744218995, −7.32855591939324898114452126191, −6.25006423499582201374278191538, −5.53524748071210440438401762146, −3.02955822023940284041418836774, −1.67464162216852216654007953540,
1.50410382751573966842658898478, 3.08016724962851297424402234432, 4.99346410689919230305140232952, 5.66799109134283366015947607349, 7.00344813414710957409297897626, 9.138326879523922468121072894503, 9.287718151533941662852735953885, 10.43736173239194276657434127838, 11.35355651811970520145798513981, 11.93107533421685191717476400679
