| L(s) = 1 | + (0.173 + 0.984i)2-s + (0.517 + 1.65i)3-s + (−0.939 + 0.342i)4-s + (−0.258 + 0.710i)5-s + (−1.53 + 0.797i)6-s + (0.777 − 1.34i)7-s + (−0.5 − 0.866i)8-s + (−2.46 + 1.71i)9-s + (−0.744 − 0.131i)10-s + (0.832 − 0.480i)11-s + (−1.05 − 1.37i)12-s + (0.416 − 0.496i)13-s + (1.46 + 0.532i)14-s + (−1.30 − 0.0594i)15-s + (0.766 − 0.642i)16-s + (6.73 − 1.18i)17-s + ⋯ |
| L(s) = 1 | + (0.122 + 0.696i)2-s + (0.298 + 0.954i)3-s + (−0.469 + 0.171i)4-s + (−0.115 + 0.317i)5-s + (−0.627 + 0.325i)6-s + (0.294 − 0.509i)7-s + (−0.176 − 0.306i)8-s + (−0.821 + 0.570i)9-s + (−0.235 − 0.0415i)10-s + (0.250 − 0.144i)11-s + (−0.303 − 0.397i)12-s + (0.115 − 0.137i)13-s + (0.390 + 0.142i)14-s + (−0.337 − 0.0153i)15-s + (0.191 − 0.160i)16-s + (1.63 − 0.287i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 114 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.184 - 0.982i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 114 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.184 - 0.982i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.729462 + 0.879312i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.729462 + 0.879312i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.173 - 0.984i)T \) |
| 3 | \( 1 + (-0.517 - 1.65i)T \) |
| 19 | \( 1 + (4.14 + 1.35i)T \) |
| good | 5 | \( 1 + (0.258 - 0.710i)T + (-3.83 - 3.21i)T^{2} \) |
| 7 | \( 1 + (-0.777 + 1.34i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (-0.832 + 0.480i)T + (5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (-0.416 + 0.496i)T + (-2.25 - 12.8i)T^{2} \) |
| 17 | \( 1 + (-6.73 + 1.18i)T + (15.9 - 5.81i)T^{2} \) |
| 23 | \( 1 + (-0.400 - 1.10i)T + (-17.6 + 14.7i)T^{2} \) |
| 29 | \( 1 + (-1.39 + 7.92i)T + (-27.2 - 9.91i)T^{2} \) |
| 31 | \( 1 + (2.63 + 1.52i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + 4.12iT - 37T^{2} \) |
| 41 | \( 1 + (4.09 - 3.43i)T + (7.11 - 40.3i)T^{2} \) |
| 43 | \( 1 + (7.34 + 2.67i)T + (32.9 + 27.6i)T^{2} \) |
| 47 | \( 1 + (3.11 + 0.548i)T + (44.1 + 16.0i)T^{2} \) |
| 53 | \( 1 + (13.6 - 4.96i)T + (40.6 - 34.0i)T^{2} \) |
| 59 | \( 1 + (-2.02 - 11.4i)T + (-55.4 + 20.1i)T^{2} \) |
| 61 | \( 1 + (-10.1 + 3.70i)T + (46.7 - 39.2i)T^{2} \) |
| 67 | \( 1 + (9.19 + 1.62i)T + (62.9 + 22.9i)T^{2} \) |
| 71 | \( 1 + (-0.0322 - 0.0117i)T + (54.3 + 45.6i)T^{2} \) |
| 73 | \( 1 + (3.04 - 2.55i)T + (12.6 - 71.8i)T^{2} \) |
| 79 | \( 1 + (0.893 + 1.06i)T + (-13.7 + 77.7i)T^{2} \) |
| 83 | \( 1 + (10.4 + 6.05i)T + (41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (-4.68 - 3.92i)T + (15.4 + 87.6i)T^{2} \) |
| 97 | \( 1 + (-9.54 + 1.68i)T + (91.1 - 33.1i)T^{2} \) |
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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
−14.25313797082788886889186664879, −13.19809185373099852823073772677, −11.67083358712823674657957585571, −10.57030123123334137049116453992, −9.610172311796264846283338001059, −8.416021669883106392291555840484, −7.41357765521036412336681041048, −5.87594332519679508753761771036, −4.58817094953926010843683071487, −3.34483724372996788864475836307,
1.61950410827600855092488691341, 3.31321013956548319010295501370, 5.16396058582476973217693566769, 6.57767324312862377559979415772, 8.120530073209348129558961756331, 8.837972739532904457400059254134, 10.22804778511427163997357574909, 11.56291898639289080631777791898, 12.38834027264263289499241841151, 12.94017169370280250183319474119
