| L(s) = 1 | + (1.15 + 0.811i)2-s + (−2.96 + 0.428i)3-s + (0.684 + 1.87i)4-s + (0.872 − 4.92i)5-s + (−3.78 − 1.91i)6-s + (−8.20 + 3.82i)7-s + (−0.732 + 2.73i)8-s + (8.63 − 2.54i)9-s + (5.00 − 4.99i)10-s + (13.1 − 11.0i)11-s + (−2.83 − 5.28i)12-s + (4.74 − 3.32i)13-s + (−12.6 − 2.22i)14-s + (−0.482 + 14.9i)15-s + (−3.06 + 2.57i)16-s + (−6.50 − 24.2i)17-s + ⋯ |
| L(s) = 1 | + (0.579 + 0.405i)2-s + (−0.989 + 0.142i)3-s + (0.171 + 0.469i)4-s + (0.174 − 0.984i)5-s + (−0.631 − 0.318i)6-s + (−1.17 + 0.546i)7-s + (−0.0915 + 0.341i)8-s + (0.959 − 0.282i)9-s + (0.500 − 0.499i)10-s + (1.19 − 1.00i)11-s + (−0.236 − 0.440i)12-s + (0.364 − 0.255i)13-s + (−0.900 − 0.158i)14-s + (−0.0321 + 0.999i)15-s + (−0.191 + 0.160i)16-s + (−0.382 − 1.42i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 270 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.779 + 0.626i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 270 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.779 + 0.626i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.33886 - 0.471702i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.33886 - 0.471702i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.15 - 0.811i)T \) |
| 3 | \( 1 + (2.96 - 0.428i)T \) |
| 5 | \( 1 + (-0.872 + 4.92i)T \) |
| good | 7 | \( 1 + (8.20 - 3.82i)T + (31.4 - 37.5i)T^{2} \) |
| 11 | \( 1 + (-13.1 + 11.0i)T + (21.0 - 119. i)T^{2} \) |
| 13 | \( 1 + (-4.74 + 3.32i)T + (57.8 - 158. i)T^{2} \) |
| 17 | \( 1 + (6.50 + 24.2i)T + (-250. + 144.5i)T^{2} \) |
| 19 | \( 1 + (-21.4 + 12.3i)T + (180.5 - 312. i)T^{2} \) |
| 23 | \( 1 + (-6.62 - 3.09i)T + (340. + 405. i)T^{2} \) |
| 29 | \( 1 + (-32.5 + 5.74i)T + (790. - 287. i)T^{2} \) |
| 31 | \( 1 + (23.9 - 8.70i)T + (736. - 617. i)T^{2} \) |
| 37 | \( 1 + (9.99 + 37.2i)T + (-1.18e3 + 684.5i)T^{2} \) |
| 41 | \( 1 + (-3.58 + 20.3i)T + (-1.57e3 - 574. i)T^{2} \) |
| 43 | \( 1 + (-80.3 - 7.02i)T + (1.82e3 + 321. i)T^{2} \) |
| 47 | \( 1 + (26.2 - 12.2i)T + (1.41e3 - 1.69e3i)T^{2} \) |
| 53 | \( 1 + (-5.42 - 5.42i)T + 2.80e3iT^{2} \) |
| 59 | \( 1 + (61.5 - 73.4i)T + (-604. - 3.42e3i)T^{2} \) |
| 61 | \( 1 + (100. + 36.4i)T + (2.85e3 + 2.39e3i)T^{2} \) |
| 67 | \( 1 + (36.0 + 51.5i)T + (-1.53e3 + 4.21e3i)T^{2} \) |
| 71 | \( 1 + (25.0 - 43.3i)T + (-2.52e3 - 4.36e3i)T^{2} \) |
| 73 | \( 1 + (-12.3 + 45.9i)T + (-4.61e3 - 2.66e3i)T^{2} \) |
| 79 | \( 1 + (-14.3 + 2.53i)T + (5.86e3 - 2.13e3i)T^{2} \) |
| 83 | \( 1 + (36.8 - 52.6i)T + (-2.35e3 - 6.47e3i)T^{2} \) |
| 89 | \( 1 + (-141. + 81.9i)T + (3.96e3 - 6.85e3i)T^{2} \) |
| 97 | \( 1 + (13.1 - 150. i)T + (-9.26e3 - 1.63e3i)T^{2} \) |
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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
−11.90294322495666475122192733444, −10.94472845385070671234192932526, −9.289909495207539853154156886175, −9.129836296772569308358907498255, −7.31548176347605743620448951979, −6.24591019302846743924427921431, −5.63263535989289926881744936276, −4.56712753646966500758925367745, −3.24055359829581566255247679162, −0.73515243484264959574035786570,
1.55170615755679627853752061134, 3.42897657790917412083203782112, 4.35976443394343789540376646524, 6.04395470437363442401329947028, 6.52681603115750451840927660317, 7.34768010600148313531054954219, 9.470392412294261698874421291021, 10.18401902176656953915767914006, 10.87600407854744911362632450858, 11.88565314423687250125918469934
