| L(s) = 1 | + (−1.39 + 0.245i)2-s + (1.87 − 0.684i)4-s + (−0.379 + 0.451i)5-s + (2.96 + 1.07i)7-s + (−2.44 + 1.41i)8-s + (0.416 − 0.722i)10-s + (6.03 + 7.19i)11-s + (2.11 − 12.0i)13-s + (−4.39 − 0.774i)14-s + (3.06 − 2.57i)16-s + (24.5 + 14.1i)17-s + (11.2 + 19.4i)19-s + (−0.403 + 1.10i)20-s + (−10.1 − 8.53i)22-s + (−3.44 − 9.46i)23-s + ⋯ |
| L(s) = 1 | + (−0.696 + 0.122i)2-s + (0.469 − 0.171i)4-s + (−0.0758 + 0.0903i)5-s + (0.423 + 0.154i)7-s + (−0.306 + 0.176i)8-s + (0.0416 − 0.0722i)10-s + (0.548 + 0.653i)11-s + (0.163 − 0.924i)13-s + (−0.313 − 0.0553i)14-s + (0.191 − 0.160i)16-s + (1.44 + 0.832i)17-s + (0.589 + 1.02i)19-s + (−0.0201 + 0.0554i)20-s + (−0.462 − 0.387i)22-s + (−0.149 − 0.411i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.870 - 0.491i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.870 - 0.491i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.11722 + 0.293780i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.11722 + 0.293780i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.39 - 0.245i)T \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + (0.379 - 0.451i)T + (-4.34 - 24.6i)T^{2} \) |
| 7 | \( 1 + (-2.96 - 1.07i)T + (37.5 + 31.4i)T^{2} \) |
| 11 | \( 1 + (-6.03 - 7.19i)T + (-21.0 + 119. i)T^{2} \) |
| 13 | \( 1 + (-2.11 + 12.0i)T + (-158. - 57.8i)T^{2} \) |
| 17 | \( 1 + (-24.5 - 14.1i)T + (144.5 + 250. i)T^{2} \) |
| 19 | \( 1 + (-11.2 - 19.4i)T + (-180.5 + 312. i)T^{2} \) |
| 23 | \( 1 + (3.44 + 9.46i)T + (-405. + 340. i)T^{2} \) |
| 29 | \( 1 + (-23.6 + 4.17i)T + (790. - 287. i)T^{2} \) |
| 31 | \( 1 + (-42.7 + 15.5i)T + (736. - 617. i)T^{2} \) |
| 37 | \( 1 + (15.7 - 27.2i)T + (-684.5 - 1.18e3i)T^{2} \) |
| 41 | \( 1 + (69.7 + 12.3i)T + (1.57e3 + 574. i)T^{2} \) |
| 43 | \( 1 + (11.1 - 9.36i)T + (321. - 1.82e3i)T^{2} \) |
| 47 | \( 1 + (-18.9 + 51.9i)T + (-1.69e3 - 1.41e3i)T^{2} \) |
| 53 | \( 1 + 25.4iT - 2.80e3T^{2} \) |
| 59 | \( 1 + (18.4 - 21.9i)T + (-604. - 3.42e3i)T^{2} \) |
| 61 | \( 1 + (106. + 38.6i)T + (2.85e3 + 2.39e3i)T^{2} \) |
| 67 | \( 1 + (-8.68 + 49.2i)T + (-4.21e3 - 1.53e3i)T^{2} \) |
| 71 | \( 1 + (-7.59 - 4.38i)T + (2.52e3 + 4.36e3i)T^{2} \) |
| 73 | \( 1 + (11.7 + 20.3i)T + (-2.66e3 + 4.61e3i)T^{2} \) |
| 79 | \( 1 + (-23.0 - 130. i)T + (-5.86e3 + 2.13e3i)T^{2} \) |
| 83 | \( 1 + (-66.1 + 11.6i)T + (6.47e3 - 2.35e3i)T^{2} \) |
| 89 | \( 1 + (62.9 - 36.3i)T + (3.96e3 - 6.85e3i)T^{2} \) |
| 97 | \( 1 + (-120. + 101. i)T + (1.63e3 - 9.26e3i)T^{2} \) |
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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
−12.37716849687658426572367770069, −11.77735740748105297736734066603, −10.38962681500636220481093774467, −9.856220170222102962322595386251, −8.404369278359103055797917807215, −7.76561867679584021205385814958, −6.43319544050412956961538482228, −5.21253951817139130528920188985, −3.39879565841197487206661970308, −1.44563432808166703110109104759,
1.13052143237212137837778095589, 3.09461557947888353551502192506, 4.76739768513786597145768875167, 6.33322024447456830867882572889, 7.44827807648898965316159688989, 8.532977212563846929116203750217, 9.429576993421748507053287317222, 10.48267459819733359298786956811, 11.64204399862586590642387806537, 12.06942735451190772374523157570
