| L(s) = 1 | + (0.173 − 0.984i)2-s + (−0.939 − 0.342i)4-s + (0.673 − 0.565i)5-s + (3.31 − 1.20i)7-s + (−0.5 + 0.866i)8-s + (−0.439 − 0.761i)10-s + (−2.73 − 2.29i)11-s + (−0.641 − 3.63i)13-s + (−0.613 − 3.47i)14-s + (0.766 + 0.642i)16-s + (3.12 + 5.41i)17-s + (−2.08 + 3.61i)19-s + (−0.826 + 0.300i)20-s + (−2.73 + 2.29i)22-s + (1.93 + 0.705i)23-s + ⋯ |
| L(s) = 1 | + (0.122 − 0.696i)2-s + (−0.469 − 0.171i)4-s + (0.301 − 0.252i)5-s + (1.25 − 0.456i)7-s + (−0.176 + 0.306i)8-s + (−0.139 − 0.240i)10-s + (−0.825 − 0.692i)11-s + (−0.177 − 1.00i)13-s + (−0.163 − 0.929i)14-s + (0.191 + 0.160i)16-s + (0.757 + 1.31i)17-s + (−0.478 + 0.828i)19-s + (−0.184 + 0.0672i)20-s + (−0.583 + 0.489i)22-s + (0.404 + 0.147i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.286 + 0.957i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.286 + 0.957i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.01710 - 0.757204i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.01710 - 0.757204i\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| good | 5 | \( 1 + (-0.673 + 0.565i)T + (0.868 - 4.92i)T^{2} \) |
| 7 | \( 1 + (-3.31 + 1.20i)T + (5.36 - 4.49i)T^{2} \) |
| 11 | \( 1 + (2.73 + 2.29i)T + (1.91 + 10.8i)T^{2} \) |
| 13 | \( 1 + (0.641 + 3.63i)T + (-12.2 + 4.44i)T^{2} \) |
| 17 | \( 1 + (-3.12 - 5.41i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (2.08 - 3.61i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-1.93 - 0.705i)T + (17.6 + 14.7i)T^{2} \) |
| 29 | \( 1 + (0.0282 - 0.160i)T + (-27.2 - 9.91i)T^{2} \) |
| 31 | \( 1 + (1.53 + 0.560i)T + (23.7 + 19.9i)T^{2} \) |
| 37 | \( 1 + (-3.85 - 6.68i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-1.33 - 7.58i)T + (-38.5 + 14.0i)T^{2} \) |
| 43 | \( 1 + (8.29 + 6.95i)T + (7.46 + 42.3i)T^{2} \) |
| 47 | \( 1 + (6.02 - 2.19i)T + (36.0 - 30.2i)T^{2} \) |
| 53 | \( 1 + 0.716T + 53T^{2} \) |
| 59 | \( 1 + (-5.35 + 4.49i)T + (10.2 - 58.1i)T^{2} \) |
| 61 | \( 1 + (-1.19 + 0.433i)T + (46.7 - 39.2i)T^{2} \) |
| 67 | \( 1 + (-0.624 - 3.54i)T + (-62.9 + 22.9i)T^{2} \) |
| 71 | \( 1 + (6.76 + 11.7i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (-1.16 + 2.01i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (1.14 - 6.51i)T + (-74.2 - 27.0i)T^{2} \) |
| 83 | \( 1 + (-0.773 + 4.38i)T + (-77.9 - 28.3i)T^{2} \) |
| 89 | \( 1 + (-4.62 + 8.00i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (8.64 + 7.25i)T + (16.8 + 95.5i)T^{2} \) |
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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
−12.78869448419297052645084960873, −11.54169123422334084035025581226, −10.68020066295033265761070240024, −9.996472851925545798804756625000, −8.379511157951755332154784417060, −7.87253428053034741798648894770, −5.82352390325018480979772847481, −4.89588655815877525823228777769, −3.38170396163347249519022023722, −1.52873340963037104889923350361,
2.36596287690234497643672623157, 4.57784777941659498859603044667, 5.34826355233230935211879246032, 6.83818532377423263228962317260, 7.74432069757878621046640440760, 8.847120417109826465186501354747, 9.885773102355795428751047799644, 11.18250721533402859762324020256, 12.08720653479349764700901999888, 13.26322270181277319969556239631
