| L(s) = 1 | + (0.893 + 0.448i)2-s + (1.02 − 1.39i)3-s + (0.597 + 0.802i)4-s + (0.664 + 2.21i)5-s + (1.54 − 0.787i)6-s + (−0.590 − 1.36i)7-s + (0.173 + 0.984i)8-s + (−0.897 − 2.86i)9-s + (−0.402 + 2.28i)10-s + (−4.28 + 1.01i)11-s + (1.73 − 0.0112i)12-s + (0.407 + 0.267i)13-s + (0.0866 − 1.48i)14-s + (3.77 + 1.34i)15-s + (−0.286 + 0.957i)16-s + (6.18 − 2.24i)17-s + ⋯ |
| L(s) = 1 | + (0.631 + 0.317i)2-s + (0.591 − 0.805i)3-s + (0.298 + 0.401i)4-s + (0.297 + 0.992i)5-s + (0.629 − 0.321i)6-s + (−0.223 − 0.517i)7-s + (0.0613 + 0.348i)8-s + (−0.299 − 0.954i)9-s + (−0.127 + 0.721i)10-s + (−1.29 + 0.305i)11-s + (0.499 − 0.00325i)12-s + (0.113 + 0.0743i)13-s + (0.0231 − 0.397i)14-s + (0.975 + 0.347i)15-s + (−0.0717 + 0.239i)16-s + (1.49 − 0.545i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.994 - 0.103i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.994 - 0.103i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.78856 + 0.0927608i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.78856 + 0.0927608i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.893 - 0.448i)T \) |
| 3 | \( 1 + (-1.02 + 1.39i)T \) |
| good | 5 | \( 1 + (-0.664 - 2.21i)T + (-4.17 + 2.74i)T^{2} \) |
| 7 | \( 1 + (0.590 + 1.36i)T + (-4.80 + 5.09i)T^{2} \) |
| 11 | \( 1 + (4.28 - 1.01i)T + (9.82 - 4.93i)T^{2} \) |
| 13 | \( 1 + (-0.407 - 0.267i)T + (5.14 + 11.9i)T^{2} \) |
| 17 | \( 1 + (-6.18 + 2.24i)T + (13.0 - 10.9i)T^{2} \) |
| 19 | \( 1 + (6.76 + 2.46i)T + (14.5 + 12.2i)T^{2} \) |
| 23 | \( 1 + (1.15 - 2.67i)T + (-15.7 - 16.7i)T^{2} \) |
| 29 | \( 1 + (-0.382 - 6.57i)T + (-28.8 + 3.36i)T^{2} \) |
| 31 | \( 1 + (1.78 - 0.208i)T + (30.1 - 7.14i)T^{2} \) |
| 37 | \( 1 + (3.51 + 2.94i)T + (6.42 + 36.4i)T^{2} \) |
| 41 | \( 1 + (7.03 - 3.53i)T + (24.4 - 32.8i)T^{2} \) |
| 43 | \( 1 + (0.304 + 0.322i)T + (-2.50 + 42.9i)T^{2} \) |
| 47 | \( 1 + (-7.76 - 0.907i)T + (45.7 + 10.8i)T^{2} \) |
| 53 | \( 1 + (-0.984 + 1.70i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-12.3 - 2.93i)T + (52.7 + 26.4i)T^{2} \) |
| 61 | \( 1 + (-4.60 + 6.18i)T + (-17.4 - 58.4i)T^{2} \) |
| 67 | \( 1 + (0.221 - 3.81i)T + (-66.5 - 7.77i)T^{2} \) |
| 71 | \( 1 + (-1.34 + 7.62i)T + (-66.7 - 24.2i)T^{2} \) |
| 73 | \( 1 + (-1.74 - 9.92i)T + (-68.5 + 24.9i)T^{2} \) |
| 79 | \( 1 + (-15.0 - 7.53i)T + (47.1 + 63.3i)T^{2} \) |
| 83 | \( 1 + (4.65 + 2.33i)T + (49.5 + 66.5i)T^{2} \) |
| 89 | \( 1 + (0.537 + 3.05i)T + (-83.6 + 30.4i)T^{2} \) |
| 97 | \( 1 + (-0.403 + 1.34i)T + (-81.0 - 53.3i)T^{2} \) |
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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
−13.07186155191622167666262719562, −12.27505588443795987814927401901, −10.90556515491489133933527325532, −10.00037476832761900652242490724, −8.425358957201750197978823064391, −7.32739046784730016265054868989, −6.75839044270831320023248518310, −5.42469448203298785662851439207, −3.52810737042808426261926807065, −2.42906702094086306517380386254,
2.34727298127201137508406041905, 3.80604515676556739339171093990, 5.09538102116409564961534786990, 5.85703196789303481507857350285, 8.022170273211190003221548709657, 8.760107063675585086702866037179, 10.03937825957337840587103295367, 10.60431272758753284434003098469, 12.14236989952110235051505957214, 12.90341313148274203515281728004
