L(s) = 1 | + (−0.698 + 0.698i)2-s + 1.02i·4-s + (0.912 − 0.244i)5-s + (2.61 − 0.407i)7-s + (−2.11 − 2.11i)8-s + (−0.466 + 0.808i)10-s + (−6.35 + 1.70i)11-s + (−3.43 − 1.08i)13-s + (−1.54 + 2.11i)14-s + 0.904·16-s − 3.73·17-s + (−0.325 + 1.21i)19-s + (0.250 + 0.933i)20-s + (3.24 − 5.62i)22-s + 0.233i·23-s + ⋯ |
L(s) = 1 | + (−0.494 + 0.494i)2-s + 0.511i·4-s + (0.407 − 0.109i)5-s + (0.988 − 0.154i)7-s + (−0.746 − 0.746i)8-s + (−0.147 + 0.255i)10-s + (−1.91 + 0.513i)11-s + (−0.953 − 0.299i)13-s + (−0.411 + 0.564i)14-s + 0.226·16-s − 0.905·17-s + (−0.0747 + 0.278i)19-s + (0.0559 + 0.208i)20-s + (0.692 − 1.19i)22-s + 0.0486i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.975 + 0.219i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.975 + 0.219i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.0474970 - 0.427592i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0474970 - 0.427592i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 + (-2.61 + 0.407i)T \) |
| 13 | \( 1 + (3.43 + 1.08i)T \) |
good | 2 | \( 1 + (0.698 - 0.698i)T - 2iT^{2} \) |
| 5 | \( 1 + (-0.912 + 0.244i)T + (4.33 - 2.5i)T^{2} \) |
| 11 | \( 1 + (6.35 - 1.70i)T + (9.52 - 5.5i)T^{2} \) |
| 17 | \( 1 + 3.73T + 17T^{2} \) |
| 19 | \( 1 + (0.325 - 1.21i)T + (-16.4 - 9.5i)T^{2} \) |
| 23 | \( 1 - 0.233iT - 23T^{2} \) |
| 29 | \( 1 + (-2.32 - 4.02i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (1.96 - 7.35i)T + (-26.8 - 15.5i)T^{2} \) |
| 37 | \( 1 + (-3.55 - 3.55i)T + 37iT^{2} \) |
| 41 | \( 1 + (2.49 - 9.29i)T + (-35.5 - 20.5i)T^{2} \) |
| 43 | \( 1 + (10.4 + 6.00i)T + (21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (-0.563 - 2.10i)T + (-40.7 + 23.5i)T^{2} \) |
| 53 | \( 1 + (2.04 + 3.53i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-5.27 + 5.27i)T - 59iT^{2} \) |
| 61 | \( 1 + (2.12 - 1.22i)T + (30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (2.63 + 9.82i)T + (-58.0 + 33.5i)T^{2} \) |
| 71 | \( 1 + (0.433 + 1.61i)T + (-61.4 + 35.5i)T^{2} \) |
| 73 | \( 1 + (7.85 + 2.10i)T + (63.2 + 36.5i)T^{2} \) |
| 79 | \( 1 + (0.942 - 1.63i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-9.95 - 9.95i)T + 83iT^{2} \) |
| 89 | \( 1 + (-2.88 + 2.88i)T - 89iT^{2} \) |
| 97 | \( 1 + (-2.41 + 0.647i)T + (84.0 - 48.5i)T^{2} \) |
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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
−10.43969698660428206781040117797, −9.857344972246420393242331500261, −8.763797900793968357089826671975, −8.023897872985876543117669835256, −7.51761859636080830743555623827, −6.62152910101595449085627183559, −5.24425335292442747369111336727, −4.72335600847743617290529590333, −3.12838989573712561673126241458, −2.01653468806632679773366332648,
0.22447587621633110328618538988, 2.10528043039621713538603933720, 2.54654232651134361293627044705, 4.53742337215321932588044325601, 5.33975543047497066115046380383, 6.06361860106226830944600424488, 7.44274113393776157266328491024, 8.225357178481370566474561047205, 9.025910358778885608249429665780, 10.01461017754258200705426162918