L(s) = 1 | + (0.5 − 0.866i)2-s + (−0.499 − 0.866i)4-s + (−2.63 − 0.189i)7-s − 0.999·8-s + (−3.44 + 1.99i)11-s − 0.0681·13-s + (−1.48 + 2.19i)14-s + (−0.5 + 0.866i)16-s + (6.34 − 3.66i)17-s + (1.76 + 1.01i)19-s + 3.98i·22-s + (−1.86 + 3.23i)23-s + (−0.0340 + 0.0590i)26-s + (1.15 + 2.38i)28-s − 0.898i·29-s + ⋯ |
L(s) = 1 | + (0.353 − 0.612i)2-s + (−0.249 − 0.433i)4-s + (−0.997 − 0.0716i)7-s − 0.353·8-s + (−1.03 + 0.600i)11-s − 0.0189·13-s + (−0.396 + 0.585i)14-s + (−0.125 + 0.216i)16-s + (1.53 − 0.888i)17-s + (0.404 + 0.233i)19-s + 0.848i·22-s + (−0.389 + 0.673i)23-s + (−0.00668 + 0.0115i)26-s + (0.218 + 0.449i)28-s − 0.166i·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.973 + 0.228i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.973 + 0.228i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.548207821\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.548207821\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.5 + 0.866i)T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (2.63 + 0.189i)T \) |
good | 11 | \( 1 + (3.44 - 1.99i)T + (5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + 0.0681T + 13T^{2} \) |
| 17 | \( 1 + (-6.34 + 3.66i)T + (8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-1.76 - 1.01i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (1.86 - 3.23i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + 0.898iT - 29T^{2} \) |
| 31 | \( 1 + (4.18 - 2.41i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-3.52 - 2.03i)T + (18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + 1.68T + 41T^{2} \) |
| 43 | \( 1 - 0.964iT - 43T^{2} \) |
| 47 | \( 1 + (-1.43 - 0.830i)T + (23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-6.61 - 11.4i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (5.32 + 9.21i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-6.51 - 3.76i)T + (30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-9.23 + 5.33i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 - 9.93iT - 71T^{2} \) |
| 73 | \( 1 + (-5.82 - 10.0i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-8.77 + 15.1i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + 14.3iT - 83T^{2} \) |
| 89 | \( 1 + (0.913 - 1.58i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 - 17.1T + 97T^{2} \) |
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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
−8.857177236733062338675585507863, −7.68197147205897028964593719879, −7.34357660792887236478359852870, −6.20106563345206112372022858181, −5.48119997035610861949422629576, −4.85121668838135435060755659605, −3.68258414495251778608112729939, −3.10720839661079125702849744909, −2.21349939972424243821909203790, −0.853752704663457373343975829840,
0.56864169516755822384818064907, 2.35398186055841636489501675538, 3.32630804528499381012007099660, 3.86932681948378655353551182120, 5.15418367884612945975936676618, 5.68591645599612183653716517439, 6.33510021538386171238508603824, 7.19856797753859328436404183672, 7.929074435414858120000880577690, 8.485965031163483589879235062818