L(s) = 1 | + (−0.886 − 0.322i)2-s + (−0.858 − 1.50i)3-s + (−0.850 − 0.713i)4-s + (−0.485 − 0.578i)5-s + (0.275 + 1.61i)6-s + (1.38 − 2.39i)7-s + (1.46 + 2.54i)8-s + (−1.52 + 2.58i)9-s + (0.243 + 0.669i)10-s + (2.28 − 1.31i)11-s + (−0.343 + 1.89i)12-s + (1.90 + 0.335i)13-s + (−1.99 + 1.67i)14-s + (−0.453 + 1.22i)15-s + (−0.0947 − 0.537i)16-s + (−0.220 + 0.606i)17-s + ⋯ |
L(s) = 1 | + (−0.626 − 0.228i)2-s + (−0.495 − 0.868i)3-s + (−0.425 − 0.356i)4-s + (−0.217 − 0.258i)5-s + (0.112 + 0.657i)6-s + (0.522 − 0.905i)7-s + (0.518 + 0.898i)8-s + (−0.508 + 0.860i)9-s + (0.0770 + 0.211i)10-s + (0.687 − 0.396i)11-s + (−0.0991 + 0.546i)12-s + (0.527 + 0.0930i)13-s + (−0.534 + 0.448i)14-s + (−0.117 + 0.317i)15-s + (−0.0236 − 0.134i)16-s + (−0.0535 + 0.147i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 57 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.235 + 0.971i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 57 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.235 + 0.971i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.320681 - 0.407729i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.320681 - 0.407729i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (0.858 + 1.50i)T \) |
| 19 | \( 1 + (1.94 - 3.90i)T \) |
good | 2 | \( 1 + (0.886 + 0.322i)T + (1.53 + 1.28i)T^{2} \) |
| 5 | \( 1 + (0.485 + 0.578i)T + (-0.868 + 4.92i)T^{2} \) |
| 7 | \( 1 + (-1.38 + 2.39i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (-2.28 + 1.31i)T + (5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (-1.90 - 0.335i)T + (12.2 + 4.44i)T^{2} \) |
| 17 | \( 1 + (0.220 - 0.606i)T + (-13.0 - 10.9i)T^{2} \) |
| 23 | \( 1 + (-5.61 + 6.69i)T + (-3.99 - 22.6i)T^{2} \) |
| 29 | \( 1 + (8.33 - 3.03i)T + (22.2 - 18.6i)T^{2} \) |
| 31 | \( 1 + (-2.10 - 1.21i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 - 6.09iT - 37T^{2} \) |
| 41 | \( 1 + (-0.930 - 5.27i)T + (-38.5 + 14.0i)T^{2} \) |
| 43 | \( 1 + (1.12 - 0.947i)T + (7.46 - 42.3i)T^{2} \) |
| 47 | \( 1 + (-3.48 - 9.56i)T + (-36.0 + 30.2i)T^{2} \) |
| 53 | \( 1 + (-4.16 - 3.49i)T + (9.20 + 52.1i)T^{2} \) |
| 59 | \( 1 + (7.06 + 2.57i)T + (45.1 + 37.9i)T^{2} \) |
| 61 | \( 1 + (1.37 + 1.15i)T + (10.5 + 60.0i)T^{2} \) |
| 67 | \( 1 + (3.35 + 9.20i)T + (-51.3 + 43.0i)T^{2} \) |
| 71 | \( 1 + (-2.08 + 1.74i)T + (12.3 - 69.9i)T^{2} \) |
| 73 | \( 1 + (-0.952 - 5.39i)T + (-68.5 + 24.9i)T^{2} \) |
| 79 | \( 1 + (-9.22 + 1.62i)T + (74.2 - 27.0i)T^{2} \) |
| 83 | \( 1 + (7.02 + 4.05i)T + (41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (0.233 - 1.32i)T + (-83.6 - 30.4i)T^{2} \) |
| 97 | \( 1 + (-4.64 + 12.7i)T + (-74.3 - 62.3i)T^{2} \) |
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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
−14.58542391820631647213219202406, −13.79987907792019109100219222124, −12.64032223884178547877118357588, −11.23092037033763777985424034217, −10.54381145175646232180093398758, −8.817898865930612983946972670165, −7.85413037305440271472375412557, −6.29257120143955330646551314430, −4.59487718742301936808130048149, −1.21369979428468023501115898218,
3.82645612123430162176176594497, 5.36927908803365504022633701681, 7.17751451054438603809819777835, 8.835858482971626779606640943395, 9.397570903658132460639261253317, 10.96375211123774369806694689136, 11.86831999586555205025389284498, 13.29263518874126178343733817565, 14.97375360044389710831562305346, 15.51433604562960849297239436575