L(s) = 1 | + (−1.63 + 0.287i)2-s + (1.35 − 1.07i)3-s + (0.700 − 0.255i)4-s + (−3.60 + 1.31i)5-s + (−1.90 + 2.15i)6-s + (2.03 − 1.69i)7-s + (1.79 − 1.03i)8-s + (0.671 − 2.92i)9-s + (5.50 − 3.17i)10-s + (0.982 − 2.70i)11-s + (0.674 − 1.10i)12-s + (−1.44 − 3.96i)13-s + (−2.82 + 3.35i)14-s + (−3.46 + 5.66i)15-s + (−3.78 + 3.17i)16-s + (−1.59 − 2.76i)17-s + ⋯ |
L(s) = 1 | + (−1.15 + 0.203i)2-s + (0.782 − 0.622i)3-s + (0.350 − 0.127i)4-s + (−1.61 + 0.586i)5-s + (−0.775 + 0.878i)6-s + (0.767 − 0.640i)7-s + (0.636 − 0.367i)8-s + (0.223 − 0.974i)9-s + (1.74 − 1.00i)10-s + (0.296 − 0.814i)11-s + (0.194 − 0.318i)12-s + (−0.399 − 1.09i)13-s + (−0.755 + 0.895i)14-s + (−0.895 + 1.46i)15-s + (−0.945 + 0.793i)16-s + (−0.386 − 0.669i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.105 + 0.994i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.105 + 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.437040 - 0.393150i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.437040 - 0.393150i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-1.35 + 1.07i)T \) |
| 7 | \( 1 + (-2.03 + 1.69i)T \) |
good | 2 | \( 1 + (1.63 - 0.287i)T + (1.87 - 0.684i)T^{2} \) |
| 5 | \( 1 + (3.60 - 1.31i)T + (3.83 - 3.21i)T^{2} \) |
| 11 | \( 1 + (-0.982 + 2.70i)T + (-8.42 - 7.07i)T^{2} \) |
| 13 | \( 1 + (1.44 + 3.96i)T + (-9.95 + 8.35i)T^{2} \) |
| 17 | \( 1 + (1.59 + 2.76i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (1.96 + 1.13i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-2.25 - 0.397i)T + (21.6 + 7.86i)T^{2} \) |
| 29 | \( 1 + (-0.356 + 0.979i)T + (-22.2 - 18.6i)T^{2} \) |
| 31 | \( 1 + (-2.96 - 8.15i)T + (-23.7 + 19.9i)T^{2} \) |
| 37 | \( 1 - 0.0244T + 37T^{2} \) |
| 41 | \( 1 + (2.99 - 1.09i)T + (31.4 - 26.3i)T^{2} \) |
| 43 | \( 1 + (0.110 + 0.627i)T + (-40.4 + 14.7i)T^{2} \) |
| 47 | \( 1 + (-10.0 - 3.67i)T + (36.0 + 30.2i)T^{2} \) |
| 53 | \( 1 + (-1.91 - 1.10i)T + (26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (8.38 + 7.03i)T + (10.2 + 58.1i)T^{2} \) |
| 61 | \( 1 + (2.60 - 7.16i)T + (-46.7 - 39.2i)T^{2} \) |
| 67 | \( 1 + (1.88 - 10.7i)T + (-62.9 - 22.9i)T^{2} \) |
| 71 | \( 1 + (-3.88 - 2.24i)T + (35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 - 2.22iT - 73T^{2} \) |
| 79 | \( 1 + (2.19 + 12.4i)T + (-74.2 + 27.0i)T^{2} \) |
| 83 | \( 1 + (-4.16 - 1.51i)T + (63.5 + 53.3i)T^{2} \) |
| 89 | \( 1 + (-2.36 + 4.09i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (13.0 - 2.29i)T + (91.1 - 33.1i)T^{2} \) |
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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
−12.17483172337619871003483622481, −11.14689052151686930968599485239, −10.38358930874613939440269562091, −8.859221332560682279608748013883, −8.218267727494411211351610502998, −7.50620085890192539354851577519, −6.89550302092536581534049409329, −4.39145897880322305563585201059, −3.14374596997651525127788888331, −0.74203180122616987456883704102,
1.99680465617021219405146051005, 4.17655805071761786817917292756, 4.73768904535494388459831040138, 7.33244093586375355514480589050, 8.122093321575942385747902379307, 8.783738461913479265019183559012, 9.466532316574605103601857706017, 10.74994815917219403455074640512, 11.56651895143564048797212939983, 12.45504279444615515185357328497