| L(s) = 1 | + (−0.0448 − 0.123i)2-s + (−1.62 − 0.587i)3-s + (1.51 − 1.27i)4-s + (−0.231 − 1.31i)5-s + (0.000699 + 0.227i)6-s + (−2.62 − 0.321i)7-s + (−0.452 − 0.261i)8-s + (2.30 + 1.91i)9-s + (−0.151 + 0.0873i)10-s + (−4.31 − 0.759i)11-s + (−3.22 + 1.18i)12-s + (1.38 − 3.81i)13-s + (0.0781 + 0.337i)14-s + (−0.393 + 2.27i)15-s + (0.676 − 3.83i)16-s + (−1.85 − 3.21i)17-s + ⋯ |
| L(s) = 1 | + (−0.0317 − 0.0871i)2-s + (−0.940 − 0.339i)3-s + (0.759 − 0.637i)4-s + (−0.103 − 0.586i)5-s + (0.000285 + 0.0927i)6-s + (−0.992 − 0.121i)7-s + (−0.159 − 0.0923i)8-s + (0.769 + 0.638i)9-s + (−0.0478 + 0.0276i)10-s + (−1.29 − 0.229i)11-s + (−0.930 + 0.341i)12-s + (0.385 − 1.05i)13-s + (0.0208 + 0.0903i)14-s + (−0.101 + 0.587i)15-s + (0.169 − 0.959i)16-s + (−0.450 − 0.780i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.496 + 0.867i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.496 + 0.867i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.395025 - 0.681248i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.395025 - 0.681248i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (1.62 + 0.587i)T \) |
| 7 | \( 1 + (2.62 + 0.321i)T \) |
| good | 2 | \( 1 + (0.0448 + 0.123i)T + (-1.53 + 1.28i)T^{2} \) |
| 5 | \( 1 + (0.231 + 1.31i)T + (-4.69 + 1.71i)T^{2} \) |
| 11 | \( 1 + (4.31 + 0.759i)T + (10.3 + 3.76i)T^{2} \) |
| 13 | \( 1 + (-1.38 + 3.81i)T + (-9.95 - 8.35i)T^{2} \) |
| 17 | \( 1 + (1.85 + 3.21i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-4.31 - 2.49i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-0.242 - 0.289i)T + (-3.99 + 22.6i)T^{2} \) |
| 29 | \( 1 + (-1.57 - 4.31i)T + (-22.2 + 18.6i)T^{2} \) |
| 31 | \( 1 + (-0.693 - 0.826i)T + (-5.38 + 30.5i)T^{2} \) |
| 37 | \( 1 + (0.172 + 0.298i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-5.09 - 1.85i)T + (31.4 + 26.3i)T^{2} \) |
| 43 | \( 1 + (0.390 - 2.21i)T + (-40.4 - 14.7i)T^{2} \) |
| 47 | \( 1 + (-9.77 - 8.19i)T + (8.16 + 46.2i)T^{2} \) |
| 53 | \( 1 + 9.19iT - 53T^{2} \) |
| 59 | \( 1 + (2.24 + 12.7i)T + (-55.4 + 20.1i)T^{2} \) |
| 61 | \( 1 + (1.14 - 1.36i)T + (-10.5 - 60.0i)T^{2} \) |
| 67 | \( 1 + (-5.40 - 1.96i)T + (51.3 + 43.0i)T^{2} \) |
| 71 | \( 1 + (-2.30 + 1.33i)T + (35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (4.63 + 2.67i)T + (36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-5.48 + 1.99i)T + (60.5 - 50.7i)T^{2} \) |
| 83 | \( 1 + (4.03 - 1.46i)T + (63.5 - 53.3i)T^{2} \) |
| 89 | \( 1 + (5.59 - 9.68i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (14.5 + 2.55i)T + (91.1 + 33.1i)T^{2} \) |
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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
−12.35804322809778220236844341470, −11.13172932123402954425622624107, −10.48051048693548117088334158394, −9.592878101272376744644558344954, −7.916713419148776754073380722484, −6.90601332491953196144716476075, −5.80739036762844943634275669098, −5.08747551265726018754046005433, −2.89260285848476323056166764961, −0.78162736925630196067782257656,
2.69595358553469473074144804604, 4.05688758985310027386362463508, 5.73961386099645896189635094461, 6.70238094424213570372316818371, 7.40479437214690072494075325608, 8.994839303147158943731104680809, 10.26213862121076822394477690960, 10.93912259693435667628176166582, 11.84186004244351563400879250108, 12.67307156511530143852480390404
