| L(s) = 1 | + (−0.910 − 0.160i)2-s + (−1.05 − 1.37i)3-s + (−1.07 − 0.391i)4-s + (−0.473 − 0.172i)5-s + (0.737 + 1.42i)6-s + (−2.46 + 0.971i)7-s + (2.51 + 1.45i)8-s + (−0.785 + 2.89i)9-s + (0.403 + 0.233i)10-s + (1.66 + 4.56i)11-s + (0.593 + 1.89i)12-s + (1.60 − 4.42i)13-s + (2.39 − 0.489i)14-s + (0.261 + 0.833i)15-s + (−0.305 − 0.256i)16-s + (−2.38 + 4.13i)17-s + ⋯ |
| L(s) = 1 | + (−0.643 − 0.113i)2-s + (−0.607 − 0.794i)3-s + (−0.537 − 0.195i)4-s + (−0.211 − 0.0771i)5-s + (0.301 + 0.580i)6-s + (−0.930 + 0.367i)7-s + (0.890 + 0.514i)8-s + (−0.261 + 0.965i)9-s + (0.127 + 0.0737i)10-s + (0.500 + 1.37i)11-s + (0.171 + 0.546i)12-s + (0.446 − 1.22i)13-s + (0.640 − 0.130i)14-s + (0.0675 + 0.215i)15-s + (−0.0763 − 0.0640i)16-s + (−0.578 + 1.00i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.228 - 0.973i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.228 - 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.109501 + 0.138158i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.109501 + 0.138158i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (1.05 + 1.37i)T \) |
| 7 | \( 1 + (2.46 - 0.971i)T \) |
| good | 2 | \( 1 + (0.910 + 0.160i)T + (1.87 + 0.684i)T^{2} \) |
| 5 | \( 1 + (0.473 + 0.172i)T + (3.83 + 3.21i)T^{2} \) |
| 11 | \( 1 + (-1.66 - 4.56i)T + (-8.42 + 7.07i)T^{2} \) |
| 13 | \( 1 + (-1.60 + 4.42i)T + (-9.95 - 8.35i)T^{2} \) |
| 17 | \( 1 + (2.38 - 4.13i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (5.51 - 3.18i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (3.53 - 0.622i)T + (21.6 - 7.86i)T^{2} \) |
| 29 | \( 1 + (-0.997 - 2.73i)T + (-22.2 + 18.6i)T^{2} \) |
| 31 | \( 1 + (0.268 - 0.737i)T + (-23.7 - 19.9i)T^{2} \) |
| 37 | \( 1 + 5.47T + 37T^{2} \) |
| 41 | \( 1 + (5.44 + 1.98i)T + (31.4 + 26.3i)T^{2} \) |
| 43 | \( 1 + (1.36 - 7.75i)T + (-40.4 - 14.7i)T^{2} \) |
| 47 | \( 1 + (-7.22 + 2.62i)T + (36.0 - 30.2i)T^{2} \) |
| 53 | \( 1 + (-4.26 + 2.46i)T + (26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (4.08 - 3.42i)T + (10.2 - 58.1i)T^{2} \) |
| 61 | \( 1 + (2.45 + 6.74i)T + (-46.7 + 39.2i)T^{2} \) |
| 67 | \( 1 + (-1.51 - 8.61i)T + (-62.9 + 22.9i)T^{2} \) |
| 71 | \( 1 + (11.3 - 6.53i)T + (35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + 9.73iT - 73T^{2} \) |
| 79 | \( 1 + (0.825 - 4.67i)T + (-74.2 - 27.0i)T^{2} \) |
| 83 | \( 1 + (2.75 - 1.00i)T + (63.5 - 53.3i)T^{2} \) |
| 89 | \( 1 + (-5.60 - 9.70i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-2.41 - 0.426i)T + (91.1 + 33.1i)T^{2} \) |
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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
−12.70767399872376179216886320354, −12.10215396107218177556825689952, −10.56178122172338968793373246980, −10.09907623557326197048628105346, −8.726777193567459114714573116768, −7.936979873134420001947396567050, −6.63889099616016027794898874814, −5.65038897868794197789867470730, −4.16424728054740344000931772690, −1.85637265541949835602434329302,
0.20926231158980982845942806674, 3.61303644127318590457690101247, 4.42334700684307671513554764165, 6.09859429967757615336251519527, 7.02178403173509847285660953017, 8.762119597147182581527300820753, 9.137818764503490414529811828997, 10.22135685857586523471604239250, 11.12843212499799031738052086820, 12.01554315625805371654752588741
