| L(s) = 1 | + (−0.266 − 1.51i)2-s + (−1.44 − 0.949i)3-s + (−0.337 + 0.122i)4-s + (3.71 − 1.35i)5-s + (−1.04 + 2.44i)6-s + (2.56 + 0.639i)7-s + (−1.26 − 2.18i)8-s + (1.19 + 2.75i)9-s + (−3.03 − 5.25i)10-s + (−0.965 − 0.351i)11-s + (0.605 + 0.142i)12-s + (−3.84 + 1.39i)13-s + (0.281 − 4.05i)14-s + (−6.66 − 1.56i)15-s + (−3.51 + 2.94i)16-s + (0.513 + 0.889i)17-s + ⋯ |
| L(s) = 1 | + (−0.188 − 1.06i)2-s + (−0.836 − 0.548i)3-s + (−0.168 + 0.0613i)4-s + (1.66 − 0.604i)5-s + (−0.428 + 0.997i)6-s + (0.970 + 0.241i)7-s + (−0.445 − 0.771i)8-s + (0.398 + 0.916i)9-s + (−0.959 − 1.66i)10-s + (−0.291 − 0.106i)11-s + (0.174 + 0.0411i)12-s + (−1.06 + 0.388i)13-s + (0.0753 − 1.08i)14-s + (−1.72 − 0.404i)15-s + (−0.878 + 0.737i)16-s + (0.124 + 0.215i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.559 + 0.828i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.559 + 0.828i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.545694 - 1.02652i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.545694 - 1.02652i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (1.44 + 0.949i)T \) |
| 7 | \( 1 + (-2.56 - 0.639i)T \) |
| good | 2 | \( 1 + (0.266 + 1.51i)T + (-1.87 + 0.684i)T^{2} \) |
| 5 | \( 1 + (-3.71 + 1.35i)T + (3.83 - 3.21i)T^{2} \) |
| 11 | \( 1 + (0.965 + 0.351i)T + (8.42 + 7.07i)T^{2} \) |
| 13 | \( 1 + (3.84 - 1.39i)T + (9.95 - 8.35i)T^{2} \) |
| 17 | \( 1 + (-0.513 - 0.889i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (2.28 - 3.96i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-0.114 + 0.650i)T + (-21.6 - 7.86i)T^{2} \) |
| 29 | \( 1 + (-6.27 - 2.28i)T + (22.2 + 18.6i)T^{2} \) |
| 31 | \( 1 + (8.69 - 3.16i)T + (23.7 - 19.9i)T^{2} \) |
| 37 | \( 1 - 3.18T + 37T^{2} \) |
| 41 | \( 1 + (5.19 - 1.89i)T + (31.4 - 26.3i)T^{2} \) |
| 43 | \( 1 + (-0.455 - 2.58i)T + (-40.4 + 14.7i)T^{2} \) |
| 47 | \( 1 + (2.95 + 1.07i)T + (36.0 + 30.2i)T^{2} \) |
| 53 | \( 1 + (-2.27 + 3.93i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (5.31 + 4.46i)T + (10.2 + 58.1i)T^{2} \) |
| 61 | \( 1 + (-12.1 - 4.43i)T + (46.7 + 39.2i)T^{2} \) |
| 67 | \( 1 + (-1.41 + 8.04i)T + (-62.9 - 22.9i)T^{2} \) |
| 71 | \( 1 + (-0.478 + 0.828i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 - 2.34T + 73T^{2} \) |
| 79 | \( 1 + (-0.996 - 5.65i)T + (-74.2 + 27.0i)T^{2} \) |
| 83 | \( 1 + (0.970 + 0.353i)T + (63.5 + 53.3i)T^{2} \) |
| 89 | \( 1 + (7.43 - 12.8i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (0.448 + 2.54i)T + (-91.1 + 33.1i)T^{2} \) |
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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
−12.35391815828163719238318573013, −11.22664835972423647311662196330, −10.37081049467005460306281905200, −9.674744975928007369716785053606, −8.414258190573490132242488943860, −6.81706819091847807399772885464, −5.72940401950174688642692071437, −4.82802928114945029076217084096, −2.25451951078372861657352285188, −1.49425406518515786249241928506,
2.39926798778280480154367036725, 4.92988188801668898404118183685, 5.58010103629746852971577995488, 6.60203689468788629842115600487, 7.44868815257190919003320561163, 8.960286726329419129833625281591, 9.983569932647479683287001616692, 10.76751561560879634432715525445, 11.70341504050197204719821240371, 13.06742148871593780570565527479
