L(s) = 1 | + (−1.19 + 2.06i)2-s + (−1.84 − 3.20i)4-s + 2.92·5-s + (2.35 + 1.20i)7-s + 4.05·8-s + (−3.48 + 6.03i)10-s + 1.35·11-s + (−0.733 + 1.26i)13-s + (−5.29 + 3.43i)14-s + (−1.13 + 1.96i)16-s + (−1.65 + 2.86i)17-s + (−1.10 − 1.91i)19-s + (−5.39 − 9.35i)20-s + (−1.61 + 2.79i)22-s − 2.62·23-s + ⋯ |
L(s) = 1 | + (−0.843 + 1.46i)2-s + (−0.924 − 1.60i)4-s + 1.30·5-s + (0.891 + 0.453i)7-s + 1.43·8-s + (−1.10 + 1.90i)10-s + 0.408·11-s + (−0.203 + 0.352i)13-s + (−1.41 + 0.919i)14-s + (−0.284 + 0.492i)16-s + (−0.401 + 0.695i)17-s + (−0.253 − 0.438i)19-s + (−1.20 − 2.09i)20-s + (−0.344 + 0.596i)22-s − 0.548·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.312 - 0.949i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.312 - 0.949i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.560977 + 0.775507i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.560977 + 0.775507i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 + (-2.35 - 1.20i)T \) |
good | 2 | \( 1 + (1.19 - 2.06i)T + (-1 - 1.73i)T^{2} \) |
| 5 | \( 1 - 2.92T + 5T^{2} \) |
| 11 | \( 1 - 1.35T + 11T^{2} \) |
| 13 | \( 1 + (0.733 - 1.26i)T + (-6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + (1.65 - 2.86i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (1.10 + 1.91i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + 2.62T + 23T^{2} \) |
| 29 | \( 1 + (0.521 + 0.903i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (1.63 + 2.83i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-5.43 - 9.41i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-0.904 + 1.56i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (2.17 + 3.76i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (-1.98 + 3.44i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-3.22 + 5.59i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (6.10 + 10.5i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (0.279 - 0.484i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (6.40 + 11.0i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + 12.9T + 71T^{2} \) |
| 73 | \( 1 + (-5.22 + 9.05i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (0.383 - 0.664i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-0.983 - 1.70i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (3.20 + 5.54i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (4.14 + 7.17i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−13.27989162941554482899117812329, −11.78299501176637596928118592554, −10.45628761113772331392337310618, −9.496905400654052657517797704382, −8.794630353896667476872635733170, −7.84696268478135087854943500408, −6.55568275325923078219013399943, −5.87313063061659091301955151689, −4.78202433516541621969463249209, −1.88669018768245945334242071333,
1.41071890215162423906895942301, 2.55588023750211615987073619322, 4.26229496477947372702106230546, 5.80428679866325997531442175820, 7.48160780264460104150801339319, 8.710160973318383140947360634180, 9.525440105459811955760557516393, 10.34180163743818108844800562573, 11.06303040117701256405759397706, 12.02830725265823682567701734939