L(s) = 1 | + (−1.30 + 0.755i)2-s + (0.140 − 0.242i)4-s + 0.775·5-s + (2.05 + 1.66i)7-s − 2.59i·8-s + (−1.01 + 0.585i)10-s + 3.84i·11-s + (−2.54 + 1.46i)13-s + (−3.94 − 0.618i)14-s + (2.24 + 3.88i)16-s + (2.69 + 4.67i)17-s + (−0.376 − 0.217i)19-s + (0.108 − 0.188i)20-s + (−2.90 − 5.02i)22-s − 0.0557i·23-s + ⋯ |
L(s) = 1 | + (−0.924 + 0.533i)2-s + (0.0700 − 0.121i)4-s + 0.346·5-s + (0.778 + 0.628i)7-s − 0.918i·8-s + (−0.320 + 0.185i)10-s + 1.15i·11-s + (−0.705 + 0.407i)13-s + (−1.05 − 0.165i)14-s + (0.560 + 0.970i)16-s + (0.654 + 1.13i)17-s + (−0.0863 − 0.0498i)19-s + (0.0243 − 0.0421i)20-s + (−0.618 − 1.07i)22-s − 0.0116i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.102 - 0.994i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.102 - 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.506619 + 0.561749i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.506619 + 0.561749i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 + (-2.05 - 1.66i)T \) |
good | 2 | \( 1 + (1.30 - 0.755i)T + (1 - 1.73i)T^{2} \) |
| 5 | \( 1 - 0.775T + 5T^{2} \) |
| 11 | \( 1 - 3.84iT - 11T^{2} \) |
| 13 | \( 1 + (2.54 - 1.46i)T + (6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + (-2.69 - 4.67i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (0.376 + 0.217i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + 0.0557iT - 23T^{2} \) |
| 29 | \( 1 + (-0.187 - 0.108i)T + (14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (-5.67 - 3.27i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-3.14 + 5.45i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (3.78 + 6.56i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-6.42 + 11.1i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (0.482 + 0.836i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-6.46 + 3.73i)T + (26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-1.56 + 2.70i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (3.01 - 1.74i)T + (30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-2.10 + 3.63i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 3.50iT - 71T^{2} \) |
| 73 | \( 1 + (7.05 - 4.07i)T + (36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-2.48 - 4.29i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-4.31 + 7.46i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (7.82 - 13.5i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-1.24 - 0.716i)T + (48.5 + 84.0i)T^{2} \) |
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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
−12.56051580069189089924334482491, −12.03766750481526750519760655974, −10.47739493587574512400707478673, −9.689657705291254644847103847945, −8.758318931216808598432432221485, −7.84361770321641859637549461277, −6.93711554271953703653340989381, −5.58399543994196542256751562337, −4.17653634802018374766490440875, −1.97917685175824363867946545380,
1.00265109611426784812380914614, 2.76903051701591595685329234915, 4.74891142152707397477515615259, 5.89367603664415413772981148397, 7.62136909348754346428887873460, 8.316322281229163248627094878838, 9.560865380021121211747960018954, 10.19041314190746080816953269966, 11.23955012391632456627275784002, 11.80855761463712147621356529204