L(s) = 1 | + 0.495·2-s − 1.75·4-s + (−1.84 − 3.19i)5-s + (0.926 − 2.47i)7-s − 1.86·8-s + (−0.915 − 1.58i)10-s + (−0.446 + 0.772i)11-s + (0.598 − 1.03i)13-s + (0.459 − 1.22i)14-s + 2.58·16-s + (0.124 + 0.216i)17-s + (1.40 − 2.43i)19-s + (3.23 + 5.60i)20-s + (−0.221 + 0.383i)22-s + (1.23 + 2.14i)23-s + ⋯ |
L(s) = 1 | + 0.350·2-s − 0.877·4-s + (−0.825 − 1.43i)5-s + (0.350 − 0.936i)7-s − 0.658·8-s + (−0.289 − 0.501i)10-s + (−0.134 + 0.233i)11-s + (0.165 − 0.287i)13-s + (0.122 − 0.328i)14-s + 0.646·16-s + (0.0303 + 0.0525i)17-s + (0.322 − 0.557i)19-s + (0.724 + 1.25i)20-s + (−0.0471 + 0.0817i)22-s + (0.258 + 0.447i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.245 + 0.969i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.245 + 0.969i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.548608 - 0.704861i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.548608 - 0.704861i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 + (-0.926 + 2.47i)T \) |
good | 2 | \( 1 - 0.495T + 2T^{2} \) |
| 5 | \( 1 + (1.84 + 3.19i)T + (-2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (0.446 - 0.772i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (-0.598 + 1.03i)T + (-6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + (-0.124 - 0.216i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-1.40 + 2.43i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-1.23 - 2.14i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (2.07 + 3.58i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 - 3.58T + 31T^{2} \) |
| 37 | \( 1 + (2.36 - 4.09i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-2.39 + 4.14i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (4.98 + 8.64i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 - 10.1T + 47T^{2} \) |
| 53 | \( 1 + (-4.94 - 8.56i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + 1.81T + 59T^{2} \) |
| 61 | \( 1 - 10.8T + 61T^{2} \) |
| 67 | \( 1 - 1.02T + 67T^{2} \) |
| 71 | \( 1 - 4.94T + 71T^{2} \) |
| 73 | \( 1 + (0.915 + 1.58i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + 1.79T + 79T^{2} \) |
| 83 | \( 1 + (6.16 + 10.6i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (-1.20 + 2.08i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-5.52 - 9.56i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−12.41205063677680396238732468891, −11.60085693060573983290134399058, −10.24229593967506130801910991769, −9.097012080537321441418444376299, −8.311983876201079429917226847678, −7.37614100374515767314471176284, −5.44467831442530836385748398724, −4.57851465913554016565264306493, −3.76545768748717158411161479561, −0.77591442529177466918848278731,
2.85252598526421973104828269099, 3.96019931308712359055381105532, 5.35760663634513773950964917575, 6.54039099138433021193336481500, 7.84107493133968333407964401189, 8.735234096776101178125199718269, 9.941422668568974061768634713241, 11.08101209411915664549597554708, 11.84767788559973961303714966901, 12.79265142101432031800126763379