| L(s) = 1 | + (−0.681 + 1.87i)2-s + (0.917 + 1.46i)3-s + (−1.50 − 1.26i)4-s + (0.291 − 1.65i)5-s + (−3.37 + 0.716i)6-s + (0.496 + 2.59i)7-s + (−0.0549 + 0.0317i)8-s + (−1.31 + 2.69i)9-s + (2.89 + 1.67i)10-s + (−3.13 + 0.553i)11-s + (0.475 − 3.37i)12-s + (0.877 + 2.41i)13-s + (−5.20 − 0.840i)14-s + (2.69 − 1.08i)15-s + (−0.705 − 4.00i)16-s + (0.934 − 1.61i)17-s + ⋯ |
| L(s) = 1 | + (−0.481 + 1.32i)2-s + (0.529 + 0.848i)3-s + (−0.753 − 0.632i)4-s + (0.130 − 0.739i)5-s + (−1.37 + 0.292i)6-s + (0.187 + 0.982i)7-s + (−0.0194 + 0.0112i)8-s + (−0.438 + 0.898i)9-s + (0.916 + 0.529i)10-s + (−0.946 + 0.166i)11-s + (0.137 − 0.974i)12-s + (0.243 + 0.668i)13-s + (−1.39 − 0.224i)14-s + (0.696 − 0.281i)15-s + (−0.176 − 1.00i)16-s + (0.226 − 0.392i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.940 - 0.340i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.940 - 0.340i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.179999 + 1.02599i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.179999 + 1.02599i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-0.917 - 1.46i)T \) |
| 7 | \( 1 + (-0.496 - 2.59i)T \) |
| good | 2 | \( 1 + (0.681 - 1.87i)T + (-1.53 - 1.28i)T^{2} \) |
| 5 | \( 1 + (-0.291 + 1.65i)T + (-4.69 - 1.71i)T^{2} \) |
| 11 | \( 1 + (3.13 - 0.553i)T + (10.3 - 3.76i)T^{2} \) |
| 13 | \( 1 + (-0.877 - 2.41i)T + (-9.95 + 8.35i)T^{2} \) |
| 17 | \( 1 + (-0.934 + 1.61i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-5.71 + 3.30i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-4.62 + 5.50i)T + (-3.99 - 22.6i)T^{2} \) |
| 29 | \( 1 + (3.25 - 8.94i)T + (-22.2 - 18.6i)T^{2} \) |
| 31 | \( 1 + (-0.942 + 1.12i)T + (-5.38 - 30.5i)T^{2} \) |
| 37 | \( 1 + (0.322 - 0.558i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-0.477 + 0.173i)T + (31.4 - 26.3i)T^{2} \) |
| 43 | \( 1 + (0.197 + 1.12i)T + (-40.4 + 14.7i)T^{2} \) |
| 47 | \( 1 + (-8.83 + 7.41i)T + (8.16 - 46.2i)T^{2} \) |
| 53 | \( 1 + 1.23iT - 53T^{2} \) |
| 59 | \( 1 + (0.0973 - 0.552i)T + (-55.4 - 20.1i)T^{2} \) |
| 61 | \( 1 + (4.58 + 5.46i)T + (-10.5 + 60.0i)T^{2} \) |
| 67 | \( 1 + (-8.34 + 3.03i)T + (51.3 - 43.0i)T^{2} \) |
| 71 | \( 1 + (5.31 + 3.07i)T + (35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (5.75 - 3.32i)T + (36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-0.814 - 0.296i)T + (60.5 + 50.7i)T^{2} \) |
| 83 | \( 1 + (-7.67 - 2.79i)T + (63.5 + 53.3i)T^{2} \) |
| 89 | \( 1 + (-3.77 - 6.54i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-9.45 + 1.66i)T + (91.1 - 33.1i)T^{2} \) |
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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
−13.28788160065569011402251561716, −12.05068776268294079021728933742, −10.81139019652738351762395813702, −9.311623535903319924460962323514, −9.018018309502545723954371987375, −8.138576961753247453274099508495, −7.00863903283022291816911234022, −5.34124970678131447278830287412, −5.00793131162734454592228122551, −2.80502605335118106364058259234,
1.14080379051812746231651563492, 2.73461615951121130081499203366, 3.59451209534956281235512031058, 5.89177963803493964019097379192, 7.34640183475640735782477526366, 8.033767672580773787011774551571, 9.434521793502790066104313553855, 10.32997509348030866166724753766, 11.05189232764797623184782685815, 11.98850678051484482201732492988
