| L(s) = 1 | + (−0.240 − 0.0424i)2-s + (−1.76 + 2.09i)3-s + (−1.82 − 0.663i)4-s + (−1.42 − 1.72i)5-s + (0.512 − 0.430i)6-s + (−1.68 + 0.970i)7-s + (0.834 + 0.481i)8-s + (−0.781 − 4.43i)9-s + (0.270 + 0.475i)10-s + (−2.11 + 3.66i)11-s + (4.60 − 2.65i)12-s + (0.816 + 0.972i)13-s + (0.445 − 0.162i)14-s + (6.12 + 0.0416i)15-s + (2.79 + 2.34i)16-s + (−2.42 − 0.427i)17-s + ⋯ |
| L(s) = 1 | + (−0.170 − 0.0300i)2-s + (−1.01 + 1.21i)3-s + (−0.911 − 0.331i)4-s + (−0.637 − 0.770i)5-s + (0.209 − 0.175i)6-s + (−0.635 + 0.366i)7-s + (0.294 + 0.170i)8-s + (−0.260 − 1.47i)9-s + (0.0854 + 0.150i)10-s + (−0.638 + 1.10i)11-s + (1.32 − 0.766i)12-s + (0.226 + 0.269i)13-s + (0.119 − 0.0433i)14-s + (1.58 + 0.0107i)15-s + (0.698 + 0.585i)16-s + (−0.587 − 0.103i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 95 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.998 - 0.0598i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 95 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.998 - 0.0598i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.00472269 + 0.157649i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.00472269 + 0.157649i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (1.42 + 1.72i)T \) |
| 19 | \( 1 + (1.64 - 4.03i)T \) |
| good | 2 | \( 1 + (0.240 + 0.0424i)T + (1.87 + 0.684i)T^{2} \) |
| 3 | \( 1 + (1.76 - 2.09i)T + (-0.520 - 2.95i)T^{2} \) |
| 7 | \( 1 + (1.68 - 0.970i)T + (3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (2.11 - 3.66i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (-0.816 - 0.972i)T + (-2.25 + 12.8i)T^{2} \) |
| 17 | \( 1 + (2.42 + 0.427i)T + (15.9 + 5.81i)T^{2} \) |
| 23 | \( 1 + (-1.75 + 4.82i)T + (-17.6 - 14.7i)T^{2} \) |
| 29 | \( 1 + (1.50 + 8.50i)T + (-27.2 + 9.91i)T^{2} \) |
| 31 | \( 1 + (2.55 + 4.41i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 - 11.0iT - 37T^{2} \) |
| 41 | \( 1 + (-1.91 - 1.60i)T + (7.11 + 40.3i)T^{2} \) |
| 43 | \( 1 + (-1.52 - 4.19i)T + (-32.9 + 27.6i)T^{2} \) |
| 47 | \( 1 + (6.66 - 1.17i)T + (44.1 - 16.0i)T^{2} \) |
| 53 | \( 1 + (0.235 - 0.648i)T + (-40.6 - 34.0i)T^{2} \) |
| 59 | \( 1 + (0.901 - 5.11i)T + (-55.4 - 20.1i)T^{2} \) |
| 61 | \( 1 + (2.63 + 0.958i)T + (46.7 + 39.2i)T^{2} \) |
| 67 | \( 1 + (-5.13 + 0.905i)T + (62.9 - 22.9i)T^{2} \) |
| 71 | \( 1 + (0.744 - 0.270i)T + (54.3 - 45.6i)T^{2} \) |
| 73 | \( 1 + (-0.910 + 1.08i)T + (-12.6 - 71.8i)T^{2} \) |
| 79 | \( 1 + (-4.64 - 3.89i)T + (13.7 + 77.7i)T^{2} \) |
| 83 | \( 1 + (10.2 - 5.92i)T + (41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (-1.71 + 1.43i)T + (15.4 - 87.6i)T^{2} \) |
| 97 | \( 1 + (8.47 + 1.49i)T + (91.1 + 33.1i)T^{2} \) |
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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
−14.95490563294572931677025781678, −13.19208828960401641270090535812, −12.36121065773058889098371348220, −11.18925921826216978800204522322, −9.996906826839490676542947472769, −9.444043576293138746803224871721, −8.172679064598987628767979429549, −6.06674975093094887343067046608, −4.82114547492735574159727029462, −4.18631456273498437557131784219,
0.22399275259444642284407505824, 3.43536321338336546101648208743, 5.39954830609665840126235995909, 6.77259231205679758108604038547, 7.57956787918853190165433607799, 8.838652091358822584985004475101, 10.64263834954988962919540143066, 11.28786167340427511520908558345, 12.67490767535661473131658246448, 13.17359251251584584511705751967
