L(s) = 1 | + (0.5 + 0.866i)3-s + (0.618 + 2.57i)7-s + (−0.499 + 0.866i)9-s + (−1.46 − 2.53i)11-s + 5.76·13-s + (0.118 + 0.205i)17-s + (−2.37 + 4.10i)19-s + (−1.91 + 1.82i)21-s + (3.38 − 5.86i)23-s − 0.999·27-s + 6.03·29-s + (5.26 + 9.12i)31-s + (1.46 − 2.53i)33-s + (−1.53 + 2.66i)37-s + (2.88 + 4.99i)39-s + ⋯ |
L(s) = 1 | + (0.288 + 0.499i)3-s + (0.233 + 0.972i)7-s + (−0.166 + 0.288i)9-s + (−0.442 − 0.765i)11-s + 1.60·13-s + (0.0287 + 0.0497i)17-s + (−0.543 + 0.941i)19-s + (−0.418 + 0.397i)21-s + (0.705 − 1.22i)23-s − 0.192·27-s + 1.12·29-s + (0.946 + 1.63i)31-s + (0.255 − 0.442i)33-s + (−0.252 + 0.437i)37-s + (0.461 + 0.800i)39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.222 - 0.974i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.222 - 0.974i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.036003437\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.036003437\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-0.5 - 0.866i)T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (-0.618 - 2.57i)T \) |
good | 11 | \( 1 + (1.46 + 2.53i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 - 5.76T + 13T^{2} \) |
| 17 | \( 1 + (-0.118 - 0.205i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (2.37 - 4.10i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-3.38 + 5.86i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 - 6.03T + 29T^{2} \) |
| 31 | \( 1 + (-5.26 - 9.12i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (1.53 - 2.66i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 - 1.06T + 41T^{2} \) |
| 43 | \( 1 + 9.79T + 43T^{2} \) |
| 47 | \( 1 + (-1.85 + 3.21i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-5.50 - 9.53i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-4.65 - 8.06i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-4.06 + 7.04i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (1.12 + 1.94i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + 11.8T + 71T^{2} \) |
| 73 | \( 1 + (-1.89 - 3.28i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (0.00873 - 0.0151i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + 7.30T + 83T^{2} \) |
| 89 | \( 1 + (8.97 - 15.5i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + 2.31T + 97T^{2} \) |
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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
−8.948634538510896341879613424343, −8.404206616298813829291704416717, −8.316178728376281510050095564987, −6.73767357764555348703428058465, −6.06954073583174139394578038250, −5.29822230019775179035457007314, −4.41164196036044628114791478645, −3.35487166493550116204749592962, −2.65997177230766060160956310451, −1.29108667765670878216908815685,
0.78067188420332712143146035688, 1.84543589739213898264942752200, 3.04598832641435671271983412744, 4.00111329159247366596544978191, 4.79550633945672582650973948119, 5.90212173951734546768450544596, 6.78468418984731833196081603330, 7.32017244533906926655554182028, 8.187959442830707021198370617648, 8.728105538437209650601215498992