| L(s) = 1 | + (−0.673 − 0.143i)2-s + (−1.39 − 0.620i)4-s + (0.00833 + 0.0392i)5-s + (−0.245 + 0.0258i)7-s + (1.96 + 1.42i)8-s − 0.0276i·10-s + (0.586 − 3.26i)11-s + (1.71 + 1.54i)13-s + (0.168 + 0.0177i)14-s + (0.924 + 1.02i)16-s + (1.32 − 4.06i)17-s + (−3.64 + 5.01i)19-s + (0.0127 − 0.0598i)20-s + (−0.862 + 2.11i)22-s + (−5.88 + 3.39i)23-s + ⋯ |
| L(s) = 1 | + (−0.476 − 0.101i)2-s + (−0.697 − 0.310i)4-s + (0.00372 + 0.0175i)5-s + (−0.0927 + 0.00975i)7-s + (0.694 + 0.504i)8-s − 0.00872i·10-s + (0.176 − 0.984i)11-s + (0.474 + 0.427i)13-s + (0.0451 + 0.00474i)14-s + (0.231 + 0.256i)16-s + (0.320 − 0.986i)17-s + (−0.836 + 1.15i)19-s + (0.00284 − 0.0133i)20-s + (−0.183 + 0.450i)22-s + (−1.22 + 0.708i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 891 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.223 + 0.974i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 891 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.223 + 0.974i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.476450 - 0.598222i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.476450 - 0.598222i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 11 | \( 1 + (-0.586 + 3.26i)T \) |
| good | 2 | \( 1 + (0.673 + 0.143i)T + (1.82 + 0.813i)T^{2} \) |
| 5 | \( 1 + (-0.00833 - 0.0392i)T + (-4.56 + 2.03i)T^{2} \) |
| 7 | \( 1 + (0.245 - 0.0258i)T + (6.84 - 1.45i)T^{2} \) |
| 13 | \( 1 + (-1.71 - 1.54i)T + (1.35 + 12.9i)T^{2} \) |
| 17 | \( 1 + (-1.32 + 4.06i)T + (-13.7 - 9.99i)T^{2} \) |
| 19 | \( 1 + (3.64 - 5.01i)T + (-5.87 - 18.0i)T^{2} \) |
| 23 | \( 1 + (5.88 - 3.39i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (0.584 + 5.56i)T + (-28.3 + 6.02i)T^{2} \) |
| 31 | \( 1 + (-3.21 + 3.56i)T + (-3.24 - 30.8i)T^{2} \) |
| 37 | \( 1 + (-3.26 + 2.36i)T + (11.4 - 35.1i)T^{2} \) |
| 41 | \( 1 + (-1.00 + 9.55i)T + (-40.1 - 8.52i)T^{2} \) |
| 43 | \( 1 + (0.893 + 0.515i)T + (21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (4.52 + 10.1i)T + (-31.4 + 34.9i)T^{2} \) |
| 53 | \( 1 + (-8.52 + 2.77i)T + (42.8 - 31.1i)T^{2} \) |
| 59 | \( 1 + (-1.13 + 2.54i)T + (-39.4 - 43.8i)T^{2} \) |
| 61 | \( 1 + (6.30 - 5.67i)T + (6.37 - 60.6i)T^{2} \) |
| 67 | \( 1 + (3.97 + 6.88i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (3.16 + 1.02i)T + (57.4 + 41.7i)T^{2} \) |
| 73 | \( 1 + (6.96 + 9.58i)T + (-22.5 + 69.4i)T^{2} \) |
| 79 | \( 1 + (-0.626 + 2.94i)T + (-72.1 - 32.1i)T^{2} \) |
| 83 | \( 1 + (-3.54 - 3.93i)T + (-8.67 + 82.5i)T^{2} \) |
| 89 | \( 1 + 8.54iT - 89T^{2} \) |
| 97 | \( 1 + (-2.96 - 0.629i)T + (88.6 + 39.4i)T^{2} \) |
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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
−9.896215788794934249026086809484, −9.017602132486238820311364260837, −8.373475557420260990900960151071, −7.61537577023156346971895110668, −6.25864948661036175788654866550, −5.63369894797862822656832730365, −4.42502947223783314687219023643, −3.56399382616620974356890824885, −1.95650013697698878031964601194, −0.48738346953709336782646230630,
1.33661158443408456874526162174, 2.98553165921723645652745981114, 4.22544792623421739674608530011, 4.84353862716968930024532269169, 6.21762357468354084349465759092, 7.04935936841959073735523965803, 8.081297195480572898354753179782, 8.605613173906604766046527963189, 9.482306444599849474002691114004, 10.23821626119153890476131178756
