| L(s) = 1 | + (0.0475 + 0.0305i)2-s + (−0.989 + 0.142i)3-s + (−0.829 − 1.81i)4-s + (1.34 + 0.396i)5-s + (−0.0514 − 0.0235i)6-s + (−1.67 + 2.04i)7-s + (0.0321 − 0.223i)8-s + (0.959 − 0.281i)9-s + (0.0521 + 0.0601i)10-s + (−2.64 − 4.12i)11-s + (1.07 + 1.67i)12-s + (5.25 − 4.55i)13-s + (−0.142 + 0.0462i)14-s + (−1.39 − 0.200i)15-s + (−2.60 + 3.00i)16-s + (−2.10 + 4.60i)17-s + ⋯ |
| L(s) = 1 | + (0.0336 + 0.0216i)2-s + (−0.571 + 0.0821i)3-s + (−0.414 − 0.908i)4-s + (0.603 + 0.177i)5-s + (−0.0210 − 0.00959i)6-s + (−0.633 + 0.774i)7-s + (0.0113 − 0.0791i)8-s + (0.319 − 0.0939i)9-s + (0.0164 + 0.0190i)10-s + (−0.798 − 1.24i)11-s + (0.311 + 0.484i)12-s + (1.45 − 1.26i)13-s + (−0.0380 + 0.0123i)14-s + (−0.359 − 0.0516i)15-s + (−0.651 + 0.752i)16-s + (−0.510 + 1.11i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.469 + 0.883i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.469 + 0.883i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.388211 - 0.645936i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.388211 - 0.645936i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (0.989 - 0.142i)T \) |
| 7 | \( 1 + (1.67 - 2.04i)T \) |
| 23 | \( 1 + (2.49 + 4.09i)T \) |
| good | 2 | \( 1 + (-0.0475 - 0.0305i)T + (0.830 + 1.81i)T^{2} \) |
| 5 | \( 1 + (-1.34 - 0.396i)T + (4.20 + 2.70i)T^{2} \) |
| 11 | \( 1 + (2.64 + 4.12i)T + (-4.56 + 10.0i)T^{2} \) |
| 13 | \( 1 + (-5.25 + 4.55i)T + (1.85 - 12.8i)T^{2} \) |
| 17 | \( 1 + (2.10 - 4.60i)T + (-11.1 - 12.8i)T^{2} \) |
| 19 | \( 1 + (2.25 + 4.94i)T + (-12.4 + 14.3i)T^{2} \) |
| 29 | \( 1 + (-2.06 + 4.52i)T + (-18.9 - 21.9i)T^{2} \) |
| 31 | \( 1 + (6.92 + 0.995i)T + (29.7 + 8.73i)T^{2} \) |
| 37 | \( 1 + (1.69 + 5.77i)T + (-31.1 + 20.0i)T^{2} \) |
| 41 | \( 1 + (1.55 - 5.29i)T + (-34.4 - 22.1i)T^{2} \) |
| 43 | \( 1 + (-4.87 + 0.700i)T + (41.2 - 12.1i)T^{2} \) |
| 47 | \( 1 + 1.42iT - 47T^{2} \) |
| 53 | \( 1 + (-9.48 - 8.21i)T + (7.54 + 52.4i)T^{2} \) |
| 59 | \( 1 + (-4.73 + 4.10i)T + (8.39 - 58.3i)T^{2} \) |
| 61 | \( 1 + (-0.654 + 4.55i)T + (-58.5 - 17.1i)T^{2} \) |
| 67 | \( 1 + (-2.14 + 3.34i)T + (-27.8 - 60.9i)T^{2} \) |
| 71 | \( 1 + (-4.34 - 2.79i)T + (29.4 + 64.5i)T^{2} \) |
| 73 | \( 1 + (2.71 - 1.23i)T + (47.8 - 55.1i)T^{2} \) |
| 79 | \( 1 + (3.80 - 3.29i)T + (11.2 - 78.1i)T^{2} \) |
| 83 | \( 1 + (2.35 - 0.690i)T + (69.8 - 44.8i)T^{2} \) |
| 89 | \( 1 + (-2.06 - 14.3i)T + (-85.3 + 25.0i)T^{2} \) |
| 97 | \( 1 + (7.20 + 2.11i)T + (81.6 + 52.4i)T^{2} \) |
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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
−10.73203268142047028128424108871, −10.01755266474443619820243347030, −8.890978337465645847861992440574, −8.310010712289936074069307353221, −6.38541233227266401607773886430, −5.95376564037909419510358243689, −5.42252087713098590473187721556, −3.88994284192999149407780798582, −2.37939121560396872746323399268, −0.47799221719707629902269284872,
1.86618936525924528962526557106, 3.62906728853340605097989705338, 4.45428087565044801816588265741, 5.62931117089188423076778154350, 6.85517187846442852869711926259, 7.40507361572254011305074869348, 8.690086009966126316485523182979, 9.551168079602718129561448491587, 10.30398621506826143228463064003, 11.36712038154994399450156682988
