| L(s) = 1 | + (−0.604 − 0.744i)2-s + (1.49 − 1.28i)3-s + (0.221 − 1.05i)4-s + (−1.03 − 3.25i)5-s + (−1.86 − 0.329i)6-s + (−0.182 + 0.528i)7-s + (−2.62 + 1.35i)8-s + (0.126 − 0.874i)9-s + (−1.80 + 2.73i)10-s + (3.75 − 1.21i)11-s + (−1.03 − 1.86i)12-s + (3.71 − 1.25i)13-s + (0.503 − 0.184i)14-s + (−5.73 − 3.52i)15-s + (0.618 + 0.270i)16-s + (−6.48 − 3.24i)17-s + ⋯ |
| L(s) = 1 | + (−0.427 − 0.526i)2-s + (0.860 − 0.744i)3-s + (0.110 − 0.528i)4-s + (−0.461 − 1.45i)5-s + (−0.760 − 0.134i)6-s + (−0.0688 + 0.199i)7-s + (−0.928 + 0.478i)8-s + (0.0422 − 0.291i)9-s + (−0.569 + 0.865i)10-s + (1.13 − 0.366i)11-s + (−0.298 − 0.537i)12-s + (1.03 − 0.347i)13-s + (0.134 − 0.0492i)14-s + (−1.48 − 0.909i)15-s + (0.154 + 0.0677i)16-s + (−1.57 − 0.785i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 503 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.978 + 0.205i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 503 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.978 + 0.205i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.144552 - 1.39193i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.144552 - 1.39193i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 503 | \( 1 + (22.2 + 2.83i)T \) |
| good | 2 | \( 1 + (0.604 + 0.744i)T + (-0.410 + 1.95i)T^{2} \) |
| 3 | \( 1 + (-1.49 + 1.28i)T + (0.430 - 2.96i)T^{2} \) |
| 5 | \( 1 + (1.03 + 3.25i)T + (-4.08 + 2.87i)T^{2} \) |
| 7 | \( 1 + (0.182 - 0.528i)T + (-5.51 - 4.31i)T^{2} \) |
| 11 | \( 1 + (-3.75 + 1.21i)T + (8.91 - 6.44i)T^{2} \) |
| 13 | \( 1 + (-3.71 + 1.25i)T + (10.3 - 7.87i)T^{2} \) |
| 17 | \( 1 + (6.48 + 3.24i)T + (10.2 + 13.5i)T^{2} \) |
| 19 | \( 1 + (-2.74 - 6.15i)T + (-12.7 + 14.1i)T^{2} \) |
| 23 | \( 1 + (-1.88 + 6.51i)T + (-19.4 - 12.2i)T^{2} \) |
| 29 | \( 1 + (0.719 - 6.02i)T + (-28.1 - 6.83i)T^{2} \) |
| 31 | \( 1 + (-0.317 - 3.89i)T + (-30.5 + 5.02i)T^{2} \) |
| 37 | \( 1 + (4.87 + 6.15i)T + (-8.49 + 36.0i)T^{2} \) |
| 41 | \( 1 + (1.46 - 7.47i)T + (-37.9 - 15.5i)T^{2} \) |
| 43 | \( 1 + (-0.662 + 3.60i)T + (-40.1 - 15.2i)T^{2} \) |
| 47 | \( 1 + (-6.47 + 0.243i)T + (46.8 - 3.52i)T^{2} \) |
| 53 | \( 1 + (1.22 - 2.75i)T + (-35.4 - 39.4i)T^{2} \) |
| 59 | \( 1 + (-2.74 + 6.59i)T + (-41.5 - 41.8i)T^{2} \) |
| 61 | \( 1 + (5.03 + 8.31i)T + (-28.2 + 54.0i)T^{2} \) |
| 67 | \( 1 + (6.22 - 3.82i)T + (30.3 - 59.7i)T^{2} \) |
| 71 | \( 1 + (1.06 + 6.74i)T + (-67.5 + 21.8i)T^{2} \) |
| 73 | \( 1 + (-8.35 - 7.99i)T + (3.19 + 72.9i)T^{2} \) |
| 79 | \( 1 + (0.0817 + 1.44i)T + (-78.4 + 8.88i)T^{2} \) |
| 83 | \( 1 + (2.41 + 4.61i)T + (-47.3 + 68.1i)T^{2} \) |
| 89 | \( 1 + (3.52 + 13.4i)T + (-77.5 + 43.6i)T^{2} \) |
| 97 | \( 1 + (1.04 + 4.44i)T + (-86.7 + 43.3i)T^{2} \) |
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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
−10.56824204390511255207840689983, −9.081469179780521940217559926604, −8.901047727412200525263829000073, −8.284499278172695378533335899082, −6.96129434935882502656094674139, −5.87375279680265905218902734305, −4.69931604434030997073340517046, −3.31758993721736075365423611627, −1.83932618734125685955929705097, −0.933742361337726126832472300639,
2.58740936980015474530833213097, 3.67623861420691880863392795461, 4.05580583509971863031395322505, 6.34928944172372969025803049304, 6.85787287095881218297140968452, 7.69246546709551188759778539196, 8.913493337332971531454980887153, 9.140373033207133668277268356582, 10.31831554898491954170974165532, 11.36211548984914560047636856536
