L(s) = 1 | + 2-s − 2.97·3-s + 4-s + 2.57·5-s − 2.97·6-s − 1.79·7-s + 8-s + 5.85·9-s + 2.57·10-s + 0.0411·11-s − 2.97·12-s − 1.04·13-s − 1.79·14-s − 7.66·15-s + 16-s − 5.67·17-s + 5.85·18-s + 5.59·19-s + 2.57·20-s + 5.33·21-s + 0.0411·22-s − 0.350·23-s − 2.97·24-s + 1.63·25-s − 1.04·26-s − 8.51·27-s − 1.79·28-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 1.71·3-s + 0.5·4-s + 1.15·5-s − 1.21·6-s − 0.677·7-s + 0.353·8-s + 1.95·9-s + 0.814·10-s + 0.0124·11-s − 0.859·12-s − 0.289·13-s − 0.478·14-s − 1.97·15-s + 0.250·16-s − 1.37·17-s + 1.38·18-s + 1.28·19-s + 0.575·20-s + 1.16·21-s + 0.00877·22-s − 0.0730·23-s − 0.607·24-s + 0.326·25-s − 0.204·26-s − 1.63·27-s − 0.338·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6002 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6002 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 3001 | \( 1+O(T) \) |
good | 3 | \( 1 + 2.97T + 3T^{2} \) |
| 5 | \( 1 - 2.57T + 5T^{2} \) |
| 7 | \( 1 + 1.79T + 7T^{2} \) |
| 11 | \( 1 - 0.0411T + 11T^{2} \) |
| 13 | \( 1 + 1.04T + 13T^{2} \) |
| 17 | \( 1 + 5.67T + 17T^{2} \) |
| 19 | \( 1 - 5.59T + 19T^{2} \) |
| 23 | \( 1 + 0.350T + 23T^{2} \) |
| 29 | \( 1 + 3.42T + 29T^{2} \) |
| 31 | \( 1 - 7.47T + 31T^{2} \) |
| 37 | \( 1 + 5.61T + 37T^{2} \) |
| 41 | \( 1 + 12.5T + 41T^{2} \) |
| 43 | \( 1 + 5.40T + 43T^{2} \) |
| 47 | \( 1 - 2.43T + 47T^{2} \) |
| 53 | \( 1 - 3.49T + 53T^{2} \) |
| 59 | \( 1 - 13.9T + 59T^{2} \) |
| 61 | \( 1 - 5.69T + 61T^{2} \) |
| 67 | \( 1 + 9.62T + 67T^{2} \) |
| 71 | \( 1 - 0.818T + 71T^{2} \) |
| 73 | \( 1 + 6.98T + 73T^{2} \) |
| 79 | \( 1 - 4.35T + 79T^{2} \) |
| 83 | \( 1 + 12.2T + 83T^{2} \) |
| 89 | \( 1 - 7.18T + 89T^{2} \) |
| 97 | \( 1 + 3.10T + 97T^{2} \) |
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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
−7.06485746868312668251536589117, −6.81834270523486610185079540971, −6.17901089824706666321001712967, −5.52566015265425319468569255917, −5.11952899074279691053949232026, −4.36535451719476082454649007293, −3.33122002421468873120704666623, −2.24685051138600368026330162778, −1.32832596294870633868823139308, 0,
1.32832596294870633868823139308, 2.24685051138600368026330162778, 3.33122002421468873120704666623, 4.36535451719476082454649007293, 5.11952899074279691053949232026, 5.52566015265425319468569255917, 6.17901089824706666321001712967, 6.81834270523486610185079540971, 7.06485746868312668251536589117