| L(s) = 1 | + 2-s − 2.69·3-s + 4-s − 1.67·5-s − 2.69·6-s − 4.07·7-s + 8-s + 4.27·9-s − 1.67·10-s − 3.51·11-s − 2.69·12-s − 2.79·13-s − 4.07·14-s + 4.53·15-s + 16-s + 1.92·17-s + 4.27·18-s + 4.48·19-s − 1.67·20-s + 10.9·21-s − 3.51·22-s + 6.38·23-s − 2.69·24-s − 2.18·25-s − 2.79·26-s − 3.44·27-s − 4.07·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.55·3-s + 0.5·4-s − 0.750·5-s − 1.10·6-s − 1.53·7-s + 0.353·8-s + 1.42·9-s − 0.531·10-s − 1.05·11-s − 0.778·12-s − 0.775·13-s − 1.08·14-s + 1.16·15-s + 0.250·16-s + 0.465·17-s + 1.00·18-s + 1.02·19-s − 0.375·20-s + 2.39·21-s − 0.748·22-s + 1.33·23-s − 0.550·24-s − 0.436·25-s − 0.548·26-s − 0.663·27-s − 0.769·28-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 | \( 1 + 2.69T + 3T^{2} \) |
| 5 | \( 1 + 1.67T + 5T^{2} \) |
| 7 | \( 1 + 4.07T + 7T^{2} \) |
| 11 | \( 1 + 3.51T + 11T^{2} \) |
| 13 | \( 1 + 2.79T + 13T^{2} \) |
| 17 | \( 1 - 1.92T + 17T^{2} \) |
| 19 | \( 1 - 4.48T + 19T^{2} \) |
| 23 | \( 1 - 6.38T + 23T^{2} \) |
| 29 | \( 1 - 0.656T + 29T^{2} \) |
| 31 | \( 1 - 8.95T + 31T^{2} \) |
| 37 | \( 1 - 0.377T + 37T^{2} \) |
| 41 | \( 1 - 4.72T + 41T^{2} \) |
| 43 | \( 1 + 0.829T + 43T^{2} \) |
| 47 | \( 1 + 0.0503T + 47T^{2} \) |
| 53 | \( 1 + 8.31T + 53T^{2} \) |
| 59 | \( 1 + 5.79T + 59T^{2} \) |
| 61 | \( 1 + 5.18T + 61T^{2} \) |
| 67 | \( 1 - 11.7T + 67T^{2} \) |
| 71 | \( 1 - 5.28T + 71T^{2} \) |
| 73 | \( 1 - 3.93T + 73T^{2} \) |
| 79 | \( 1 - 1.66T + 79T^{2} \) |
| 83 | \( 1 + 6.93T + 83T^{2} \) |
| 89 | \( 1 - 11.6T + 89T^{2} \) |
| 97 | \( 1 + 17.6T + 97T^{2} \) |
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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
−7.44843950279409367413699535839, −6.78515858085407725854079066439, −6.26871515888270572521548325554, −5.45985466460222512549200399267, −5.02568484205204422724159947487, −4.24845068120534193326705267311, −3.23057884621205225382547581242, −2.72116496425743158426190524630, −0.927895768099122487047609069778, 0,
0.927895768099122487047609069778, 2.72116496425743158426190524630, 3.23057884621205225382547581242, 4.24845068120534193326705267311, 5.02568484205204422724159947487, 5.45985466460222512549200399267, 6.26871515888270572521548325554, 6.78515858085407725854079066439, 7.44843950279409367413699535839
