| L(s) = 1 | − 1.18·2-s − 0.607·4-s + 4.43·5-s − 7-s + 3.07·8-s − 5.23·10-s − 5.41·11-s − 2.38·13-s + 1.18·14-s − 2.41·16-s + 2.36·17-s − 2.75·19-s − 2.69·20-s + 6.39·22-s − 5.73·23-s + 14.6·25-s + 2.81·26-s + 0.607·28-s − 3.90·29-s + 2.83·31-s − 3.30·32-s − 2.78·34-s − 4.43·35-s − 3.58·37-s + 3.25·38-s + 13.6·40-s − 0.707·41-s + ⋯ |
| L(s) = 1 | − 0.834·2-s − 0.303·4-s + 1.98·5-s − 0.377·7-s + 1.08·8-s − 1.65·10-s − 1.63·11-s − 0.661·13-s + 0.315·14-s − 0.604·16-s + 0.572·17-s − 0.632·19-s − 0.602·20-s + 1.36·22-s − 1.19·23-s + 2.93·25-s + 0.551·26-s + 0.114·28-s − 0.725·29-s + 0.509·31-s − 0.583·32-s − 0.477·34-s − 0.750·35-s − 0.589·37-s + 0.527·38-s + 2.15·40-s − 0.110·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8001 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8001 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 7 | \( 1 + T \) |
| 127 | \( 1 + T \) |
| good | 2 | \( 1 + 1.18T + 2T^{2} \) |
| 5 | \( 1 - 4.43T + 5T^{2} \) |
| 11 | \( 1 + 5.41T + 11T^{2} \) |
| 13 | \( 1 + 2.38T + 13T^{2} \) |
| 17 | \( 1 - 2.36T + 17T^{2} \) |
| 19 | \( 1 + 2.75T + 19T^{2} \) |
| 23 | \( 1 + 5.73T + 23T^{2} \) |
| 29 | \( 1 + 3.90T + 29T^{2} \) |
| 31 | \( 1 - 2.83T + 31T^{2} \) |
| 37 | \( 1 + 3.58T + 37T^{2} \) |
| 41 | \( 1 + 0.707T + 41T^{2} \) |
| 43 | \( 1 - 9.07T + 43T^{2} \) |
| 47 | \( 1 - 4.92T + 47T^{2} \) |
| 53 | \( 1 - 10.8T + 53T^{2} \) |
| 59 | \( 1 - 8.70T + 59T^{2} \) |
| 61 | \( 1 - 15.4T + 61T^{2} \) |
| 67 | \( 1 - 5.37T + 67T^{2} \) |
| 71 | \( 1 + 8.79T + 71T^{2} \) |
| 73 | \( 1 + 14.5T + 73T^{2} \) |
| 79 | \( 1 + 13.5T + 79T^{2} \) |
| 83 | \( 1 + 5.85T + 83T^{2} \) |
| 89 | \( 1 - 5.77T + 89T^{2} \) |
| 97 | \( 1 - 13.8T + 97T^{2} \) |
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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
−7.50582328821994848720363834019, −6.98326093595861854495764693752, −5.90124016050090547710334333033, −5.54710185984756811595306201402, −4.93548756603028870646867939392, −3.94837432797236099325285836928, −2.51323680077382289971597742197, −2.32440059651922780510710434731, −1.20377590485720792860619938233, 0,
1.20377590485720792860619938233, 2.32440059651922780510710434731, 2.51323680077382289971597742197, 3.94837432797236099325285836928, 4.93548756603028870646867939392, 5.54710185984756811595306201402, 5.90124016050090547710334333033, 6.98326093595861854495764693752, 7.50582328821994848720363834019
