| L(s) = 1 | + 3-s − 2.71·5-s + 5.14·7-s + 9-s − 1.40·11-s + 2.82·13-s − 2.71·15-s + 0.388·17-s − 2.47·19-s + 5.14·21-s − 2.05·23-s + 2.35·25-s + 27-s − 9.76·29-s − 5.79·31-s − 1.40·33-s − 13.9·35-s − 6.17·37-s + 2.82·39-s − 4.11·41-s − 8.58·43-s − 2.71·45-s + 8.78·47-s + 19.4·49-s + 0.388·51-s − 2.59·53-s + 3.81·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 1.21·5-s + 1.94·7-s + 0.333·9-s − 0.423·11-s + 0.783·13-s − 0.700·15-s + 0.0943·17-s − 0.568·19-s + 1.12·21-s − 0.429·23-s + 0.471·25-s + 0.192·27-s − 1.81·29-s − 1.04·31-s − 0.244·33-s − 2.35·35-s − 1.01·37-s + 0.452·39-s − 0.643·41-s − 1.30·43-s − 0.404·45-s + 1.28·47-s + 2.77·49-s + 0.0544·51-s − 0.356·53-s + 0.514·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8016 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8016 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 - T \) |
| 167 | \( 1 + T \) |
| good | 5 | \( 1 + 2.71T + 5T^{2} \) |
| 7 | \( 1 - 5.14T + 7T^{2} \) |
| 11 | \( 1 + 1.40T + 11T^{2} \) |
| 13 | \( 1 - 2.82T + 13T^{2} \) |
| 17 | \( 1 - 0.388T + 17T^{2} \) |
| 19 | \( 1 + 2.47T + 19T^{2} \) |
| 23 | \( 1 + 2.05T + 23T^{2} \) |
| 29 | \( 1 + 9.76T + 29T^{2} \) |
| 31 | \( 1 + 5.79T + 31T^{2} \) |
| 37 | \( 1 + 6.17T + 37T^{2} \) |
| 41 | \( 1 + 4.11T + 41T^{2} \) |
| 43 | \( 1 + 8.58T + 43T^{2} \) |
| 47 | \( 1 - 8.78T + 47T^{2} \) |
| 53 | \( 1 + 2.59T + 53T^{2} \) |
| 59 | \( 1 + 13.6T + 59T^{2} \) |
| 61 | \( 1 + 4.03T + 61T^{2} \) |
| 67 | \( 1 + 11.5T + 67T^{2} \) |
| 71 | \( 1 + 16.1T + 71T^{2} \) |
| 73 | \( 1 - 10.4T + 73T^{2} \) |
| 79 | \( 1 + 7.27T + 79T^{2} \) |
| 83 | \( 1 - 11.5T + 83T^{2} \) |
| 89 | \( 1 - 1.50T + 89T^{2} \) |
| 97 | \( 1 - 6.01T + 97T^{2} \) |
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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
−7.64201145429801806456522604821, −7.24511408860041109362849569424, −6.02710310383599077307773730332, −5.22073758297088365103750733821, −4.54699432783245403158824405385, −3.90662942767569729932205725655, −3.31230503834023189195231836984, −2.01484290660410208911895311810, −1.51439200578073031517558226092, 0,
1.51439200578073031517558226092, 2.01484290660410208911895311810, 3.31230503834023189195231836984, 3.90662942767569729932205725655, 4.54699432783245403158824405385, 5.22073758297088365103750733821, 6.02710310383599077307773730332, 7.24511408860041109362849569424, 7.64201145429801806456522604821
