| L(s) = 1 | + 1.56·2-s − 1.47·3-s − 5.56·4-s − 5·5-s − 2.31·6-s + 22.4·7-s − 21.1·8-s − 24.8·9-s − 7.80·10-s − 11·11-s + 8.23·12-s + 56.6·13-s + 35.0·14-s + 7.39·15-s + 11.4·16-s + 58.0·17-s − 38.7·18-s + 19·19-s + 27.8·20-s − 33.2·21-s − 17.1·22-s − 171.·23-s + 31.3·24-s + 25·25-s + 88.4·26-s + 76.6·27-s − 124.·28-s + ⋯ |
| L(s) = 1 | + 0.551·2-s − 0.284·3-s − 0.695·4-s − 0.447·5-s − 0.157·6-s + 1.21·7-s − 0.935·8-s − 0.918·9-s − 0.246·10-s − 0.301·11-s + 0.198·12-s + 1.20·13-s + 0.669·14-s + 0.127·15-s + 0.178·16-s + 0.828·17-s − 0.507·18-s + 0.229·19-s + 0.310·20-s − 0.345·21-s − 0.166·22-s − 1.55·23-s + 0.266·24-s + 0.200·25-s + 0.667·26-s + 0.546·27-s − 0.842·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + 5T \) |
| 11 | \( 1 + 11T \) |
| 19 | \( 1 - 19T \) |
| good | 2 | \( 1 - 1.56T + 8T^{2} \) |
| 3 | \( 1 + 1.47T + 27T^{2} \) |
| 7 | \( 1 - 22.4T + 343T^{2} \) |
| 13 | \( 1 - 56.6T + 2.19e3T^{2} \) |
| 17 | \( 1 - 58.0T + 4.91e3T^{2} \) |
| 23 | \( 1 + 171.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 236.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 311.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 342.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 54.6T + 6.89e4T^{2} \) |
| 43 | \( 1 + 30.8T + 7.95e4T^{2} \) |
| 47 | \( 1 + 452.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 2.82T + 1.48e5T^{2} \) |
| 59 | \( 1 + 177.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 44.8T + 2.26e5T^{2} \) |
| 67 | \( 1 + 602.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 503.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 650.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 1.35e3T + 4.93e5T^{2} \) |
| 83 | \( 1 - 583.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 449.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.59e3T + 9.12e5T^{2} \) |
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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
−8.896343575370696482966859840544, −8.256661009004753741350111771511, −7.74919156864906118384900309871, −6.16630578035398481924935875111, −5.60487344394633616100823766610, −4.74861549029500547576727292024, −3.92270803717426413255896735064, −2.94920533062799774515461353058, −1.31439434763604914879348110058, 0,
1.31439434763604914879348110058, 2.94920533062799774515461353058, 3.92270803717426413255896735064, 4.74861549029500547576727292024, 5.60487344394633616100823766610, 6.16630578035398481924935875111, 7.74919156864906118384900309871, 8.256661009004753741350111771511, 8.896343575370696482966859840544
