| L(s) = 1 | + 1.41·2-s + 1.73i·3-s + 2.00·4-s + 2.44i·6-s + (6.91 − 1.07i)7-s + 2.82·8-s − 2.99·9-s − 9.92·11-s + 3.46i·12-s + 20.2i·13-s + (9.78 − 1.51i)14-s + 4.00·16-s + 18.9i·17-s − 4.24·18-s − 4.68i·19-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577i·3-s + 0.500·4-s + 0.408i·6-s + (0.988 − 0.152i)7-s + 0.353·8-s − 0.333·9-s − 0.902·11-s + 0.288i·12-s + 1.55i·13-s + (0.698 − 0.108i)14-s + 0.250·16-s + 1.11i·17-s − 0.235·18-s − 0.246i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.152 - 0.988i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.152 - 0.988i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(2.824636112\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.824636112\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 1.41T \) |
| 3 | \( 1 - 1.73iT \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (-6.91 + 1.07i)T \) |
| good | 11 | \( 1 + 9.92T + 121T^{2} \) |
| 13 | \( 1 - 20.2iT - 169T^{2} \) |
| 17 | \( 1 - 18.9iT - 289T^{2} \) |
| 19 | \( 1 + 4.68iT - 361T^{2} \) |
| 23 | \( 1 + 25.4T + 529T^{2} \) |
| 29 | \( 1 - 15.9T + 841T^{2} \) |
| 31 | \( 1 - 20.9iT - 961T^{2} \) |
| 37 | \( 1 - 30.1T + 1.36e3T^{2} \) |
| 41 | \( 1 - 42.2iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 49.7T + 1.84e3T^{2} \) |
| 47 | \( 1 + 2.91iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 11.1T + 2.80e3T^{2} \) |
| 59 | \( 1 - 63.6iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 89.2iT - 3.72e3T^{2} \) |
| 67 | \( 1 - 13.5T + 4.48e3T^{2} \) |
| 71 | \( 1 + 50.9T + 5.04e3T^{2} \) |
| 73 | \( 1 + 78.9iT - 5.32e3T^{2} \) |
| 79 | \( 1 + 69.8T + 6.24e3T^{2} \) |
| 83 | \( 1 + 68.8iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 156. iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 154. iT - 9.40e3T^{2} \) |
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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
−10.23158179240501935221337799352, −9.083734086259469157315026680717, −8.247198490588922712332820394822, −7.46167637705270615300046276297, −6.35625073214519108521161437774, −5.52562674230672852877374615187, −4.47395193461909654874474672694, −4.13784780717820482452434452240, −2.68803048836779401122687116045, −1.61592324063940631463221896070,
0.66573400045749296840051128660, 2.17293704038846049592568220629, 2.95464747770672959746654994923, 4.30763801159353768194897012943, 5.37053205976598256710563948777, 5.73995610708240810136128450837, 7.02537058687598433166501551861, 7.967868721597453540883068803082, 8.118617756111175483749093829938, 9.593490623668793724814393325810
