| L(s) = 1 | + i·2-s − i·3-s − 4-s + 0.516i·5-s + 6-s − i·8-s − 9-s − 0.516·10-s + (−3.09 − 1.19i)11-s + i·12-s + 2.41·13-s + 0.516·15-s + 16-s − 7.29·17-s − i·18-s + 2.57·19-s + ⋯ |
| L(s) = 1 | + 0.707i·2-s − 0.577i·3-s − 0.5·4-s + 0.230i·5-s + 0.408·6-s − 0.353i·8-s − 0.333·9-s − 0.163·10-s + (−0.933 − 0.359i)11-s + 0.288i·12-s + 0.669·13-s + 0.133·15-s + 0.250·16-s − 1.77·17-s − 0.235i·18-s + 0.590·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3234 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.500 - 0.865i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3234 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.500 - 0.865i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.367848777\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.367848777\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - iT \) |
| 3 | \( 1 + iT \) |
| 7 | \( 1 \) |
| 11 | \( 1 + (3.09 + 1.19i)T \) |
| good | 5 | \( 1 - 0.516iT - 5T^{2} \) |
| 13 | \( 1 - 2.41T + 13T^{2} \) |
| 17 | \( 1 + 7.29T + 17T^{2} \) |
| 19 | \( 1 - 2.57T + 19T^{2} \) |
| 23 | \( 1 + 1.88T + 23T^{2} \) |
| 29 | \( 1 - 1.33iT - 29T^{2} \) |
| 31 | \( 1 - 4.58iT - 31T^{2} \) |
| 37 | \( 1 - 6.34T + 37T^{2} \) |
| 41 | \( 1 - 2.14T + 41T^{2} \) |
| 43 | \( 1 - 2.68iT - 43T^{2} \) |
| 47 | \( 1 - 1.14iT - 47T^{2} \) |
| 53 | \( 1 - 7.15T + 53T^{2} \) |
| 59 | \( 1 + 1.00iT - 59T^{2} \) |
| 61 | \( 1 - 4.50T + 61T^{2} \) |
| 67 | \( 1 - 8.00T + 67T^{2} \) |
| 71 | \( 1 - 9.05T + 71T^{2} \) |
| 73 | \( 1 + 5.82T + 73T^{2} \) |
| 79 | \( 1 - 17.5iT - 79T^{2} \) |
| 83 | \( 1 - 3.86T + 83T^{2} \) |
| 89 | \( 1 + 14.3iT - 89T^{2} \) |
| 97 | \( 1 + 3.02iT - 97T^{2} \) |
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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
−8.540878758586712651637199302651, −8.067672862924088279969060318997, −7.14262967293765464192978103714, −6.65871107565710973885923922883, −5.89546594288855183020243005018, −5.13868857635715326619190740275, −4.25893463865468192183605421085, −3.16077184496987195155068436342, −2.26089639709259051485464953989, −0.841213349418739325380445296691,
0.56901665270509346579560801092, 2.07163043403455673218670051800, 2.80140759654342243613704945571, 3.89740522996909566855335478328, 4.51723338220567280937040990203, 5.25934525857094711699331345860, 6.09977450548123933135862368337, 7.08420141979506378651895526900, 8.069667353785940508936559410522, 8.665254500451587736687257240408
