| L(s) = 1 | + 0.732i·3-s − i·5-s + 2.73·7-s + 2.46·9-s + 2i·11-s + 3.46i·13-s + 0.732·15-s − 3.46·17-s − 7.46i·19-s + 2i·21-s − 4.19·23-s − 25-s + 4i·27-s − 6.92i·29-s − 1.46·31-s + ⋯ |
| L(s) = 1 | + 0.422i·3-s − 0.447i·5-s + 1.03·7-s + 0.821·9-s + 0.603i·11-s + 0.960i·13-s + 0.189·15-s − 0.840·17-s − 1.71i·19-s + 0.436i·21-s − 0.874·23-s − 0.200·25-s + 0.769i·27-s − 1.28i·29-s − 0.262·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.965 - 0.258i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 160 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.965 - 0.258i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.23634 + 0.162767i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.23634 + 0.162767i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + iT \) |
| good | 3 | \( 1 - 0.732iT - 3T^{2} \) |
| 7 | \( 1 - 2.73T + 7T^{2} \) |
| 11 | \( 1 - 2iT - 11T^{2} \) |
| 13 | \( 1 - 3.46iT - 13T^{2} \) |
| 17 | \( 1 + 3.46T + 17T^{2} \) |
| 19 | \( 1 + 7.46iT - 19T^{2} \) |
| 23 | \( 1 + 4.19T + 23T^{2} \) |
| 29 | \( 1 + 6.92iT - 29T^{2} \) |
| 31 | \( 1 + 1.46T + 31T^{2} \) |
| 37 | \( 1 + 2iT - 37T^{2} \) |
| 41 | \( 1 + 5.46T + 41T^{2} \) |
| 43 | \( 1 - 8.73iT - 43T^{2} \) |
| 47 | \( 1 + 6.73T + 47T^{2} \) |
| 53 | \( 1 - 4.53iT - 53T^{2} \) |
| 59 | \( 1 - 0.535iT - 59T^{2} \) |
| 61 | \( 1 - 4.92iT - 61T^{2} \) |
| 67 | \( 1 + 7.26iT - 67T^{2} \) |
| 71 | \( 1 - 1.46T + 71T^{2} \) |
| 73 | \( 1 - 0.535T + 73T^{2} \) |
| 79 | \( 1 - 14.9T + 79T^{2} \) |
| 83 | \( 1 + 4.73iT - 83T^{2} \) |
| 89 | \( 1 + 4.92T + 89T^{2} \) |
| 97 | \( 1 - 6.39T + 97T^{2} \) |
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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
−12.99261874947934116780120102815, −11.78484344171774851052503764376, −11.04621370999344565072059040098, −9.786769929919333392744217173543, −8.976072389013114517911942541056, −7.75994293805868588706149594218, −6.62208857896194916224533283059, −4.82806032479817054719576933332, −4.31977732382427068268194869951, −1.93937176132492931752819808343,
1.77346204202998070979589408106, 3.69548444200860688675306498822, 5.23635057618917991931870911009, 6.52102605241615628405615221240, 7.72398672422804204707184318497, 8.431501346196820914516609083895, 10.05799221630167176620421404623, 10.81119608004655359653125875504, 11.89500129536260901508899527712, 12.82879660579266998281532307671
