L(s) = 1 | + 2.93·5-s + 4.82i·11-s + 2.93i·13-s − 7.07·17-s − 5.86i·19-s − 2i·23-s + 3.58·25-s + 0.828i·29-s + 5.86i·31-s − 5.41·37-s − 1.21·41-s − 4.48·43-s − 5.86·47-s + 7.07i·53-s + 14.1i·55-s + ⋯ |
L(s) = 1 | + 1.31·5-s + 1.45i·11-s + 0.812i·13-s − 1.71·17-s − 1.34i·19-s − 0.417i·23-s + 0.717·25-s + 0.153i·29-s + 1.05i·31-s − 0.890·37-s − 0.189·41-s − 0.683·43-s − 0.854·47-s + 0.971i·53-s + 1.90i·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7056 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.860 - 0.508i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7056 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.860 - 0.508i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.062968064\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.062968064\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
good | 5 | \( 1 - 2.93T + 5T^{2} \) |
| 11 | \( 1 - 4.82iT - 11T^{2} \) |
| 13 | \( 1 - 2.93iT - 13T^{2} \) |
| 17 | \( 1 + 7.07T + 17T^{2} \) |
| 19 | \( 1 + 5.86iT - 19T^{2} \) |
| 23 | \( 1 + 2iT - 23T^{2} \) |
| 29 | \( 1 - 0.828iT - 29T^{2} \) |
| 31 | \( 1 - 5.86iT - 31T^{2} \) |
| 37 | \( 1 + 5.41T + 37T^{2} \) |
| 41 | \( 1 + 1.21T + 41T^{2} \) |
| 43 | \( 1 + 4.48T + 43T^{2} \) |
| 47 | \( 1 + 5.86T + 47T^{2} \) |
| 53 | \( 1 - 7.07iT - 53T^{2} \) |
| 59 | \( 1 + 5.86T + 59T^{2} \) |
| 61 | \( 1 - 1.21iT - 61T^{2} \) |
| 67 | \( 1 - 8.48T + 67T^{2} \) |
| 71 | \( 1 - 0.828iT - 71T^{2} \) |
| 73 | \( 1 - 7.07iT - 73T^{2} \) |
| 79 | \( 1 + 1.65T + 79T^{2} \) |
| 83 | \( 1 - 11.7T + 83T^{2} \) |
| 89 | \( 1 + 11.2T + 89T^{2} \) |
| 97 | \( 1 + 7.07iT - 97T^{2} \) |
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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
−8.414701539781055863048404028743, −7.23520391685886007434347205969, −6.69712527399416536089036999644, −6.44363736333176773157916560369, −5.21825490896335843960177299660, −4.80327709205191955623009993370, −4.11361385221057290597896058277, −2.77269385001125963418677357324, −2.12488819413898098326702039720, −1.52751249273918298430174049297,
0.22121468390301595157909422831, 1.52579825895698779457666885764, 2.25977559065271480881644531148, 3.19069248639252601088189347410, 3.91352171300998701677698260003, 5.05769339439136114007281728160, 5.61453436421438465226649151862, 6.22501990936437933274499245487, 6.65917136763507031112512928275, 7.82634831998153547132515519601