| L(s) = 1 | − i·2-s − 4-s + (1.80 + 1.93i)7-s + i·8-s − 3.87i·11-s − 1.60i·13-s + (1.93 − 1.80i)14-s + 16-s − 8.11·17-s − 2.63i·19-s − 3.87·22-s + 5.47i·23-s − 1.60·26-s + (−1.80 − 1.93i)28-s − 5.47i·29-s + ⋯ |
| L(s) = 1 | − 0.707i·2-s − 0.5·4-s + (0.681 + 0.732i)7-s + 0.353i·8-s − 1.16i·11-s − 0.445i·13-s + (0.517 − 0.481i)14-s + 0.250·16-s − 1.96·17-s − 0.605i·19-s − 0.825·22-s + 1.14i·23-s − 0.314·26-s + (−0.340 − 0.366i)28-s − 1.01i·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.978 - 0.204i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.978 - 0.204i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5928019746\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5928019746\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (-1.80 - 1.93i)T \) |
| good | 11 | \( 1 + 3.87iT - 11T^{2} \) |
| 13 | \( 1 + 1.60iT - 13T^{2} \) |
| 17 | \( 1 + 8.11T + 17T^{2} \) |
| 19 | \( 1 + 2.63iT - 19T^{2} \) |
| 23 | \( 1 - 5.47iT - 23T^{2} \) |
| 29 | \( 1 + 5.47iT - 29T^{2} \) |
| 31 | \( 1 - 3.73iT - 31T^{2} \) |
| 37 | \( 1 + 4.51T + 37T^{2} \) |
| 41 | \( 1 + 1.60T + 41T^{2} \) |
| 43 | \( 1 - 10.1T + 43T^{2} \) |
| 47 | \( 1 + 11.1T + 47T^{2} \) |
| 53 | \( 1 + 2.26iT - 53T^{2} \) |
| 59 | \( 1 - 4.61T + 59T^{2} \) |
| 61 | \( 1 + 11.8iT - 61T^{2} \) |
| 67 | \( 1 + 6.90T + 67T^{2} \) |
| 71 | \( 1 + 2.63iT - 71T^{2} \) |
| 73 | \( 1 + 13.7iT - 73T^{2} \) |
| 79 | \( 1 + 8.01T + 79T^{2} \) |
| 83 | \( 1 + 3.20T + 83T^{2} \) |
| 89 | \( 1 + 17.8T + 89T^{2} \) |
| 97 | \( 1 - 8.68iT - 97T^{2} \) |
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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
−8.497751271920286767905458231942, −7.79869104699280922849865508559, −6.69557778184902040666056458348, −5.85857800286351492991808867492, −5.12124781545003139721157885657, −4.36304818353270871408320408356, −3.32997362348800033108465860902, −2.51523141336406952461333943888, −1.60369212421406493496746736545, −0.17738316155944236902699729960,
1.48159398043644526969639088222, 2.47447128797894892784832642815, 4.08469112682378284418297515869, 4.38988686497188581273259350441, 5.16006982357308299414547106856, 6.25679117146348184913465837860, 7.01505414955916984231233224994, 7.32230419111969905971812361048, 8.393900552898553229663129321660, 8.808520902915412005997042444467
