| L(s) = 1 | + (−0.432 + 2.19i)5-s + 2.86·11-s − i·13-s + 5.52i·17-s − 3.52·19-s − 7.52i·23-s + (−4.62 − 1.89i)25-s + 6.77·29-s + 5.72·31-s + 3.72i·37-s + 10.1·41-s − 5.52i·43-s + 8.65i·47-s + 7·49-s + 6.77i·53-s + ⋯ |
| L(s) = 1 | + (−0.193 + 0.981i)5-s + 0.863·11-s − 0.277i·13-s + 1.33i·17-s − 0.808·19-s − 1.56i·23-s + (−0.925 − 0.379i)25-s + 1.25·29-s + 1.02·31-s + 0.613i·37-s + 1.58·41-s − 0.842i·43-s + 1.26i·47-s + 49-s + 0.930i·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.193 - 0.981i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4680 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.193 - 0.981i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.801061854\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.801061854\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (0.432 - 2.19i)T \) |
| 13 | \( 1 + iT \) |
| good | 7 | \( 1 - 7T^{2} \) |
| 11 | \( 1 - 2.86T + 11T^{2} \) |
| 17 | \( 1 - 5.52iT - 17T^{2} \) |
| 19 | \( 1 + 3.52T + 19T^{2} \) |
| 23 | \( 1 + 7.52iT - 23T^{2} \) |
| 29 | \( 1 - 6.77T + 29T^{2} \) |
| 31 | \( 1 - 5.72T + 31T^{2} \) |
| 37 | \( 1 - 3.72iT - 37T^{2} \) |
| 41 | \( 1 - 10.1T + 41T^{2} \) |
| 43 | \( 1 + 5.52iT - 43T^{2} \) |
| 47 | \( 1 - 8.65iT - 47T^{2} \) |
| 53 | \( 1 - 6.77iT - 53T^{2} \) |
| 59 | \( 1 - 0.593T + 59T^{2} \) |
| 61 | \( 1 + 5.25T + 61T^{2} \) |
| 67 | \( 1 - 10.5iT - 67T^{2} \) |
| 71 | \( 1 - 2.38T + 71T^{2} \) |
| 73 | \( 1 - 5.45iT - 73T^{2} \) |
| 79 | \( 1 + 2.47T + 79T^{2} \) |
| 83 | \( 1 + 8.11iT - 83T^{2} \) |
| 89 | \( 1 + 14.1T + 89T^{2} \) |
| 97 | \( 1 + 6iT - 97T^{2} \) |
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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
−8.425120743073341628868273339490, −7.79400609123136420726740042124, −6.83739152639280979666889543239, −6.35304221367034421979273785517, −5.85481012593241506975413612539, −4.38394205765901716660828495813, −4.13842918912213600175080120516, −2.97956226764813729100427102126, −2.33132382598715334318289327529, −1.03386278589994126815067191048,
0.59243593757709679059700618896, 1.54079912051774152703859307490, 2.65458032746447290560620701111, 3.75685407349228408381358467755, 4.43441508361631149398344838610, 5.09840545324127403708726290904, 5.93020585797601907393633195346, 6.72220424177298598955374694829, 7.47260936783346338712087983172, 8.200723662411147341530368792946
