L(s) = 1 | + (1 − 1.73i)3-s + (−0.5 − 0.866i)5-s + (−0.499 − 0.866i)9-s + (2 − 3.46i)11-s + 13-s − 1.99·15-s + (1 − 1.73i)17-s + (0.5 + 0.866i)19-s + (3.5 + 6.06i)23-s + (2 − 3.46i)25-s + 4.00·27-s − 5·29-s + (4.5 − 7.79i)31-s + (−3.99 − 6.92i)33-s + (1 + 1.73i)37-s + ⋯ |
L(s) = 1 | + (0.577 − 0.999i)3-s + (−0.223 − 0.387i)5-s + (−0.166 − 0.288i)9-s + (0.603 − 1.04i)11-s + 0.277·13-s − 0.516·15-s + (0.242 − 0.420i)17-s + (0.114 + 0.198i)19-s + (0.729 + 1.26i)23-s + (0.400 − 0.692i)25-s + 0.769·27-s − 0.928·29-s + (0.808 − 1.39i)31-s + (−0.696 − 1.20i)33-s + (0.164 + 0.284i)37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2548 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.386 + 0.922i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2548 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.386 + 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.240260509\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.240260509\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
| 13 | \( 1 - T \) |
good | 3 | \( 1 + (-1 + 1.73i)T + (-1.5 - 2.59i)T^{2} \) |
| 5 | \( 1 + (0.5 + 0.866i)T + (-2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (-2 + 3.46i)T + (-5.5 - 9.52i)T^{2} \) |
| 17 | \( 1 + (-1 + 1.73i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-0.5 - 0.866i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-3.5 - 6.06i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + 5T + 29T^{2} \) |
| 31 | \( 1 + (-4.5 + 7.79i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-1 - 1.73i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 - 2T + 41T^{2} \) |
| 43 | \( 1 - T + 43T^{2} \) |
| 47 | \( 1 + (4.5 + 7.79i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (1.5 - 2.59i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (7 + 12.1i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (5 - 8.66i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 14T + 71T^{2} \) |
| 73 | \( 1 + (1.5 - 2.59i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (2.5 + 4.33i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 - 5T + 83T^{2} \) |
| 89 | \( 1 + (-4.5 - 7.79i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + T + 97T^{2} \) |
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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
−8.576557333304831194821038880185, −7.86503275641902387232717258353, −7.33172431260927536375271535051, −6.41395353529576368830825707878, −5.71998758098083371334693730987, −4.67166847417759882508252958057, −3.62155780516508405477539417710, −2.82659703759217071234806557988, −1.65098321519603733686082269949, −0.75395473796557944257600146978,
1.40487108238202274894544768303, 2.78669386246033289589965643685, 3.46543002298796179756053140952, 4.38152068970085890237994398879, 4.85481365142780101930846655449, 6.12892221193525003179226695600, 6.91388535701494826473542892400, 7.59615207306682079744560021213, 8.680865787335957949403779501525, 9.092496202876055546141401646199