L(s) = 1 | + (−0.5 + 0.866i)3-s + (−1.5 − 2.59i)5-s + (−2 + 1.73i)7-s + (1 + 1.73i)9-s + (1.5 − 2.59i)11-s + 2·13-s + 3·15-s + (−1.5 + 2.59i)17-s + (0.5 + 0.866i)19-s + (−0.499 − 2.59i)21-s + (−1.5 − 2.59i)23-s + (−2 + 3.46i)25-s − 5·27-s − 6·29-s + (3.5 − 6.06i)31-s + ⋯ |
L(s) = 1 | + (−0.288 + 0.499i)3-s + (−0.670 − 1.16i)5-s + (−0.755 + 0.654i)7-s + (0.333 + 0.577i)9-s + (0.452 − 0.783i)11-s + 0.554·13-s + 0.774·15-s + (−0.363 + 0.630i)17-s + (0.114 + 0.198i)19-s + (−0.109 − 0.566i)21-s + (−0.312 − 0.541i)23-s + (−0.400 + 0.692i)25-s − 0.962·27-s − 1.11·29-s + (0.628 − 1.08i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 28 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.991 - 0.126i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 28 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.991 - 0.126i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.597311 + 0.0379054i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.597311 + 0.0379054i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 + (2 - 1.73i)T \) |
good | 3 | \( 1 + (0.5 - 0.866i)T + (-1.5 - 2.59i)T^{2} \) |
| 5 | \( 1 + (1.5 + 2.59i)T + (-2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (-1.5 + 2.59i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 - 2T + 13T^{2} \) |
| 17 | \( 1 + (1.5 - 2.59i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-0.5 - 0.866i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (1.5 + 2.59i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + 6T + 29T^{2} \) |
| 31 | \( 1 + (-3.5 + 6.06i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-0.5 - 0.866i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 - 6T + 41T^{2} \) |
| 43 | \( 1 + 4T + 43T^{2} \) |
| 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 | \( 1 + (4.5 - 7.79i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-0.5 - 0.866i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-3.5 + 6.06i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 + (-0.5 + 0.866i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-6.5 - 11.2i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 - 12T + 83T^{2} \) |
| 89 | \( 1 + (7.5 + 12.9i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + 10T + 97T^{2} \) |
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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
−16.83240693466222331245255824153, −16.22593751048373275065867264340, −15.34293187880635901845287645069, −13.40005263692909520970142355282, −12.33996226575045532080751057895, −11.04513563684512874615408261657, −9.378231197034825499779622084931, −8.202084321555248126345415607740, −5.88053399239330996514645357185, −4.16290105624966053923484697414,
3.68820621838161426787695488127, 6.59765781472494882013944777804, 7.32269192359830248395197349962, 9.616510756305673496977057334962, 11.02460749959139304874986304333, 12.20439698718194328052721890579, 13.54814894465822888147983726342, 14.93563706892529383383121625487, 15.95920936655884041627259306018, 17.52539665179351686539607658141