L(s) = 1 | + (−0.5 − 0.866i)3-s + (−0.380 − 0.219i)5-s + (2.02 + 1.70i)7-s + (−0.499 + 0.866i)9-s + (−1.83 + 1.05i)11-s + 3.84i·13-s + 0.438i·15-s + (−4.89 + 2.82i)17-s + (1.48 − 2.57i)19-s + (0.464 − 2.60i)21-s + (−4.13 − 2.38i)23-s + (−2.40 − 4.16i)25-s + 0.999·27-s − 7.02·29-s + (−3.71 − 6.43i)31-s + ⋯ |
L(s) = 1 | + (−0.288 − 0.499i)3-s + (−0.170 − 0.0981i)5-s + (0.764 + 0.644i)7-s + (−0.166 + 0.288i)9-s + (−0.552 + 0.318i)11-s + 1.06i·13-s + 0.113i·15-s + (−1.18 + 0.684i)17-s + (0.341 − 0.591i)19-s + (0.101 − 0.568i)21-s + (−0.861 − 0.497i)23-s + (−0.480 − 0.832i)25-s + 0.192·27-s − 1.30·29-s + (−0.666 − 1.15i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.677 - 0.735i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.677 - 0.735i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.4989318195\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4989318195\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (0.5 + 0.866i)T \) |
| 7 | \( 1 + (-2.02 - 1.70i)T \) |
good | 5 | \( 1 + (0.380 + 0.219i)T + (2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (1.83 - 1.05i)T + (5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 - 3.84iT - 13T^{2} \) |
| 17 | \( 1 + (4.89 - 2.82i)T + (8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-1.48 + 2.57i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (4.13 + 2.38i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + 7.02T + 29T^{2} \) |
| 31 | \( 1 + (3.71 + 6.43i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (2.64 - 4.57i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 - 6.81iT - 41T^{2} \) |
| 43 | \( 1 - 4.38iT - 43T^{2} \) |
| 47 | \( 1 + (0.844 - 1.46i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-5.35 - 9.27i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (4.05 + 7.03i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (5.35 + 3.09i)T + (30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-6.79 + 3.92i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 - 1.16iT - 71T^{2} \) |
| 73 | \( 1 + (8.69 - 5.01i)T + (36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-13.4 - 7.79i)T + (39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + 5.49T + 83T^{2} \) |
| 89 | \( 1 + (9.02 + 5.20i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 - 2.22iT - 97T^{2} \) |
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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
−9.816449864860648111595789706804, −9.011094552412730937418351519018, −8.204914177136978730827888091389, −7.58051360073034121382235820827, −6.55926814403362612804549878898, −5.86882736102748069842648850036, −4.80391861304740289876177929604, −4.14060401552591275900189937078, −2.44354951110861199444382502010, −1.76036016625971485532555233889,
0.19918339917908269558996081354, 1.89007662242200476272781832417, 3.35980829258018654199859940335, 4.08540545459923934492897625832, 5.28461218004257683965359472130, 5.60887159618244079472509527520, 7.07986326336282007022006613918, 7.61600057066581918858274438669, 8.506154811826060798091943457543, 9.347037199445148740108486096992