L(s) = 1 | + (0.204 + 1.39i)2-s + (−0.539 + 0.743i)3-s + (−1.91 + 0.573i)4-s + (0.529 − 1.63i)5-s + (−1.15 − 0.603i)6-s + (1.93 − 1.40i)7-s + (−1.19 − 2.56i)8-s + (0.666 + 2.05i)9-s + (2.39 + 0.407i)10-s + (−2.65 − 1.98i)11-s + (0.608 − 1.73i)12-s + (−1.52 + 0.497i)13-s + (2.36 + 2.41i)14-s + (0.925 + 1.27i)15-s + (3.34 − 2.19i)16-s + (−5.74 − 1.86i)17-s + ⋯ |
L(s) = 1 | + (0.144 + 0.989i)2-s + (−0.311 + 0.428i)3-s + (−0.957 + 0.286i)4-s + (0.236 − 0.729i)5-s + (−0.469 − 0.246i)6-s + (0.730 − 0.531i)7-s + (−0.422 − 0.906i)8-s + (0.222 + 0.683i)9-s + (0.755 + 0.128i)10-s + (−0.800 − 0.599i)11-s + (0.175 − 0.500i)12-s + (−0.424 + 0.137i)13-s + (0.631 + 0.646i)14-s + (0.239 + 0.328i)15-s + (0.835 − 0.549i)16-s + (−1.39 − 0.453i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 44 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.324 - 0.945i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 44 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.324 - 0.945i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.619486 + 0.442391i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.619486 + 0.442391i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.204 - 1.39i)T \) |
| 11 | \( 1 + (2.65 + 1.98i)T \) |
good | 3 | \( 1 + (0.539 - 0.743i)T + (-0.927 - 2.85i)T^{2} \) |
| 5 | \( 1 + (-0.529 + 1.63i)T + (-4.04 - 2.93i)T^{2} \) |
| 7 | \( 1 + (-1.93 + 1.40i)T + (2.16 - 6.65i)T^{2} \) |
| 13 | \( 1 + (1.52 - 0.497i)T + (10.5 - 7.64i)T^{2} \) |
| 17 | \( 1 + (5.74 + 1.86i)T + (13.7 + 9.99i)T^{2} \) |
| 19 | \( 1 + (-1.07 - 0.779i)T + (5.87 + 18.0i)T^{2} \) |
| 23 | \( 1 - 7.25iT - 23T^{2} \) |
| 29 | \( 1 + (0.318 + 0.439i)T + (-8.96 + 27.5i)T^{2} \) |
| 31 | \( 1 + (-7.82 + 2.54i)T + (25.0 - 18.2i)T^{2} \) |
| 37 | \( 1 + (0.299 - 0.217i)T + (11.4 - 35.1i)T^{2} \) |
| 41 | \( 1 + (-2.99 + 4.11i)T + (-12.6 - 38.9i)T^{2} \) |
| 43 | \( 1 + 9.45T + 43T^{2} \) |
| 47 | \( 1 + (-2.35 + 3.24i)T + (-14.5 - 44.6i)T^{2} \) |
| 53 | \( 1 + (0.765 + 2.35i)T + (-42.8 + 31.1i)T^{2} \) |
| 59 | \( 1 + (-4.14 - 5.70i)T + (-18.2 + 56.1i)T^{2} \) |
| 61 | \( 1 + (-9.41 - 3.05i)T + (49.3 + 35.8i)T^{2} \) |
| 67 | \( 1 - 4.79iT - 67T^{2} \) |
| 71 | \( 1 + (3.34 + 1.08i)T + (57.4 + 41.7i)T^{2} \) |
| 73 | \( 1 + (1.23 + 1.70i)T + (-22.5 + 69.4i)T^{2} \) |
| 79 | \( 1 + (0.797 + 2.45i)T + (-63.9 + 46.4i)T^{2} \) |
| 83 | \( 1 + (-1.22 + 3.77i)T + (-67.1 - 48.7i)T^{2} \) |
| 89 | \( 1 + 8.45T + 89T^{2} \) |
| 97 | \( 1 + (2.57 + 7.91i)T + (-78.4 + 57.0i)T^{2} \) |
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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
−16.16692060450789056611903312425, −15.33316604893244421196485778639, −13.75946092575663644411795227874, −13.23055616689476412265281444112, −11.40939859388480613670136030010, −9.968301492202687700821637584192, −8.551602194997631995278879448563, −7.38084116022951475226201610418, −5.42921536950679246297502834054, −4.54091099639989253225350124466,
2.43970731718250358503347518610, 4.77211385045028668663464353930, 6.55718498931181233110651247168, 8.451998456557875807436910900311, 9.998171371152420931189467185418, 11.05788500286960806575559991781, 12.17439349740895176457031830641, 13.07872019983712583752725447062, 14.49290098463456369681285914499, 15.30035676842707206175264127957