| L(s) = 1 | + (1.35 + 0.906i)3-s + (−6.70 + 1.33i)5-s + (−0.886 − 0.176i)7-s + (−2.42 − 5.85i)9-s + (−3.73 − 5.59i)11-s + (10.5 − 10.5i)13-s + (−10.2 − 4.26i)15-s + (−14.7 − 8.37i)17-s + (12.9 − 31.3i)19-s + (−1.04 − 1.04i)21-s + (−15.4 + 10.3i)23-s + (20.0 − 8.30i)25-s + (4.88 − 24.5i)27-s + (4.13 + 20.7i)29-s + (−21.1 + 31.6i)31-s + ⋯ |
| L(s) = 1 | + (0.451 + 0.302i)3-s + (−1.34 + 0.266i)5-s + (−0.126 − 0.0251i)7-s + (−0.269 − 0.650i)9-s + (−0.339 − 0.508i)11-s + (0.812 − 0.812i)13-s + (−0.686 − 0.284i)15-s + (−0.870 − 0.492i)17-s + (0.683 − 1.64i)19-s + (−0.0496 − 0.0496i)21-s + (−0.672 + 0.449i)23-s + (0.802 − 0.332i)25-s + (0.180 − 0.908i)27-s + (0.142 + 0.716i)29-s + (−0.681 + 1.02i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 272 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.355 + 0.934i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 272 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.355 + 0.934i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.433643 - 0.628680i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.433643 - 0.628680i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 17 | \( 1 + (14.7 + 8.37i)T \) |
| good | 3 | \( 1 + (-1.35 - 0.906i)T + (3.44 + 8.31i)T^{2} \) |
| 5 | \( 1 + (6.70 - 1.33i)T + (23.0 - 9.56i)T^{2} \) |
| 7 | \( 1 + (0.886 + 0.176i)T + (45.2 + 18.7i)T^{2} \) |
| 11 | \( 1 + (3.73 + 5.59i)T + (-46.3 + 111. i)T^{2} \) |
| 13 | \( 1 + (-10.5 + 10.5i)T - 169iT^{2} \) |
| 19 | \( 1 + (-12.9 + 31.3i)T + (-255. - 255. i)T^{2} \) |
| 23 | \( 1 + (15.4 - 10.3i)T + (202. - 488. i)T^{2} \) |
| 29 | \( 1 + (-4.13 - 20.7i)T + (-776. + 321. i)T^{2} \) |
| 31 | \( 1 + (21.1 - 31.6i)T + (-367. - 887. i)T^{2} \) |
| 37 | \( 1 + (33.3 + 22.2i)T + (523. + 1.26e3i)T^{2} \) |
| 41 | \( 1 + (3.70 + 0.736i)T + (1.55e3 + 643. i)T^{2} \) |
| 43 | \( 1 + (-5.21 - 12.5i)T + (-1.30e3 + 1.30e3i)T^{2} \) |
| 47 | \( 1 + (-9.20 + 9.20i)T - 2.20e3iT^{2} \) |
| 53 | \( 1 + (-0.763 + 1.84i)T + (-1.98e3 - 1.98e3i)T^{2} \) |
| 59 | \( 1 + (31.7 - 13.1i)T + (2.46e3 - 2.46e3i)T^{2} \) |
| 61 | \( 1 + (-6.92 + 34.8i)T + (-3.43e3 - 1.42e3i)T^{2} \) |
| 67 | \( 1 + 31.9iT - 4.48e3T^{2} \) |
| 71 | \( 1 + (86.1 + 57.5i)T + (1.92e3 + 4.65e3i)T^{2} \) |
| 73 | \( 1 + (-61.7 + 12.2i)T + (4.92e3 - 2.03e3i)T^{2} \) |
| 79 | \( 1 + (-36.2 - 54.3i)T + (-2.38e3 + 5.76e3i)T^{2} \) |
| 83 | \( 1 + (-39.7 - 16.4i)T + (4.87e3 + 4.87e3i)T^{2} \) |
| 89 | \( 1 + (-49.5 - 49.5i)T + 7.92e3iT^{2} \) |
| 97 | \( 1 + (-24.9 - 125. i)T + (-8.69e3 + 3.60e3i)T^{2} \) |
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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
−11.30667301180300952138903294901, −10.69347719013518818031792887583, −9.243059983960579334572478272760, −8.563570856947031006503695532179, −7.58071212547414015320965486491, −6.57119962384686520531413170993, −5.08386938707512573557937727791, −3.68416842242828170950321486481, −3.04984553642867002172162113628, −0.35877449835020870898570540015,
1.91559433182606250765000862161, 3.60344056637972000994876795231, 4.51106618791193123346127363712, 6.04187875583167410246933265404, 7.40393898503082256760657034347, 8.075336759118818928663622756273, 8.785712624321681578193000877625, 10.13735002549766830696828640306, 11.21779047851350785342038617199, 11.94518557923281370071557236113
