| L(s) = 1 | + (−0.471 + 1.29i)2-s + (0.0766 + 0.0643i)4-s + (−0.459 + 2.60i)5-s + (1.90 + 1.83i)7-s + (−2.50 + 1.44i)8-s + (−3.15 − 1.82i)10-s + (0.681 − 0.120i)11-s + (−0.171 − 0.472i)13-s + (−3.27 + 1.61i)14-s + (−0.658 − 3.73i)16-s + (−2.42 + 4.19i)17-s + (1.03 − 0.597i)19-s + (−0.202 + 0.170i)20-s + (−0.165 + 0.939i)22-s + (4.74 − 5.65i)23-s + ⋯ |
| L(s) = 1 | + (−0.333 + 0.915i)2-s + (0.0383 + 0.0321i)4-s + (−0.205 + 1.16i)5-s + (0.721 + 0.692i)7-s + (−0.886 + 0.511i)8-s + (−0.998 − 0.576i)10-s + (0.205 − 0.0362i)11-s + (−0.0476 − 0.130i)13-s + (−0.874 + 0.430i)14-s + (−0.164 − 0.933i)16-s + (−0.587 + 1.01i)17-s + (0.237 − 0.137i)19-s + (−0.0453 + 0.0380i)20-s + (−0.0353 + 0.200i)22-s + (0.990 − 1.18i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.994 - 0.101i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.994 - 0.101i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0615433 + 1.21033i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0615433 + 1.21033i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 + (-1.90 - 1.83i)T \) |
| good | 2 | \( 1 + (0.471 - 1.29i)T + (-1.53 - 1.28i)T^{2} \) |
| 5 | \( 1 + (0.459 - 2.60i)T + (-4.69 - 1.71i)T^{2} \) |
| 11 | \( 1 + (-0.681 + 0.120i)T + (10.3 - 3.76i)T^{2} \) |
| 13 | \( 1 + (0.171 + 0.472i)T + (-9.95 + 8.35i)T^{2} \) |
| 17 | \( 1 + (2.42 - 4.19i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-1.03 + 0.597i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-4.74 + 5.65i)T + (-3.99 - 22.6i)T^{2} \) |
| 29 | \( 1 + (-2.19 + 6.03i)T + (-22.2 - 18.6i)T^{2} \) |
| 31 | \( 1 + (5.17 - 6.16i)T + (-5.38 - 30.5i)T^{2} \) |
| 37 | \( 1 + (3.71 - 6.43i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-3.25 + 1.18i)T + (31.4 - 26.3i)T^{2} \) |
| 43 | \( 1 + (1.81 + 10.2i)T + (-40.4 + 14.7i)T^{2} \) |
| 47 | \( 1 + (2.93 - 2.45i)T + (8.16 - 46.2i)T^{2} \) |
| 53 | \( 1 - 10.3iT - 53T^{2} \) |
| 59 | \( 1 + (-1.13 + 6.41i)T + (-55.4 - 20.1i)T^{2} \) |
| 61 | \( 1 + (2.36 + 2.81i)T + (-10.5 + 60.0i)T^{2} \) |
| 67 | \( 1 + (3.24 - 1.17i)T + (51.3 - 43.0i)T^{2} \) |
| 71 | \( 1 + (0.286 + 0.165i)T + (35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (3.13 - 1.81i)T + (36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-9.71 - 3.53i)T + (60.5 + 50.7i)T^{2} \) |
| 83 | \( 1 + (-12.8 - 4.69i)T + (63.5 + 53.3i)T^{2} \) |
| 89 | \( 1 + (4.84 + 8.39i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-3.19 + 0.563i)T + (91.1 - 33.1i)T^{2} \) |
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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
−11.06977174270891233784229931486, −10.45167033767225123457346302342, −9.000106844680852975229061213120, −8.448126858623770005323166179814, −7.51750046935576296246893506135, −6.72188466759858240492415660824, −6.05668013731768230610855311388, −4.87215342535771387398644087074, −3.30963726881155760048121167674, −2.26494187524115592766565143351,
0.78508029347627182461220253220, 1.81392062247230189149826987857, 3.38562756200171437042201798472, 4.57048854935444757758470014904, 5.41600257586906668098443904745, 6.86691244765067242848291334093, 7.75858018839990163932568375955, 8.987846758825776889086841174863, 9.333071257818041409397325656471, 10.44987238067559494343763803376
