| L(s) = 1 | + (0.841 + 0.540i)3-s + (−1.77 + 3.89i)5-s + (−1.59 − 0.469i)7-s + (0.415 + 0.909i)9-s + (1.76 − 2.04i)11-s + (−4.30 + 1.26i)13-s + (−3.60 + 2.31i)15-s + (0.201 + 1.40i)17-s + (−0.0468 + 0.325i)19-s + (−1.09 − 1.25i)21-s + (−2.48 + 4.10i)23-s + (−8.72 − 10.0i)25-s + (−0.142 + 0.989i)27-s + (0.0394 + 0.274i)29-s + (−5.73 + 3.68i)31-s + ⋯ |
| L(s) = 1 | + (0.485 + 0.312i)3-s + (−0.795 + 1.74i)5-s + (−0.604 − 0.177i)7-s + (0.138 + 0.303i)9-s + (0.533 − 0.615i)11-s + (−1.19 + 0.350i)13-s + (−0.929 + 0.597i)15-s + (0.0488 + 0.339i)17-s + (−0.0107 + 0.0747i)19-s + (−0.238 − 0.274i)21-s + (−0.517 + 0.855i)23-s + (−1.74 − 2.01i)25-s + (−0.0273 + 0.190i)27-s + (0.00731 + 0.0509i)29-s + (−1.02 + 0.661i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.861 - 0.507i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.861 - 0.507i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.247819 + 0.908464i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.247819 + 0.908464i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (-0.841 - 0.540i)T \) |
| 23 | \( 1 + (2.48 - 4.10i)T \) |
| good | 5 | \( 1 + (1.77 - 3.89i)T + (-3.27 - 3.77i)T^{2} \) |
| 7 | \( 1 + (1.59 + 0.469i)T + (5.88 + 3.78i)T^{2} \) |
| 11 | \( 1 + (-1.76 + 2.04i)T + (-1.56 - 10.8i)T^{2} \) |
| 13 | \( 1 + (4.30 - 1.26i)T + (10.9 - 7.02i)T^{2} \) |
| 17 | \( 1 + (-0.201 - 1.40i)T + (-16.3 + 4.78i)T^{2} \) |
| 19 | \( 1 + (0.0468 - 0.325i)T + (-18.2 - 5.35i)T^{2} \) |
| 29 | \( 1 + (-0.0394 - 0.274i)T + (-27.8 + 8.17i)T^{2} \) |
| 31 | \( 1 + (5.73 - 3.68i)T + (12.8 - 28.1i)T^{2} \) |
| 37 | \( 1 + (1.12 + 2.46i)T + (-24.2 + 27.9i)T^{2} \) |
| 41 | \( 1 + (2.74 - 6.00i)T + (-26.8 - 30.9i)T^{2} \) |
| 43 | \( 1 + (-6.44 - 4.13i)T + (17.8 + 39.1i)T^{2} \) |
| 47 | \( 1 - 9.55T + 47T^{2} \) |
| 53 | \( 1 + (-9.58 - 2.81i)T + (44.5 + 28.6i)T^{2} \) |
| 59 | \( 1 + (-5.58 + 1.63i)T + (49.6 - 31.8i)T^{2} \) |
| 61 | \( 1 + (-3.46 + 2.22i)T + (25.3 - 55.4i)T^{2} \) |
| 67 | \( 1 + (-7.14 - 8.24i)T + (-9.53 + 66.3i)T^{2} \) |
| 71 | \( 1 + (2.00 + 2.31i)T + (-10.1 + 70.2i)T^{2} \) |
| 73 | \( 1 + (2.18 - 15.1i)T + (-70.0 - 20.5i)T^{2} \) |
| 79 | \( 1 + (5.03 - 1.47i)T + (66.4 - 42.7i)T^{2} \) |
| 83 | \( 1 + (-3.38 - 7.40i)T + (-54.3 + 62.7i)T^{2} \) |
| 89 | \( 1 + (12.2 + 7.86i)T + (36.9 + 80.9i)T^{2} \) |
| 97 | \( 1 + (2.19 - 4.81i)T + (-63.5 - 73.3i)T^{2} \) |
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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
−11.11652887458017615402612209783, −10.22054607045799915450738139971, −9.619758803003483873478329693552, −8.431825953545331652813547595806, −7.35030916832031143610315133403, −6.93709860862948820478415551033, −5.79437527329839967767352356603, −4.08425948383376782013896022912, −3.43219560009895680939988152156, −2.45789969468856441812455759079,
0.49225480776396991779453696646, 2.17079047603922036711155609649, 3.77440290095341238389247788434, 4.63397222573693672227755017035, 5.62934463401717090056995604305, 7.09752248212812204146161781259, 7.76409834341844648968757626307, 8.783211248978404137169854854658, 9.268163776396793606235402457759, 10.15263370538314257942229899969
