| L(s) = 1 | + (0.900 − 0.433i)2-s + (−1.16 − 1.28i)3-s + (0.623 − 0.781i)4-s + (1.63 + 1.52i)5-s + (−1.60 − 0.655i)6-s + (−2.40 − 1.11i)7-s + (0.222 − 0.974i)8-s + (−0.306 + 2.98i)9-s + (2.13 + 0.658i)10-s + (−3.40 − 2.32i)11-s + (−1.72 + 0.105i)12-s + (−1.28 − 0.876i)13-s + (−2.64 + 0.0407i)14-s + (0.0535 − 3.87i)15-s + (−0.222 − 0.974i)16-s + (−1.51 + 0.229i)17-s + ⋯ |
| L(s) = 1 | + (0.637 − 0.306i)2-s + (−0.669 − 0.742i)3-s + (0.311 − 0.390i)4-s + (0.732 + 0.679i)5-s + (−0.654 − 0.267i)6-s + (−0.907 − 0.419i)7-s + (0.0786 − 0.344i)8-s + (−0.102 + 0.994i)9-s + (0.675 + 0.208i)10-s + (−1.02 − 0.700i)11-s + (−0.499 + 0.0304i)12-s + (−0.356 − 0.243i)13-s + (−0.707 + 0.0108i)14-s + (0.0138 − 0.999i)15-s + (−0.0556 − 0.243i)16-s + (−0.368 + 0.0555i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.990 + 0.138i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.990 + 0.138i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0659788 - 0.951361i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0659788 - 0.951361i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.900 + 0.433i)T \) |
| 3 | \( 1 + (1.16 + 1.28i)T \) |
| 7 | \( 1 + (2.40 + 1.11i)T \) |
| good | 5 | \( 1 + (-1.63 - 1.52i)T + (0.373 + 4.98i)T^{2} \) |
| 11 | \( 1 + (3.40 + 2.32i)T + (4.01 + 10.2i)T^{2} \) |
| 13 | \( 1 + (1.28 + 0.876i)T + (4.74 + 12.1i)T^{2} \) |
| 17 | \( 1 + (1.51 - 0.229i)T + (16.2 - 5.01i)T^{2} \) |
| 19 | \( 1 + (2.94 + 5.10i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (0.680 + 1.73i)T + (-16.8 + 15.6i)T^{2} \) |
| 29 | \( 1 + (0.710 - 0.107i)T + (27.7 - 8.54i)T^{2} \) |
| 31 | \( 1 + 7.55T + 31T^{2} \) |
| 37 | \( 1 + (-2.87 + 7.33i)T + (-27.1 - 25.1i)T^{2} \) |
| 41 | \( 1 + (-9.16 + 2.82i)T + (33.8 - 23.0i)T^{2} \) |
| 43 | \( 1 + (-4.42 - 1.36i)T + (35.5 + 24.2i)T^{2} \) |
| 47 | \( 1 + (11.0 - 5.32i)T + (29.3 - 36.7i)T^{2} \) |
| 53 | \( 1 + (0.397 + 1.01i)T + (-38.8 + 36.0i)T^{2} \) |
| 59 | \( 1 + (-2.41 - 10.5i)T + (-53.1 + 25.5i)T^{2} \) |
| 61 | \( 1 + (-2.15 - 2.69i)T + (-13.5 + 59.4i)T^{2} \) |
| 67 | \( 1 + 7.50T + 67T^{2} \) |
| 71 | \( 1 + (-5.97 + 7.49i)T + (-15.7 - 69.2i)T^{2} \) |
| 73 | \( 1 + (13.7 - 9.38i)T + (26.6 - 67.9i)T^{2} \) |
| 79 | \( 1 - 12.0T + 79T^{2} \) |
| 83 | \( 1 + (-6.89 + 4.70i)T + (30.3 - 77.2i)T^{2} \) |
| 89 | \( 1 + (0.680 + 9.08i)T + (-88.0 + 13.2i)T^{2} \) |
| 97 | \( 1 + (4.58 - 7.94i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−10.10990713560453445183702754357, −9.051366128517373829543957336393, −7.65354968524575815118598228446, −6.93880319427412716576350139486, −6.11876727180394244037513970643, −5.61131311098886543320931238827, −4.40833488395805468216213886461, −2.91432961930817659770223355587, −2.26786510160185648238794104922, −0.36371119526174310568096138805,
2.09354115437980811899846107955, 3.44631065538368081751327708469, 4.54433443389457208995373572101, 5.33066878327178548346090229789, 5.93762264055917692559356641350, 6.76251920736899451272590512490, 7.955821252292409199152614962816, 9.149368533706609399373957646757, 9.704419390994975507713198156283, 10.41365811629790845235809940345
