| L(s) = 1 | + (2.91 + 0.945i)2-s + (−1.86 − 2.35i)3-s + (4.34 + 3.15i)4-s + (−6.31 + 2.05i)5-s + (−3.20 − 8.60i)6-s + (2.47 + 1.80i)7-s + (2.45 + 3.37i)8-s + (−2.05 + 8.76i)9-s − 20.3·10-s + (10.9 − 1.01i)11-s + (−0.678 − 16.0i)12-s + (5.01 − 15.4i)13-s + (5.50 + 7.58i)14-s + (16.5 + 11.0i)15-s + (−2.68 − 8.25i)16-s + (0.766 − 0.248i)17-s + ⋯ |
| L(s) = 1 | + (1.45 + 0.472i)2-s + (−0.621 − 0.783i)3-s + (1.08 + 0.788i)4-s + (−1.26 + 0.410i)5-s + (−0.533 − 1.43i)6-s + (0.353 + 0.257i)7-s + (0.306 + 0.422i)8-s + (−0.227 + 0.973i)9-s − 2.03·10-s + (0.995 − 0.0924i)11-s + (−0.0565 − 1.34i)12-s + (0.386 − 1.18i)13-s + (0.393 + 0.541i)14-s + (1.10 + 0.734i)15-s + (−0.167 − 0.515i)16-s + (0.0450 − 0.0146i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.983 - 0.179i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.983 - 0.179i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.48370 + 0.134296i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.48370 + 0.134296i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (1.86 + 2.35i)T \) |
| 11 | \( 1 + (-10.9 + 1.01i)T \) |
| good | 2 | \( 1 + (-2.91 - 0.945i)T + (3.23 + 2.35i)T^{2} \) |
| 5 | \( 1 + (6.31 - 2.05i)T + (20.2 - 14.6i)T^{2} \) |
| 7 | \( 1 + (-2.47 - 1.80i)T + (15.1 + 46.6i)T^{2} \) |
| 13 | \( 1 + (-5.01 + 15.4i)T + (-136. - 99.3i)T^{2} \) |
| 17 | \( 1 + (-0.766 + 0.248i)T + (233. - 169. i)T^{2} \) |
| 19 | \( 1 + (16.7 - 12.1i)T + (111. - 343. i)T^{2} \) |
| 23 | \( 1 - 27.3iT - 529T^{2} \) |
| 29 | \( 1 + (2.22 - 3.06i)T + (-259. - 799. i)T^{2} \) |
| 31 | \( 1 + (6.42 - 19.7i)T + (-777. - 564. i)T^{2} \) |
| 37 | \( 1 + (31.1 + 22.6i)T + (423. + 1.30e3i)T^{2} \) |
| 41 | \( 1 + (-7.86 - 10.8i)T + (-519. + 1.59e3i)T^{2} \) |
| 43 | \( 1 - 43.4T + 1.84e3T^{2} \) |
| 47 | \( 1 + (11.6 + 16.0i)T + (-682. + 2.10e3i)T^{2} \) |
| 53 | \( 1 + (16.8 + 5.46i)T + (2.27e3 + 1.65e3i)T^{2} \) |
| 59 | \( 1 + (-25.5 + 35.2i)T + (-1.07e3 - 3.31e3i)T^{2} \) |
| 61 | \( 1 + (-3.29 - 10.1i)T + (-3.01e3 + 2.18e3i)T^{2} \) |
| 67 | \( 1 - 72.2T + 4.48e3T^{2} \) |
| 71 | \( 1 + (2.44 - 0.794i)T + (4.07e3 - 2.96e3i)T^{2} \) |
| 73 | \( 1 + (-36.7 - 26.7i)T + (1.64e3 + 5.06e3i)T^{2} \) |
| 79 | \( 1 + (30.3 - 93.4i)T + (-5.04e3 - 3.66e3i)T^{2} \) |
| 83 | \( 1 + (-30.3 + 9.85i)T + (5.57e3 - 4.04e3i)T^{2} \) |
| 89 | \( 1 - 18.5iT - 7.92e3T^{2} \) |
| 97 | \( 1 + (19.5 - 60.2i)T + (-7.61e3 - 5.53e3i)T^{2} \) |
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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
−16.13729454383026636737223325443, −15.17732316039617530393464838263, −14.18990023158930555094541191794, −12.82756837654583371360583002277, −11.98318547718054298928785836038, −11.06889887792736954698234138459, −8.030640762009016661583054760140, −6.83628932455559411481363187657, −5.49834461611893522119616692867, −3.72148618207764170458922966311,
3.98771700021863636673754337924, 4.56294848408930895543033444433, 6.48453345258323560754757685647, 8.845069496206329559085515422287, 10.96708208893856504509838963311, 11.66485079110216993030844417624, 12.50925896138867728211268200411, 14.19721296638072793555515334261, 15.10387369331872754405015804178, 16.14589732785526656804634737306
