| L(s) = 1 | + (−2.73 − 0.889i)2-s + (−2.95 − 0.494i)3-s + (3.46 + 2.51i)4-s + (4.70 − 1.52i)5-s + (7.65 + 3.98i)6-s + (9.04 + 6.57i)7-s + (−0.474 − 0.653i)8-s + (8.51 + 2.92i)9-s − 14.2·10-s + (−9.00 − 9.15i)12-s + (−5.22 + 16.0i)13-s + (−18.9 − 26.0i)14-s + (−14.6 + 2.19i)15-s + (−4.57 − 14.0i)16-s + (0.0595 − 0.0193i)17-s + (−20.6 − 15.5i)18-s + ⋯ |
| L(s) = 1 | + (−1.36 − 0.444i)2-s + (−0.986 − 0.164i)3-s + (0.865 + 0.629i)4-s + (0.941 − 0.305i)5-s + (1.27 + 0.664i)6-s + (1.29 + 0.938i)7-s + (−0.0593 − 0.0816i)8-s + (0.945 + 0.325i)9-s − 1.42·10-s + (−0.750 − 0.763i)12-s + (−0.401 + 1.23i)13-s + (−1.35 − 1.85i)14-s + (−0.978 + 0.146i)15-s + (−0.285 − 0.879i)16-s + (0.00350 − 0.00113i)17-s + (−1.14 − 0.865i)18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.503 - 0.864i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.503 - 0.864i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.547579 + 0.314791i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.547579 + 0.314791i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (2.95 + 0.494i)T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 + (2.73 + 0.889i)T + (3.23 + 2.35i)T^{2} \) |
| 5 | \( 1 + (-4.70 + 1.52i)T + (20.2 - 14.6i)T^{2} \) |
| 7 | \( 1 + (-9.04 - 6.57i)T + (15.1 + 46.6i)T^{2} \) |
| 13 | \( 1 + (5.22 - 16.0i)T + (-136. - 99.3i)T^{2} \) |
| 17 | \( 1 + (-0.0595 + 0.0193i)T + (233. - 169. i)T^{2} \) |
| 19 | \( 1 + (4.35 - 3.16i)T + (111. - 343. i)T^{2} \) |
| 23 | \( 1 + 8.69iT - 529T^{2} \) |
| 29 | \( 1 + (29.3 - 40.4i)T + (-259. - 799. i)T^{2} \) |
| 31 | \( 1 + (7.59 - 23.3i)T + (-777. - 564. i)T^{2} \) |
| 37 | \( 1 + (-31.7 - 23.0i)T + (423. + 1.30e3i)T^{2} \) |
| 41 | \( 1 + (25.8 + 35.5i)T + (-519. + 1.59e3i)T^{2} \) |
| 43 | \( 1 - 0.201T + 1.84e3T^{2} \) |
| 47 | \( 1 + (-8.76 - 12.0i)T + (-682. + 2.10e3i)T^{2} \) |
| 53 | \( 1 + (93.1 + 30.2i)T + (2.27e3 + 1.65e3i)T^{2} \) |
| 59 | \( 1 + (8.51 - 11.7i)T + (-1.07e3 - 3.31e3i)T^{2} \) |
| 61 | \( 1 + (-9.06 - 27.8i)T + (-3.01e3 + 2.18e3i)T^{2} \) |
| 67 | \( 1 - 82.8T + 4.48e3T^{2} \) |
| 71 | \( 1 + (25.3 - 8.25i)T + (4.07e3 - 2.96e3i)T^{2} \) |
| 73 | \( 1 + (11.3 + 8.21i)T + (1.64e3 + 5.06e3i)T^{2} \) |
| 79 | \( 1 + (-15.5 + 47.7i)T + (-5.04e3 - 3.66e3i)T^{2} \) |
| 83 | \( 1 + (-57.2 + 18.6i)T + (5.57e3 - 4.04e3i)T^{2} \) |
| 89 | \( 1 - 40.3iT - 7.92e3T^{2} \) |
| 97 | \( 1 + (9.52 - 29.3i)T + (-7.61e3 - 5.53e3i)T^{2} \) |
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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
−11.24409662770650175149895038555, −10.47338086135995064725893565541, −9.454093828319260433282554854312, −8.888247352202721192691086841393, −7.83570934643950869976551717981, −6.72432449335132546111259799489, −5.46203799614952081854106085723, −4.77805977319131977008210978644, −2.04232832771135769846658422665, −1.50646281420689383609255831773,
0.53559198882069974049752786965, 1.80630376564064128128906432460, 4.24665951575748721440408310776, 5.46637264658297102852930450837, 6.39290871615811394677215317743, 7.53927454049463283177272048828, 7.987008849704843451769198663817, 9.547789361698146497839914244683, 10.01655241865963247947204064049, 10.86261145796232209160949327307
