| L(s) = 1 | + (1.68 − 0.405i)3-s + (3.00 + 0.529i)5-s + (−0.251 − 0.435i)7-s + (2.67 − 1.36i)9-s + (−2.58 − 1.49i)11-s + (0.364 − 1.00i)13-s + (5.27 − 0.325i)15-s + (−0.866 + 1.03i)17-s + (1.71 + 4.00i)19-s + (−0.599 − 0.630i)21-s + (−8.83 + 1.55i)23-s + (4.03 + 1.46i)25-s + (3.94 − 3.38i)27-s + (−3.43 + 2.88i)29-s + (3.58 − 2.06i)31-s + ⋯ |
| L(s) = 1 | + (0.972 − 0.234i)3-s + (1.34 + 0.236i)5-s + (−0.0949 − 0.164i)7-s + (0.890 − 0.455i)9-s + (−0.779 − 0.450i)11-s + (0.101 − 0.277i)13-s + (1.36 − 0.0841i)15-s + (−0.210 + 0.250i)17-s + (0.394 + 0.918i)19-s + (−0.130 − 0.137i)21-s + (−1.84 + 0.324i)23-s + (0.807 + 0.293i)25-s + (0.759 − 0.650i)27-s + (−0.638 + 0.535i)29-s + (0.643 − 0.371i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 456 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.973 + 0.228i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 456 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.973 + 0.228i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.23687 - 0.258782i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.23687 - 0.258782i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (-1.68 + 0.405i)T \) |
| 19 | \( 1 + (-1.71 - 4.00i)T \) |
| good | 5 | \( 1 + (-3.00 - 0.529i)T + (4.69 + 1.71i)T^{2} \) |
| 7 | \( 1 + (0.251 + 0.435i)T + (-3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (2.58 + 1.49i)T + (5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (-0.364 + 1.00i)T + (-9.95 - 8.35i)T^{2} \) |
| 17 | \( 1 + (0.866 - 1.03i)T + (-2.95 - 16.7i)T^{2} \) |
| 23 | \( 1 + (8.83 - 1.55i)T + (21.6 - 7.86i)T^{2} \) |
| 29 | \( 1 + (3.43 - 2.88i)T + (5.03 - 28.5i)T^{2} \) |
| 31 | \( 1 + (-3.58 + 2.06i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + 1.40iT - 37T^{2} \) |
| 41 | \( 1 + (-4.19 + 1.52i)T + (31.4 - 26.3i)T^{2} \) |
| 43 | \( 1 + (0.539 - 3.05i)T + (-40.4 - 14.7i)T^{2} \) |
| 47 | \( 1 + (1.56 + 1.86i)T + (-8.16 + 46.2i)T^{2} \) |
| 53 | \( 1 + (-1.97 - 11.2i)T + (-49.8 + 18.1i)T^{2} \) |
| 59 | \( 1 + (8.56 + 7.18i)T + (10.2 + 58.1i)T^{2} \) |
| 61 | \( 1 + (-1.52 - 8.62i)T + (-57.3 + 20.8i)T^{2} \) |
| 67 | \( 1 + (6.94 + 8.27i)T + (-11.6 + 65.9i)T^{2} \) |
| 71 | \( 1 + (2.02 - 11.5i)T + (-66.7 - 24.2i)T^{2} \) |
| 73 | \( 1 + (-9.45 + 3.44i)T + (55.9 - 46.9i)T^{2} \) |
| 79 | \( 1 + (-0.310 - 0.852i)T + (-60.5 + 50.7i)T^{2} \) |
| 83 | \( 1 + (-6.19 + 3.57i)T + (41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (13.7 + 5.00i)T + (68.1 + 57.2i)T^{2} \) |
| 97 | \( 1 + (9.45 - 11.2i)T + (-16.8 - 95.5i)T^{2} \) |
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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
−10.67414866115214027305349885925, −10.02869087034079355618034385464, −9.366895351123366540321271343700, −8.272704877530779912164585142264, −7.55030293241044669093450498470, −6.27838142496728374209534150971, −5.57446554585855596318021615196, −3.94537521571522597041431989991, −2.74304090983598166146421506508, −1.71377492627439386721937539311,
1.91105416941547272722669855292, 2.73391381132640820883294220360, 4.30476730017411846095280290840, 5.33217501362587062445255386535, 6.42562220195972361574470834408, 7.58457431318711602583238377962, 8.527092777340032753798995592432, 9.514949680733792061052318077956, 9.852036528542061666993384799262, 10.80737404679403420891031664247
