| L(s) = 1 | + (0.188 + 0.826i)2-s + (−0.0129 − 0.0119i)3-s + (1.15 − 0.556i)4-s + (−3.39 + 0.512i)5-s + (0.00747 − 0.0129i)6-s + (−0.134 − 0.232i)7-s + (1.73 + 2.17i)8-s + (−0.224 − 2.99i)9-s + (−1.06 − 2.71i)10-s + (−2.96 − 1.42i)11-s + (−0.0216 − 0.00666i)12-s + (−0.736 + 1.87i)13-s + (0.167 − 0.155i)14-s + (0.0500 + 0.0341i)15-s + (0.129 − 0.161i)16-s + (6.37 + 0.960i)17-s + ⋯ |
| L(s) = 1 | + (0.133 + 0.584i)2-s + (−0.00746 − 0.00692i)3-s + (0.577 − 0.278i)4-s + (−1.51 + 0.229i)5-s + (0.00305 − 0.00528i)6-s + (−0.0508 − 0.0880i)7-s + (0.613 + 0.768i)8-s + (−0.0747 − 0.997i)9-s + (−0.336 − 0.857i)10-s + (−0.894 − 0.430i)11-s + (−0.00623 − 0.00192i)12-s + (−0.204 + 0.520i)13-s + (0.0446 − 0.0414i)14-s + (0.0129 + 0.00881i)15-s + (0.0322 − 0.0404i)16-s + (1.54 + 0.233i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.857 - 0.514i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.857 - 0.514i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.769080 + 0.213138i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.769080 + 0.213138i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 43 | \( 1 + (4.01 + 5.18i)T \) |
| good | 2 | \( 1 + (-0.188 - 0.826i)T + (-1.80 + 0.867i)T^{2} \) |
| 3 | \( 1 + (0.0129 + 0.0119i)T + (0.224 + 2.99i)T^{2} \) |
| 5 | \( 1 + (3.39 - 0.512i)T + (4.77 - 1.47i)T^{2} \) |
| 7 | \( 1 + (0.134 + 0.232i)T + (-3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (2.96 + 1.42i)T + (6.85 + 8.60i)T^{2} \) |
| 13 | \( 1 + (0.736 - 1.87i)T + (-9.52 - 8.84i)T^{2} \) |
| 17 | \( 1 + (-6.37 - 0.960i)T + (16.2 + 5.01i)T^{2} \) |
| 19 | \( 1 + (0.449 - 5.99i)T + (-18.7 - 2.83i)T^{2} \) |
| 23 | \( 1 + (1.83 - 1.25i)T + (8.40 - 21.4i)T^{2} \) |
| 29 | \( 1 + (-1.75 + 1.63i)T + (2.16 - 28.9i)T^{2} \) |
| 31 | \( 1 + (4.93 + 1.52i)T + (25.6 + 17.4i)T^{2} \) |
| 37 | \( 1 + (-2.63 + 4.56i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (0.643 + 2.81i)T + (-36.9 + 17.7i)T^{2} \) |
| 47 | \( 1 + (5.31 - 2.56i)T + (29.3 - 36.7i)T^{2} \) |
| 53 | \( 1 + (-2.34 - 5.97i)T + (-38.8 + 36.0i)T^{2} \) |
| 59 | \( 1 + (-8.36 + 10.4i)T + (-13.1 - 57.5i)T^{2} \) |
| 61 | \( 1 + (9.50 - 2.93i)T + (50.4 - 34.3i)T^{2} \) |
| 67 | \( 1 + (0.950 - 12.6i)T + (-66.2 - 9.98i)T^{2} \) |
| 71 | \( 1 + (4.98 + 3.40i)T + (25.9 + 66.0i)T^{2} \) |
| 73 | \( 1 + (-0.609 + 1.55i)T + (-53.5 - 49.6i)T^{2} \) |
| 79 | \( 1 + (-6.97 - 12.0i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (5.60 + 5.19i)T + (6.20 + 82.7i)T^{2} \) |
| 89 | \( 1 + (1.32 + 1.23i)T + (6.65 + 88.7i)T^{2} \) |
| 97 | \( 1 + (-2.36 - 1.13i)T + (60.4 + 75.8i)T^{2} \) |
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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
−16.01187373706816386090014790560, −15.05304986242327805311567279648, −14.32723998236206583005643924187, −12.32374161895736394167348059426, −11.52722667411031890609602540181, −10.23266760642578425059424261129, −8.152075516225040642416897040102, −7.29092140710799427239364958072, −5.76965005998965698879881645177, −3.64735500018803253957244067901,
3.02817159373934181191224135386, 4.82003467776036744184842805854, 7.37618891774067609623104052879, 8.064465185726210142267160785387, 10.28169892285607282831575736646, 11.29664459352253528223080781012, 12.24462184449001221333277607918, 13.17173066950531623174649144725, 15.03207855865339642475973709692, 15.97485525950707828691481746814
