| L(s) = 1 | + (−1.71 − 1.37i)2-s + (0.978 + 1.42i)3-s + (0.630 + 2.76i)4-s + (0.322 + 4.30i)5-s + (0.278 − 3.79i)6-s + (−2.62 − 0.338i)7-s + (0.795 − 1.65i)8-s + (−1.08 + 2.79i)9-s + (5.34 − 7.83i)10-s + (1.29 − 0.508i)11-s + (−3.33 + 3.60i)12-s + (−1.64 + 0.643i)13-s + (4.04 + 4.17i)14-s + (−5.83 + 4.66i)15-s + (1.47 − 0.710i)16-s + (3.40 − 1.05i)17-s + ⋯ |
| L(s) = 1 | + (−1.21 − 0.969i)2-s + (0.564 + 0.825i)3-s + (0.315 + 1.38i)4-s + (0.144 + 1.92i)5-s + (0.113 − 1.55i)6-s + (−0.991 − 0.127i)7-s + (0.281 − 0.583i)8-s + (−0.362 + 0.932i)9-s + (1.68 − 2.47i)10-s + (0.390 − 0.153i)11-s + (−0.961 + 1.04i)12-s + (−0.454 + 0.178i)13-s + (1.08 + 1.11i)14-s + (−1.50 + 1.20i)15-s + (0.369 − 0.177i)16-s + (0.826 − 0.255i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.637 - 0.770i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.637 - 0.770i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.227150 + 0.482515i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.227150 + 0.482515i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-0.978 - 1.42i)T \) |
| 7 | \( 1 + (2.62 + 0.338i)T \) |
| good | 2 | \( 1 + (1.71 + 1.37i)T + (0.445 + 1.94i)T^{2} \) |
| 5 | \( 1 + (-0.322 - 4.30i)T + (-4.94 + 0.745i)T^{2} \) |
| 11 | \( 1 + (-1.29 + 0.508i)T + (8.06 - 7.48i)T^{2} \) |
| 13 | \( 1 + (1.64 - 0.643i)T + (9.52 - 8.84i)T^{2} \) |
| 17 | \( 1 + (-3.40 + 1.05i)T + (14.0 - 9.57i)T^{2} \) |
| 19 | \( 1 + (6.82 + 3.93i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-0.105 - 0.113i)T + (-1.71 + 22.9i)T^{2} \) |
| 29 | \( 1 + (-0.317 - 1.02i)T + (-23.9 + 16.3i)T^{2} \) |
| 31 | \( 1 + 6.23iT - 31T^{2} \) |
| 37 | \( 1 + (-5.31 - 4.93i)T + (2.76 + 36.8i)T^{2} \) |
| 41 | \( 1 + (2.63 - 1.79i)T + (14.9 - 38.1i)T^{2} \) |
| 43 | \( 1 + (-3.44 - 2.34i)T + (15.7 + 40.0i)T^{2} \) |
| 47 | \( 1 + (4.44 - 5.56i)T + (-10.4 - 45.8i)T^{2} \) |
| 53 | \( 1 + (-8.10 - 8.73i)T + (-3.96 + 52.8i)T^{2} \) |
| 59 | \( 1 + (3.91 - 1.88i)T + (36.7 - 46.1i)T^{2} \) |
| 61 | \( 1 + (-4.45 - 1.01i)T + (54.9 + 26.4i)T^{2} \) |
| 67 | \( 1 - 2.35T + 67T^{2} \) |
| 71 | \( 1 + (2.63 - 0.601i)T + (63.9 - 30.8i)T^{2} \) |
| 73 | \( 1 + (-0.546 - 0.214i)T + (53.5 + 49.6i)T^{2} \) |
| 79 | \( 1 - 0.529T + 79T^{2} \) |
| 83 | \( 1 + (4.19 - 10.6i)T + (-60.8 - 56.4i)T^{2} \) |
| 89 | \( 1 + (6.08 - 0.917i)T + (85.0 - 26.2i)T^{2} \) |
| 97 | \( 1 + (9.87 - 5.69i)T + (48.5 - 84.0i)T^{2} \) |
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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
−11.00554900021418814727069784106, −10.40412220266620808468181154049, −9.777150424331458974047166401888, −9.244889639974267775269277010485, −8.033345186187956908364560742488, −7.09585808177068079781303468987, −6.06333329016856575994420184138, −3.99158635144471881550118678405, −2.93780697604006684921063015396, −2.46806147189442880028479510520,
0.45124582803882080754419098622, 1.75316230824687666225136269179, 3.86184302180112369032524374534, 5.55099632903110646332519069703, 6.31528701637701377324365837101, 7.33587895330151347027916656452, 8.369199430769228674670033606682, 8.645553592152028961659559222906, 9.521268448647823435349264421492, 10.10977743397546570149947776119
