| L(s) = 1 | + (0.703 + 1.22i)2-s + (0.956 − 2.31i)3-s + (−1.01 + 1.72i)4-s + (−0.300 + 0.152i)5-s + (3.50 − 0.450i)6-s + (3.46 − 2.12i)7-s + (−2.82 − 0.0266i)8-s + (−2.30 − 2.30i)9-s + (−0.398 − 0.260i)10-s + (−0.267 + 3.40i)11-s + (3.01 + 3.98i)12-s + (−2.06 + 0.495i)13-s + (5.04 + 2.76i)14-s + (0.0661 + 0.839i)15-s + (−1.95 − 3.48i)16-s + (−0.262 − 0.307i)17-s + ⋯ |
| L(s) = 1 | + (0.497 + 0.867i)2-s + (0.552 − 1.33i)3-s + (−0.505 + 0.862i)4-s + (−0.134 + 0.0684i)5-s + (1.43 − 0.183i)6-s + (1.31 − 0.803i)7-s + (−0.999 − 0.00941i)8-s + (−0.766 − 0.766i)9-s + (−0.126 − 0.0824i)10-s + (−0.0807 + 1.02i)11-s + (0.871 + 1.15i)12-s + (−0.572 + 0.137i)13-s + (1.34 + 0.737i)14-s + (0.0170 + 0.216i)15-s + (−0.489 − 0.872i)16-s + (−0.0637 − 0.0746i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 164 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.987 - 0.158i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 164 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.987 - 0.158i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.63310 + 0.130071i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.63310 + 0.130071i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.703 - 1.22i)T \) |
| 41 | \( 1 + (-3.05 + 5.62i)T \) |
| good | 3 | \( 1 + (-0.956 + 2.31i)T + (-2.12 - 2.12i)T^{2} \) |
| 5 | \( 1 + (0.300 - 0.152i)T + (2.93 - 4.04i)T^{2} \) |
| 7 | \( 1 + (-3.46 + 2.12i)T + (3.17 - 6.23i)T^{2} \) |
| 11 | \( 1 + (0.267 - 3.40i)T + (-10.8 - 1.72i)T^{2} \) |
| 13 | \( 1 + (2.06 - 0.495i)T + (11.5 - 5.90i)T^{2} \) |
| 17 | \( 1 + (0.262 + 0.307i)T + (-2.65 + 16.7i)T^{2} \) |
| 19 | \( 1 + (0.0388 - 0.161i)T + (-16.9 - 8.62i)T^{2} \) |
| 23 | \( 1 + (2.58 - 1.87i)T + (7.10 - 21.8i)T^{2} \) |
| 29 | \( 1 + (5.71 - 6.68i)T + (-4.53 - 28.6i)T^{2} \) |
| 31 | \( 1 + (3.33 + 10.2i)T + (-25.0 + 18.2i)T^{2} \) |
| 37 | \( 1 + (1.96 - 6.04i)T + (-29.9 - 21.7i)T^{2} \) |
| 43 | \( 1 + (-6.05 - 0.958i)T + (40.8 + 13.2i)T^{2} \) |
| 47 | \( 1 + (5.56 + 3.41i)T + (21.3 + 41.8i)T^{2} \) |
| 53 | \( 1 + (-3.52 - 3.01i)T + (8.29 + 52.3i)T^{2} \) |
| 59 | \( 1 + (0.573 + 0.788i)T + (-18.2 + 56.1i)T^{2} \) |
| 61 | \( 1 + (-4.52 + 0.716i)T + (58.0 - 18.8i)T^{2} \) |
| 67 | \( 1 + (-11.4 + 0.900i)T + (66.1 - 10.4i)T^{2} \) |
| 71 | \( 1 + (-8.75 - 0.688i)T + (70.1 + 11.1i)T^{2} \) |
| 73 | \( 1 + (-5.21 + 5.21i)T - 73iT^{2} \) |
| 79 | \( 1 + (13.9 + 5.79i)T + (55.8 + 55.8i)T^{2} \) |
| 83 | \( 1 - 9.27iT - 83T^{2} \) |
| 89 | \( 1 + (-8.67 - 14.1i)T + (-40.4 + 79.2i)T^{2} \) |
| 97 | \( 1 + (7.56 - 0.595i)T + (95.8 - 15.1i)T^{2} \) |
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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
−13.09636163108933780882661194403, −12.25393292381671409949858620801, −11.25787152224241158783929769996, −9.484674624037153937521983741035, −8.102070232953866033391100893234, −7.51953221386927453979088143182, −6.96994891910384758365003186666, −5.32043700639733942079619087716, −4.02598918453582138108324300511, −1.99756949925919584941411195970,
2.38239953802124089642584809234, 3.77905768679386899977083150733, 4.81896826798692813658765538274, 5.72263054607236707042098010021, 8.196816235440100840547441015730, 8.929593591427639189336060298442, 9.933473916613284887664768249847, 10.92101302854762948848591192286, 11.58211005471152760180420319641, 12.67768841800713982542056977121
