| L(s) = 1 | + (−1.55 − 1.30i)2-s + (1.19 + 1.24i)3-s + (0.368 + 2.08i)4-s + (0.441 + 2.50i)5-s + (−0.234 − 3.50i)6-s + (−0.472 − 2.60i)7-s + (0.123 − 0.213i)8-s + (−0.123 + 2.99i)9-s + (2.58 − 4.46i)10-s + (−0.994 + 5.64i)11-s + (−2.16 + 2.96i)12-s + (0.494 + 2.80i)13-s + (−2.66 + 4.66i)14-s + (−2.59 + 3.55i)15-s + (3.51 − 1.27i)16-s + (2.41 − 4.18i)17-s + ⋯ |
| L(s) = 1 | + (−1.09 − 0.922i)2-s + (0.692 + 0.721i)3-s + (0.184 + 1.04i)4-s + (0.197 + 1.11i)5-s + (−0.0956 − 1.43i)6-s + (−0.178 − 0.983i)7-s + (0.0436 − 0.0755i)8-s + (−0.0411 + 0.999i)9-s + (0.816 − 1.41i)10-s + (−0.299 + 1.70i)11-s + (−0.626 + 0.856i)12-s + (0.137 + 0.778i)13-s + (−0.711 + 1.24i)14-s + (−0.671 + 0.917i)15-s + (0.879 − 0.319i)16-s + (0.586 − 1.01i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.847 - 0.530i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.847 - 0.530i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.781913 + 0.224585i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.781913 + 0.224585i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-1.19 - 1.24i)T \) |
| 7 | \( 1 + (0.472 + 2.60i)T \) |
| good | 2 | \( 1 + (1.55 + 1.30i)T + (0.347 + 1.96i)T^{2} \) |
| 5 | \( 1 + (-0.441 - 2.50i)T + (-4.69 + 1.71i)T^{2} \) |
| 11 | \( 1 + (0.994 - 5.64i)T + (-10.3 - 3.76i)T^{2} \) |
| 13 | \( 1 + (-0.494 - 2.80i)T + (-12.2 + 4.44i)T^{2} \) |
| 17 | \( 1 + (-2.41 + 4.18i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (0.277 + 0.481i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-2.10 + 1.76i)T + (3.99 - 22.6i)T^{2} \) |
| 29 | \( 1 + (-0.164 + 0.930i)T + (-27.2 - 9.91i)T^{2} \) |
| 31 | \( 1 + (-0.867 - 4.92i)T + (-29.1 + 10.6i)T^{2} \) |
| 37 | \( 1 - 8.16T + 37T^{2} \) |
| 41 | \( 1 + (1.92 + 10.8i)T + (-38.5 + 14.0i)T^{2} \) |
| 43 | \( 1 + (4.80 + 4.02i)T + (7.46 + 42.3i)T^{2} \) |
| 47 | \( 1 + (0.517 - 2.93i)T + (-44.1 - 16.0i)T^{2} \) |
| 53 | \( 1 + (-3.07 - 5.32i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (8.92 + 3.24i)T + (45.1 + 37.9i)T^{2} \) |
| 61 | \( 1 + (-0.889 + 5.04i)T + (-57.3 - 20.8i)T^{2} \) |
| 67 | \( 1 + (-7.29 + 6.12i)T + (11.6 - 65.9i)T^{2} \) |
| 71 | \( 1 + (5.28 + 9.14i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + 0.456T + 73T^{2} \) |
| 79 | \( 1 + (-2.52 - 2.11i)T + (13.7 + 77.7i)T^{2} \) |
| 83 | \( 1 + (-0.441 + 2.50i)T + (-77.9 - 28.3i)T^{2} \) |
| 89 | \( 1 + (-3.34 - 5.79i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-4.52 - 3.79i)T + (16.8 + 95.5i)T^{2} \) |
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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
−12.36048134214890002731076104322, −11.12442261250899388045235571453, −10.42499583198553692805365232646, −9.867420519982950532927906746180, −9.142224436525744978864033156031, −7.69764432555264946327504770094, −6.97058510309194526394185124382, −4.70526554401298436465362139332, −3.25623482756583419321267462862, −2.13953343078642880039358210185,
1.04199615663741356608308188239, 3.19830744154706267420678338395, 5.64927692305721779732799170963, 6.23727003986174478350962869035, 7.921395054006350475448982679102, 8.360746221446313510778122459236, 8.973347263335171947640387723574, 9.926827418503288547902092475362, 11.57332213802543530451286608253, 12.90417401322916474115877086931
