| L(s) = 1 | + (0.797 − 0.175i)2-s + (1.70 + 0.278i)3-s + (−1.21 + 0.559i)4-s + (0.456 + 0.126i)5-s + (1.41 − 0.0781i)6-s + (2.07 − 1.96i)7-s + (−2.16 + 1.64i)8-s + (2.84 + 0.951i)9-s + (0.386 + 0.0209i)10-s + (−2.28 − 2.68i)11-s + (−2.22 + 0.620i)12-s + (1.74 + 5.18i)13-s + (1.30 − 1.92i)14-s + (0.745 + 0.343i)15-s + (0.288 − 0.339i)16-s + (−2.42 + 2.56i)17-s + ⋯ |
| L(s) = 1 | + (0.563 − 0.124i)2-s + (0.987 + 0.160i)3-s + (−0.605 + 0.279i)4-s + (0.204 + 0.0567i)5-s + (0.576 − 0.0319i)6-s + (0.782 − 0.741i)7-s + (−0.765 + 0.582i)8-s + (0.948 + 0.317i)9-s + (0.122 + 0.00662i)10-s + (−0.688 − 0.810i)11-s + (−0.642 + 0.179i)12-s + (0.484 + 1.43i)13-s + (0.349 − 0.514i)14-s + (0.192 + 0.0887i)15-s + (0.0720 − 0.0847i)16-s + (−0.589 + 0.622i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.995 - 0.0987i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.995 - 0.0987i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.79948 + 0.0890294i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.79948 + 0.0890294i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-1.70 - 0.278i)T \) |
| 59 | \( 1 + (3.95 - 6.58i)T \) |
| good | 2 | \( 1 + (-0.797 + 0.175i)T + (1.81 - 0.839i)T^{2} \) |
| 5 | \( 1 + (-0.456 - 0.126i)T + (4.28 + 2.57i)T^{2} \) |
| 7 | \( 1 + (-2.07 + 1.96i)T + (0.378 - 6.98i)T^{2} \) |
| 11 | \( 1 + (2.28 + 2.68i)T + (-1.77 + 10.8i)T^{2} \) |
| 13 | \( 1 + (-1.74 - 5.18i)T + (-10.3 + 7.86i)T^{2} \) |
| 17 | \( 1 + (2.42 - 2.56i)T + (-0.920 - 16.9i)T^{2} \) |
| 19 | \( 1 + (2.57 + 6.46i)T + (-13.7 + 13.0i)T^{2} \) |
| 23 | \( 1 + (9.33 - 1.01i)T + (22.4 - 4.94i)T^{2} \) |
| 29 | \( 1 + (-1.13 + 5.17i)T + (-26.3 - 12.1i)T^{2} \) |
| 31 | \( 1 + (-4.78 - 1.90i)T + (22.5 + 21.3i)T^{2} \) |
| 37 | \( 1 + (0.0730 - 0.0961i)T + (-9.89 - 35.6i)T^{2} \) |
| 41 | \( 1 + (0.354 - 3.25i)T + (-40.0 - 8.81i)T^{2} \) |
| 43 | \( 1 + (-4.72 - 4.01i)T + (6.95 + 42.4i)T^{2} \) |
| 47 | \( 1 + (0.590 + 2.12i)T + (-40.2 + 24.2i)T^{2} \) |
| 53 | \( 1 + (0.396 - 0.0215i)T + (52.6 - 5.73i)T^{2} \) |
| 61 | \( 1 + (-0.313 - 1.42i)T + (-55.3 + 25.6i)T^{2} \) |
| 67 | \( 1 + (-2.39 - 3.15i)T + (-17.9 + 64.5i)T^{2} \) |
| 71 | \( 1 + (-10.4 + 2.91i)T + (60.8 - 36.6i)T^{2} \) |
| 73 | \( 1 + (-8.06 - 5.46i)T + (27.0 + 67.8i)T^{2} \) |
| 79 | \( 1 + (2.04 + 12.4i)T + (-74.8 + 25.2i)T^{2} \) |
| 83 | \( 1 + (-2.74 - 5.17i)T + (-46.5 + 68.6i)T^{2} \) |
| 89 | \( 1 + (-11.6 - 2.56i)T + (80.7 + 37.3i)T^{2} \) |
| 97 | \( 1 + (10.9 - 7.44i)T + (35.9 - 90.1i)T^{2} \) |
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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
−13.19748692998430929282875185267, −11.79704239029240100207063966359, −10.77992752355219313864752428156, −9.566038616926927674767756591658, −8.516165520193485414724749151542, −7.927169151010191463998839386479, −6.30675341891709233039350559358, −4.54735604336973120203467488561, −3.99211841478442904039517491087, −2.32969974318220212062691116008,
2.13382543776025880052791428887, 3.76956864394489782271296401903, 5.03200117347835654231104978845, 6.06809486049859150009596101970, 7.85600385933213888091746233402, 8.438552081558708519891657979859, 9.650006392492455912821442852646, 10.41330299125238200555532855216, 12.23016496531143371478173037495, 12.77483601623103748180122833080
