| L(s) = 1 | + (0.317 + 1.93i)2-s + (−1.70 + 0.290i)3-s + (−1.75 + 0.589i)4-s + (−2.92 − 1.98i)5-s + (−1.10 − 3.21i)6-s + (−2.73 − 0.602i)7-s + (0.140 + 0.265i)8-s + (2.83 − 0.993i)9-s + (2.91 − 6.29i)10-s + (−2.01 − 1.52i)11-s + (2.81 − 1.51i)12-s + (−2.03 + 3.37i)13-s + (0.297 − 5.49i)14-s + (5.57 + 2.53i)15-s + (−3.40 + 2.59i)16-s + (1.04 + 4.76i)17-s + ⋯ |
| L(s) = 1 | + (0.224 + 1.36i)2-s + (−0.985 + 0.167i)3-s + (−0.875 + 0.294i)4-s + (−1.30 − 0.886i)5-s + (−0.451 − 1.31i)6-s + (−1.03 − 0.227i)7-s + (0.0497 + 0.0938i)8-s + (0.943 − 0.331i)9-s + (0.920 − 1.98i)10-s + (−0.606 − 0.461i)11-s + (0.813 − 0.437i)12-s + (−0.563 + 0.936i)13-s + (0.0795 − 1.46i)14-s + (1.43 + 0.654i)15-s + (−0.852 + 0.647i)16-s + (0.254 + 1.15i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.356 + 0.934i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.356 + 0.934i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0577257 - 0.0837877i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0577257 - 0.0837877i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (1.70 - 0.290i)T \) |
| 59 | \( 1 + (7.29 + 2.40i)T \) |
| good | 2 | \( 1 + (-0.317 - 1.93i)T + (-1.89 + 0.638i)T^{2} \) |
| 5 | \( 1 + (2.92 + 1.98i)T + (1.85 + 4.64i)T^{2} \) |
| 7 | \( 1 + (2.73 + 0.602i)T + (6.35 + 2.93i)T^{2} \) |
| 11 | \( 1 + (2.01 + 1.52i)T + (2.94 + 10.5i)T^{2} \) |
| 13 | \( 1 + (2.03 - 3.37i)T + (-6.08 - 11.4i)T^{2} \) |
| 17 | \( 1 + (-1.04 - 4.76i)T + (-15.4 + 7.13i)T^{2} \) |
| 19 | \( 1 + (5.48 + 0.596i)T + (18.5 + 4.08i)T^{2} \) |
| 23 | \( 1 + (-5.39 + 6.34i)T + (-3.72 - 22.6i)T^{2} \) |
| 29 | \( 1 + (4.96 + 0.814i)T + (27.4 + 9.25i)T^{2} \) |
| 31 | \( 1 + (0.0516 + 0.474i)T + (-30.2 + 6.66i)T^{2} \) |
| 37 | \( 1 + (-4.35 - 2.31i)T + (20.7 + 30.6i)T^{2} \) |
| 41 | \( 1 + (5.04 - 4.28i)T + (6.63 - 40.4i)T^{2} \) |
| 43 | \( 1 + (1.09 + 1.44i)T + (-11.5 + 41.4i)T^{2} \) |
| 47 | \( 1 + (-2.14 - 3.16i)T + (-17.3 + 43.6i)T^{2} \) |
| 53 | \( 1 + (-2.23 - 4.83i)T + (-34.3 + 40.3i)T^{2} \) |
| 61 | \( 1 + (-10.4 + 1.71i)T + (57.8 - 19.4i)T^{2} \) |
| 67 | \( 1 + (2.77 - 1.47i)T + (37.5 - 55.4i)T^{2} \) |
| 71 | \( 1 + (-2.09 + 1.41i)T + (26.2 - 65.9i)T^{2} \) |
| 73 | \( 1 + (4.66 + 0.252i)T + (72.5 + 7.89i)T^{2} \) |
| 79 | \( 1 + (-0.406 + 1.46i)T + (-67.6 - 40.7i)T^{2} \) |
| 83 | \( 1 + (-0.830 + 0.787i)T + (4.49 - 82.8i)T^{2} \) |
| 89 | \( 1 + (-1.85 + 11.3i)T + (-84.3 - 28.4i)T^{2} \) |
| 97 | \( 1 + (11.0 - 0.598i)T + (96.4 - 10.4i)T^{2} \) |
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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
−13.07045693951936178696725877388, −12.69257621006731364204956017053, −11.49275445977248146828914516297, −10.52679730584805387241492339942, −9.004737006736589915083357882229, −8.025292226095388640570026995746, −6.93521922693506160247128999128, −6.15766231871261058688868439408, −4.85279801867260755590325893334, −4.07753612546509581121885907706,
0.093060363895267442057991324735, 2.72989021282127768327969650532, 3.78445260187798750058275835570, 5.19373839807526203549437783047, 6.89299937154062702326616764182, 7.53560464314313735278883629783, 9.602593209341582122156072648754, 10.45093152799674982307404982661, 11.15079727089347502497376989300, 11.90402106830483397404815393404
