| 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
−11.90402106830483397404815393404, −11.15079727089347502497376989300, −10.45093152799674982307404982661, −9.602593209341582122156072648754, −7.53560464314313735278883629783, −6.89299937154062702326616764182, −5.19373839807526203549437783047, −3.78445260187798750058275835570, −2.72989021282127768327969650532, −0.093060363895267442057991324735,
4.07753612546509581121885907706, 4.85279801867260755590325893334, 6.15766231871261058688868439408, 6.93521922693506160247128999128, 8.025292226095388640570026995746, 9.004737006736589915083357882229, 10.52679730584805387241492339942, 11.49275445977248146828914516297, 12.69257621006731364204956017053, 13.07045693951936178696725877388
