| L(s) = 1 | + (−0.601 + 1.73i)2-s + (0.530 + 1.02i)3-s + (−1.09 − 0.857i)4-s + (−1.45 + 0.138i)5-s + (−2.10 + 0.303i)6-s + (−2.03 + 1.69i)7-s + (−0.948 + 0.609i)8-s + (0.962 − 1.35i)9-s + (0.633 − 2.61i)10-s + (−0.0712 + 0.0246i)11-s + (0.303 − 1.57i)12-s + (1.89 + 6.45i)13-s + (−1.72 − 4.55i)14-s + (−0.913 − 1.42i)15-s + (−1.14 − 4.71i)16-s + (−0.122 − 0.0489i)17-s + ⋯ |
| L(s) = 1 | + (−0.425 + 1.22i)2-s + (0.306 + 0.594i)3-s + (−0.545 − 0.428i)4-s + (−0.649 + 0.0620i)5-s + (−0.860 + 0.123i)6-s + (−0.767 + 0.640i)7-s + (−0.335 + 0.215i)8-s + (0.320 − 0.450i)9-s + (0.200 − 0.825i)10-s + (−0.0214 + 0.00743i)11-s + (0.0877 − 0.455i)12-s + (0.525 + 1.79i)13-s + (−0.461 − 1.21i)14-s + (−0.235 − 0.366i)15-s + (−0.285 − 1.17i)16-s + (−0.0296 − 0.0118i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 161 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.988 - 0.148i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 161 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.988 - 0.148i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0614640 + 0.825012i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0614640 + 0.825012i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 + (2.03 - 1.69i)T \) |
| 23 | \( 1 + (3.15 + 3.61i)T \) |
| good | 2 | \( 1 + (0.601 - 1.73i)T + (-1.57 - 1.23i)T^{2} \) |
| 3 | \( 1 + (-0.530 - 1.02i)T + (-1.74 + 2.44i)T^{2} \) |
| 5 | \( 1 + (1.45 - 0.138i)T + (4.90 - 0.946i)T^{2} \) |
| 11 | \( 1 + (0.0712 - 0.0246i)T + (8.64 - 6.79i)T^{2} \) |
| 13 | \( 1 + (-1.89 - 6.45i)T + (-10.9 + 7.02i)T^{2} \) |
| 17 | \( 1 + (0.122 + 0.0489i)T + (12.3 + 11.7i)T^{2} \) |
| 19 | \( 1 + (-6.22 + 2.49i)T + (13.7 - 13.1i)T^{2} \) |
| 29 | \( 1 + (-1.08 - 7.51i)T + (-27.8 + 8.17i)T^{2} \) |
| 31 | \( 1 + (-8.46 + 0.403i)T + (30.8 - 2.94i)T^{2} \) |
| 37 | \( 1 + (-4.96 - 3.53i)T + (12.1 + 34.9i)T^{2} \) |
| 41 | \( 1 + (-0.923 - 0.421i)T + (26.8 + 30.9i)T^{2} \) |
| 43 | \( 1 + (2.48 - 3.86i)T + (-17.8 - 39.1i)T^{2} \) |
| 47 | \( 1 + (0.735 - 0.424i)T + (23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (6.26 + 6.57i)T + (-2.52 + 52.9i)T^{2} \) |
| 59 | \( 1 + (-5.69 - 1.38i)T + (52.4 + 27.0i)T^{2} \) |
| 61 | \( 1 + (1.85 + 0.957i)T + (35.3 + 49.6i)T^{2} \) |
| 67 | \( 1 + (2.00 + 10.3i)T + (-62.2 + 24.9i)T^{2} \) |
| 71 | \( 1 + (2.67 + 3.08i)T + (-10.1 + 70.2i)T^{2} \) |
| 73 | \( 1 + (-0.126 + 0.160i)T + (-17.2 - 70.9i)T^{2} \) |
| 79 | \( 1 + (-4.20 + 4.40i)T + (-3.75 - 78.9i)T^{2} \) |
| 83 | \( 1 + (4.32 + 9.47i)T + (-54.3 + 62.7i)T^{2} \) |
| 89 | \( 1 + (0.572 - 12.0i)T + (-88.5 - 8.45i)T^{2} \) |
| 97 | \( 1 + (0.934 - 2.04i)T + (-63.5 - 73.3i)T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−13.73808802523294591372169199502, −12.17572491446894516427831760901, −11.51852609650035285899008910598, −9.743808544752799041819930947853, −9.168437006637021286143972857834, −8.253184921928359619208424539453, −6.93662293134136347444805053019, −6.27820590993704384633329865643, −4.63389646948726101146096958465, −3.20636708761911581915962979194,
0.935131479174652699274597432744, 2.82915703925132000836328278093, 3.88817646950641330017945664041, 5.98065274108846857437833468019, 7.54416040598786220170437131269, 8.180416400824753117572401901969, 9.828372405496682667939735731783, 10.27158276183574085101598203279, 11.44748291906350347793702855161, 12.30871624546155329949727955291
