| L(s) = 1 | + 2.34·2-s + 0.609·3-s + 3.51·4-s − 1.66·5-s + 1.43·6-s − 1.77·7-s + 3.55·8-s − 2.62·9-s − 3.91·10-s − 0.606·11-s + 2.14·12-s + 0.813·13-s − 4.15·14-s − 1.01·15-s + 1.31·16-s + 4.59·17-s − 6.17·18-s + 3.35·19-s − 5.86·20-s − 1.08·21-s − 1.42·22-s + 2.48·23-s + 2.16·24-s − 2.21·25-s + 1.91·26-s − 3.43·27-s − 6.22·28-s + ⋯ |
| L(s) = 1 | + 1.66·2-s + 0.352·3-s + 1.75·4-s − 0.746·5-s + 0.584·6-s − 0.669·7-s + 1.25·8-s − 0.876·9-s − 1.23·10-s − 0.182·11-s + 0.618·12-s + 0.225·13-s − 1.11·14-s − 0.262·15-s + 0.329·16-s + 1.11·17-s − 1.45·18-s + 0.770·19-s − 1.31·20-s − 0.235·21-s − 0.303·22-s + 0.518·23-s + 0.442·24-s − 0.443·25-s + 0.374·26-s − 0.660·27-s − 1.17·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 163 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 163 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.394847808\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.394847808\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 163 | \( 1 - T \) |
| good | 2 | \( 1 - 2.34T + 2T^{2} \) |
| 3 | \( 1 - 0.609T + 3T^{2} \) |
| 5 | \( 1 + 1.66T + 5T^{2} \) |
| 7 | \( 1 + 1.77T + 7T^{2} \) |
| 11 | \( 1 + 0.606T + 11T^{2} \) |
| 13 | \( 1 - 0.813T + 13T^{2} \) |
| 17 | \( 1 - 4.59T + 17T^{2} \) |
| 19 | \( 1 - 3.35T + 19T^{2} \) |
| 23 | \( 1 - 2.48T + 23T^{2} \) |
| 29 | \( 1 - 5.39T + 29T^{2} \) |
| 31 | \( 1 - 2.66T + 31T^{2} \) |
| 37 | \( 1 + 1.68T + 37T^{2} \) |
| 41 | \( 1 + 1.10T + 41T^{2} \) |
| 43 | \( 1 + 12.2T + 43T^{2} \) |
| 47 | \( 1 - 8.62T + 47T^{2} \) |
| 53 | \( 1 - 3.09T + 53T^{2} \) |
| 59 | \( 1 + 8.36T + 59T^{2} \) |
| 61 | \( 1 - 13.3T + 61T^{2} \) |
| 67 | \( 1 - 6.84T + 67T^{2} \) |
| 71 | \( 1 + 0.971T + 71T^{2} \) |
| 73 | \( 1 + 6.33T + 73T^{2} \) |
| 79 | \( 1 + 12.8T + 79T^{2} \) |
| 83 | \( 1 + 12.5T + 83T^{2} \) |
| 89 | \( 1 - 3.92T + 89T^{2} \) |
| 97 | \( 1 - 16.3T + 97T^{2} \) |
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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
−12.98105387615794628533975929168, −11.97333894807139936553980997793, −11.47658344754211236395847841363, −10.04677476801400375161918303798, −8.546974237554506148767157442851, −7.35011284552128731698395957192, −6.13043721653694831106622057632, −5.09621504393955739036992433568, −3.65044867129466979333847764461, −2.93393844430639228261608141677,
2.93393844430639228261608141677, 3.65044867129466979333847764461, 5.09621504393955739036992433568, 6.13043721653694831106622057632, 7.35011284552128731698395957192, 8.546974237554506148767157442851, 10.04677476801400375161918303798, 11.47658344754211236395847841363, 11.97333894807139936553980997793, 12.98105387615794628533975929168
