| L(s) = 1 | + (−1.42 − 1.11i)2-s + (−0.659 + 0.925i)3-s + (0.301 + 1.24i)4-s + (2.15 − 0.415i)5-s + (1.97 − 0.579i)6-s + (−1.48 + 2.19i)7-s + (−0.542 + 1.18i)8-s + (0.558 + 1.61i)9-s + (−3.52 − 1.81i)10-s + (4.13 − 3.25i)11-s + (−1.34 − 0.540i)12-s + (2.70 − 1.74i)13-s + (4.56 − 1.46i)14-s + (−1.03 + 2.26i)15-s + (4.37 − 2.25i)16-s + (2.17 + 2.07i)17-s + ⋯ |
| L(s) = 1 | + (−1.00 − 0.791i)2-s + (−0.380 + 0.534i)3-s + (0.150 + 0.621i)4-s + (0.963 − 0.185i)5-s + (0.806 − 0.236i)6-s + (−0.559 + 0.828i)7-s + (−0.191 + 0.419i)8-s + (0.186 + 0.538i)9-s + (−1.11 − 0.575i)10-s + (1.24 − 0.981i)11-s + (−0.389 − 0.155i)12-s + (0.751 − 0.482i)13-s + (1.21 − 0.390i)14-s + (−0.267 + 0.585i)15-s + (1.09 − 0.563i)16-s + (0.528 + 0.503i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 161 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.978 + 0.207i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 161 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.978 + 0.207i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.714624 - 0.0748209i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.714624 - 0.0748209i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 + (1.48 - 2.19i)T \) |
| 23 | \( 1 + (-2.78 - 3.90i)T \) |
| good | 2 | \( 1 + (1.42 + 1.11i)T + (0.471 + 1.94i)T^{2} \) |
| 3 | \( 1 + (0.659 - 0.925i)T + (-0.981 - 2.83i)T^{2} \) |
| 5 | \( 1 + (-2.15 + 0.415i)T + (4.64 - 1.85i)T^{2} \) |
| 11 | \( 1 + (-4.13 + 3.25i)T + (2.59 - 10.6i)T^{2} \) |
| 13 | \( 1 + (-2.70 + 1.74i)T + (5.40 - 11.8i)T^{2} \) |
| 17 | \( 1 + (-2.17 - 2.07i)T + (0.808 + 16.9i)T^{2} \) |
| 19 | \( 1 + (-0.648 + 0.617i)T + (0.904 - 18.9i)T^{2} \) |
| 29 | \( 1 + (-4.62 + 1.35i)T + (24.3 - 15.6i)T^{2} \) |
| 31 | \( 1 + (9.86 - 0.942i)T + (30.4 - 5.86i)T^{2} \) |
| 37 | \( 1 + (0.733 + 2.11i)T + (-29.0 + 22.8i)T^{2} \) |
| 41 | \( 1 + (-6.26 - 7.23i)T + (-5.83 + 40.5i)T^{2} \) |
| 43 | \( 1 + (0.00349 + 0.00765i)T + (-28.1 + 32.4i)T^{2} \) |
| 47 | \( 1 + (4.50 - 7.80i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-0.474 + 9.95i)T + (-52.7 - 5.03i)T^{2} \) |
| 59 | \( 1 + (5.69 + 2.93i)T + (34.2 + 48.0i)T^{2} \) |
| 61 | \( 1 + (2.62 + 3.68i)T + (-19.9 + 57.6i)T^{2} \) |
| 67 | \( 1 + (1.71 - 0.685i)T + (48.4 - 46.2i)T^{2} \) |
| 71 | \( 1 + (-2.37 + 16.5i)T + (-68.1 - 20.0i)T^{2} \) |
| 73 | \( 1 + (-0.623 - 2.56i)T + (-64.8 + 33.4i)T^{2} \) |
| 79 | \( 1 + (-0.559 - 11.7i)T + (-78.6 + 7.50i)T^{2} \) |
| 83 | \( 1 + (-4.07 + 4.69i)T + (-11.8 - 82.1i)T^{2} \) |
| 89 | \( 1 + (3.25 + 0.311i)T + (87.3 + 16.8i)T^{2} \) |
| 97 | \( 1 + (3.04 + 3.51i)T + (-13.8 + 96.0i)T^{2} \) |
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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
−12.61661820232391351306352637277, −11.38536836306160506536128737219, −10.81464093820611047436181010258, −9.633478241680483175750362400278, −9.270756014285484590211702686124, −8.182414688737405239161394167396, −6.11370043722142014154529048901, −5.44292869224371995534088444098, −3.31120638797342206677867698079, −1.58323130784852748233819378285,
1.25037358023856603031010923515, 3.86243033979875722887584513068, 6.02963077257583866809900727909, 6.77544566141063097332934540082, 7.30874179410672001461174927592, 9.041027818819889816461145689129, 9.560213445692024731159157694722, 10.51662003906684958351131478505, 12.04613039992410764190821030622, 12.87161799928844202470677426961
