| L(s) = 1 | + (1.32 − 0.506i)2-s + (0.323 + 0.133i)3-s + (1.48 − 1.33i)4-s + (−0.705 + 0.359i)5-s + (0.494 + 0.0131i)6-s + (2.28 + 3.73i)7-s + (1.28 − 2.51i)8-s + (−2.03 − 2.03i)9-s + (−0.750 + 0.832i)10-s + (−2.94 − 0.231i)11-s + (0.659 − 0.233i)12-s + (−1.05 − 4.39i)13-s + (4.91 + 3.77i)14-s + (−0.276 + 0.0217i)15-s + (0.423 − 3.97i)16-s + (−2.76 + 2.36i)17-s + ⋯ |
| L(s) = 1 | + (0.933 − 0.358i)2-s + (0.186 + 0.0773i)3-s + (0.743 − 0.668i)4-s + (−0.315 + 0.160i)5-s + (0.201 + 0.00536i)6-s + (0.865 + 1.41i)7-s + (0.454 − 0.890i)8-s + (−0.678 − 0.678i)9-s + (−0.237 + 0.263i)10-s + (−0.886 − 0.0697i)11-s + (0.190 − 0.0672i)12-s + (−0.292 − 1.21i)13-s + (1.31 + 1.00i)14-s + (−0.0713 + 0.00561i)15-s + (0.105 − 0.994i)16-s + (−0.671 + 0.573i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 164 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.930 + 0.367i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 164 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.930 + 0.367i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.85427 - 0.352575i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.85427 - 0.352575i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.32 + 0.506i)T \) |
| 41 | \( 1 + (-1.56 + 6.20i)T \) |
| good | 3 | \( 1 + (-0.323 - 0.133i)T + (2.12 + 2.12i)T^{2} \) |
| 5 | \( 1 + (0.705 - 0.359i)T + (2.93 - 4.04i)T^{2} \) |
| 7 | \( 1 + (-2.28 - 3.73i)T + (-3.17 + 6.23i)T^{2} \) |
| 11 | \( 1 + (2.94 + 0.231i)T + (10.8 + 1.72i)T^{2} \) |
| 13 | \( 1 + (1.05 + 4.39i)T + (-11.5 + 5.90i)T^{2} \) |
| 17 | \( 1 + (2.76 - 2.36i)T + (2.65 - 16.7i)T^{2} \) |
| 19 | \( 1 + (-4.50 - 1.08i)T + (16.9 + 8.62i)T^{2} \) |
| 23 | \( 1 + (6.50 - 4.72i)T + (7.10 - 21.8i)T^{2} \) |
| 29 | \( 1 + (-5.31 - 4.54i)T + (4.53 + 28.6i)T^{2} \) |
| 31 | \( 1 + (-0.131 - 0.403i)T + (-25.0 + 18.2i)T^{2} \) |
| 37 | \( 1 + (-2.21 + 6.82i)T + (-29.9 - 21.7i)T^{2} \) |
| 43 | \( 1 + (3.65 + 0.579i)T + (40.8 + 13.2i)T^{2} \) |
| 47 | \( 1 + (-4.08 + 6.66i)T + (-21.3 - 41.8i)T^{2} \) |
| 53 | \( 1 + (1.61 - 1.88i)T + (-8.29 - 52.3i)T^{2} \) |
| 59 | \( 1 + (-2.29 - 3.16i)T + (-18.2 + 56.1i)T^{2} \) |
| 61 | \( 1 + (-12.3 + 1.96i)T + (58.0 - 18.8i)T^{2} \) |
| 67 | \( 1 + (-0.516 - 6.55i)T + (-66.1 + 10.4i)T^{2} \) |
| 71 | \( 1 + (-0.111 + 1.41i)T + (-70.1 - 11.1i)T^{2} \) |
| 73 | \( 1 + (-2.08 + 2.08i)T - 73iT^{2} \) |
| 79 | \( 1 + (-3.24 + 7.83i)T + (-55.8 - 55.8i)T^{2} \) |
| 83 | \( 1 - 5.01iT - 83T^{2} \) |
| 89 | \( 1 + (5.31 - 3.25i)T + (40.4 - 79.2i)T^{2} \) |
| 97 | \( 1 + (-0.157 - 1.99i)T + (-95.8 + 15.1i)T^{2} \) |
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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
−12.58753110216212020024445316776, −11.91117545055026561552364743174, −11.14069114046863767363380434503, −9.976667557892382821817819040956, −8.616253787180718520582683224915, −7.58642987548622015478805650161, −5.78019290914039428478876931694, −5.32011454757303002344590573034, −3.52844745006874629625571951689, −2.34653971281941295269609747430,
2.43420253201938227937763172737, 4.27336323473933368225681544203, 4.90403089301919973255655949029, 6.56714534426092687857712812894, 7.73005662597124915747600400762, 8.205718417062188119933524359717, 10.13946414439910485692641022752, 11.28379502528012100054909294143, 11.80832229772447438615525164906, 13.28600939663946304424273542959
