L(s) = 1 | + (−0.642 + 0.766i)2-s + (−0.173 − 0.984i)4-s + (1.45 − 4.00i)5-s + (−3.39 − 1.23i)7-s + (0.866 + 0.500i)8-s + (2.13 + 3.69i)10-s + (−1.05 + 1.83i)11-s + (2.84 − 0.500i)13-s + (3.12 − 1.80i)14-s + (−0.939 + 0.342i)16-s + (−0.0263 − 0.00463i)17-s + (−2.07 − 2.47i)19-s + (−4.19 − 0.740i)20-s + (−0.723 − 1.98i)22-s + (−2.57 + 1.48i)23-s + ⋯ |
L(s) = 1 | + (−0.454 + 0.541i)2-s + (−0.0868 − 0.492i)4-s + (0.652 − 1.79i)5-s + (−1.28 − 0.466i)7-s + (0.306 + 0.176i)8-s + (0.674 + 1.16i)10-s + (−0.319 + 0.552i)11-s + (0.787 − 0.138i)13-s + (0.835 − 0.482i)14-s + (−0.234 + 0.0855i)16-s + (−0.00638 − 0.00112i)17-s + (−0.476 − 0.567i)19-s + (−0.939 − 0.165i)20-s + (−0.154 − 0.424i)22-s + (−0.536 + 0.309i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 666 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.560 + 0.827i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 666 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.560 + 0.827i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.334155 - 0.629919i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.334155 - 0.629919i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.642 - 0.766i)T \) |
| 3 | \( 1 \) |
| 37 | \( 1 + (4.49 + 4.09i)T \) |
good | 5 | \( 1 + (-1.45 + 4.00i)T + (-3.83 - 3.21i)T^{2} \) |
| 7 | \( 1 + (3.39 + 1.23i)T + (5.36 + 4.49i)T^{2} \) |
| 11 | \( 1 + (1.05 - 1.83i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (-2.84 + 0.500i)T + (12.2 - 4.44i)T^{2} \) |
| 17 | \( 1 + (0.0263 + 0.00463i)T + (15.9 + 5.81i)T^{2} \) |
| 19 | \( 1 + (2.07 + 2.47i)T + (-3.29 + 18.7i)T^{2} \) |
| 23 | \( 1 + (2.57 - 1.48i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (4.96 + 2.86i)T + (14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 - 6.76iT - 31T^{2} \) |
| 41 | \( 1 + (0.259 + 1.46i)T + (-38.5 + 14.0i)T^{2} \) |
| 43 | \( 1 + 5.53iT - 43T^{2} \) |
| 47 | \( 1 + (-1.30 - 2.26i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-1.79 + 0.652i)T + (40.6 - 34.0i)T^{2} \) |
| 59 | \( 1 + (1.92 + 5.28i)T + (-45.1 + 37.9i)T^{2} \) |
| 61 | \( 1 + (5.65 - 0.996i)T + (57.3 - 20.8i)T^{2} \) |
| 67 | \( 1 + (6.50 + 2.36i)T + (51.3 + 43.0i)T^{2} \) |
| 71 | \( 1 + (-7.10 + 5.96i)T + (12.3 - 69.9i)T^{2} \) |
| 73 | \( 1 - 16.2T + 73T^{2} \) |
| 79 | \( 1 + (0.484 - 1.32i)T + (-60.5 - 50.7i)T^{2} \) |
| 83 | \( 1 + (-0.294 + 1.67i)T + (-77.9 - 28.3i)T^{2} \) |
| 89 | \( 1 + (2.82 + 7.77i)T + (-68.1 + 57.2i)T^{2} \) |
| 97 | \( 1 + (-5.65 + 3.26i)T + (48.5 - 84.0i)T^{2} \) |
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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
−9.921084997584740592968970922805, −9.242395947510552465178049206677, −8.699728900139901281077897574783, −7.68860433655589297615717081021, −6.58900583767742178081651760621, −5.76705483489460115931633467096, −4.90657956079575490120439221821, −3.79021518191474511792319508309, −1.85613112869709213880136028256, −0.42104885896602155377247312217,
2.11115873107271349391616793604, 3.05751545129879217897027002828, 3.71889521796337368214768978177, 5.87210641159737037369707634644, 6.30111497617667418558101750256, 7.21842877009367913748146555556, 8.336481860913834700665925532606, 9.433838584064814844468620603587, 9.990302545629250524351918911202, 10.75115103989673416862271449923