L(s) = 1 | + (0.766 − 0.642i)2-s + (0.173 − 0.984i)4-s + (0.766 + 0.642i)5-s + (−0.5 − 0.866i)8-s + 10-s + (0.766 − 0.642i)11-s + (−0.939 + 0.342i)13-s + (−0.939 − 0.342i)16-s + 17-s + 19-s + (0.766 − 0.642i)20-s + (0.173 − 0.984i)22-s + (−0.939 + 0.342i)23-s + (0.173 + 0.984i)25-s + (−0.5 + 0.866i)26-s + ⋯ |
L(s) = 1 | + (0.766 − 0.642i)2-s + (0.173 − 0.984i)4-s + (0.766 + 0.642i)5-s + (−0.5 − 0.866i)8-s + 10-s + (0.766 − 0.642i)11-s + (−0.939 + 0.342i)13-s + (−0.939 − 0.342i)16-s + 17-s + 19-s + (0.766 − 0.642i)20-s + (0.173 − 0.984i)22-s + (−0.939 + 0.342i)23-s + (0.173 + 0.984i)25-s + (−0.5 + 0.866i)26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.545 - 0.838i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.545 - 0.838i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.684522775 - 0.9139449847i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.684522775 - 0.9139449847i\) |
\(L(1)\) |
\(\approx\) |
\(1.570577210 - 0.5910230526i\) |
\(L(1)\) |
\(\approx\) |
\(1.570577210 - 0.5910230526i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 \) |
good | 2 | \( 1 + (0.766 - 0.642i)T \) |
| 5 | \( 1 + (0.766 + 0.642i)T \) |
| 11 | \( 1 + (0.766 - 0.642i)T \) |
| 13 | \( 1 + (-0.939 + 0.342i)T \) |
| 17 | \( 1 + T \) |
| 19 | \( 1 + T \) |
| 23 | \( 1 + (-0.939 + 0.342i)T \) |
| 29 | \( 1 + (-0.939 - 0.342i)T \) |
| 31 | \( 1 + (0.173 - 0.984i)T \) |
| 37 | \( 1 + (-0.5 - 0.866i)T \) |
| 41 | \( 1 + (-0.939 + 0.342i)T \) |
| 43 | \( 1 + (0.173 + 0.984i)T \) |
| 47 | \( 1 + (0.173 + 0.984i)T \) |
| 53 | \( 1 + (-0.5 - 0.866i)T \) |
| 59 | \( 1 + (-0.939 + 0.342i)T \) |
| 61 | \( 1 + (0.173 + 0.984i)T \) |
| 67 | \( 1 + (0.766 + 0.642i)T \) |
| 71 | \( 1 + (-0.5 + 0.866i)T \) |
| 73 | \( 1 + (-0.5 + 0.866i)T \) |
| 79 | \( 1 + (0.766 - 0.642i)T \) |
| 83 | \( 1 + (-0.939 - 0.342i)T \) |
| 89 | \( 1 + T \) |
| 97 | \( 1 + (0.173 + 0.984i)T \) |
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\(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−27.15487946387516100314719836224, −25.980621540350973656206791151352, −25.13954997129887916663211184141, −24.55504165432280665438752894765, −23.60285064664992326082036817566, −22.3632177028071958193389939206, −21.88333167559893381952487335118, −20.6377066531345436819297639576, −20.04099170541004998211776653734, −18.27915874163396126963709956028, −17.21922019740964726926713947546, −16.7090038662211470722013645869, −15.5086157189298583474995624892, −14.39387285673949719754988782915, −13.74922802270636586697981562626, −12.41702551214979229886315726448, −12.03903158527541083815344983554, −10.13345631875588707825486479563, −9.12158715666793622787641130569, −7.851497450143045019000984162745, −6.77299449560103740542578545187, −5.5452734426576343839512044388, −4.79827244293101398874804024280, −3.38590370701298534874872058178, −1.84704416649776700366415900418,
1.526879207148818731763326419889, 2.77974489525715623672161556856, 3.86343521907362467718261913640, 5.385767102778221797585692680815, 6.18698353855869258668686850451, 7.43030704907919391985738897788, 9.42265976204002803125464696695, 9.97951536419003007018070126593, 11.27080609411698364096231765733, 12.02678124174112003532438881351, 13.322621333595743906485926476410, 14.22138305394219732881767113197, 14.72816481109999461253040379561, 16.18207162594141978243806053481, 17.383184088332911459287930998124, 18.60524757015851619839999807773, 19.30811240522119907317577511708, 20.43205467566517366226742847613, 21.47244105392368730783090578880, 22.12168795023795266282096068784, 22.82044033262753428860014226006, 24.17361930528095755482000829842, 24.77896327416030831895355006632, 26.01195921429748562530278030185, 27.084289362395617805177218669726