L(s) = 1 | + (−0.955 + 0.294i)5-s + (−0.0747 + 0.997i)11-s + (−0.0747 + 0.997i)13-s + (−0.365 + 0.930i)17-s + (−0.5 + 0.866i)19-s + (0.988 + 0.149i)23-s + (0.826 − 0.563i)25-s + (0.365 − 0.930i)29-s + 31-s + (−0.988 + 0.149i)37-s + (0.733 + 0.680i)41-s + (0.733 − 0.680i)43-s + (−0.900 + 0.433i)47-s + (−0.988 − 0.149i)53-s + (−0.222 − 0.974i)55-s + ⋯ |
L(s) = 1 | + (−0.955 + 0.294i)5-s + (−0.0747 + 0.997i)11-s + (−0.0747 + 0.997i)13-s + (−0.365 + 0.930i)17-s + (−0.5 + 0.866i)19-s + (0.988 + 0.149i)23-s + (0.826 − 0.563i)25-s + (0.365 − 0.930i)29-s + 31-s + (−0.988 + 0.149i)37-s + (0.733 + 0.680i)41-s + (0.733 − 0.680i)43-s + (−0.900 + 0.433i)47-s + (−0.988 − 0.149i)53-s + (−0.222 − 0.974i)55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.960 + 0.277i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.960 + 0.277i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.09410314754 + 0.6644263211i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.09410314754 + 0.6644263211i\) |
\(L(1)\) |
\(\approx\) |
\(0.7510365973 + 0.2513979193i\) |
\(L(1)\) |
\(\approx\) |
\(0.7510365973 + 0.2513979193i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
good | 5 | \( 1 + (-0.955 + 0.294i)T \) |
| 11 | \( 1 + (-0.0747 + 0.997i)T \) |
| 13 | \( 1 + (-0.0747 + 0.997i)T \) |
| 17 | \( 1 + (-0.365 + 0.930i)T \) |
| 19 | \( 1 + (-0.5 + 0.866i)T \) |
| 23 | \( 1 + (0.988 + 0.149i)T \) |
| 29 | \( 1 + (0.365 - 0.930i)T \) |
| 31 | \( 1 + T \) |
| 37 | \( 1 + (-0.988 + 0.149i)T \) |
| 41 | \( 1 + (0.733 + 0.680i)T \) |
| 43 | \( 1 + (0.733 - 0.680i)T \) |
| 47 | \( 1 + (-0.900 + 0.433i)T \) |
| 53 | \( 1 + (-0.988 - 0.149i)T \) |
| 59 | \( 1 + (-0.222 - 0.974i)T \) |
| 61 | \( 1 + (-0.623 - 0.781i)T \) |
| 67 | \( 1 - T \) |
| 71 | \( 1 + (-0.623 + 0.781i)T \) |
| 73 | \( 1 + (-0.0747 - 0.997i)T \) |
| 79 | \( 1 - T \) |
| 83 | \( 1 + (0.0747 + 0.997i)T \) |
| 89 | \( 1 + (-0.826 + 0.563i)T \) |
| 97 | \( 1 + (0.5 + 0.866i)T \) |
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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
−19.64509997807191965388869753940, −19.43696347113914283338853153941, −18.49378373743268970534811206369, −17.70388029860570816273831626883, −16.91274375933095863623840267356, −16.02794914133590411086732619816, −15.635932032475428344611006898199, −14.84506674733038049842777456830, −13.92240292283008828090245896445, −13.102052949188393669766807566767, −12.4896254566453106845719570864, −11.5448154263568993615218227547, −11.00410131292287441013060418258, −10.27115815434792026235731749211, −8.88814684398888170055006181768, −8.71421138499674879990385892238, −7.63745189887925075338484024204, −7.031716734278582351104090100366, −5.98835339649072337169220152694, −5.00269502494412071070376461168, −4.4335234200254868402615836223, −3.16664834402208692692280697308, −2.840705417344010288423095758046, −1.148952103851435135694491767732, −0.27010856096100319035594025361,
1.41236259247039749761840787274, 2.34581113578707875813024847455, 3.42335327885697938630815337756, 4.30616878785495244049531492181, 4.75434057869649677421184997982, 6.18993752213167220427387839351, 6.77260414233220923342308645562, 7.649816413147445987210654128418, 8.299758834506713243351193625620, 9.20106965209162365126186337059, 10.10360429675886089113162208296, 10.85105070176850062410201298907, 11.63697859860986989293281247682, 12.32343753076939528136586562125, 12.94722047973026510923870254127, 14.06712572139246976864879044913, 14.74304517563125960770090952946, 15.38831542271609066047617380732, 15.998723002444272239963411599331, 17.01282419753932438845062514896, 17.49162064409942557825463203729, 18.56583833365588814158535447413, 19.25767543679430729121441932501, 19.54493863320250850126581073839, 20.70835701216903622420630256369