L(s) = 1 | + (0.179 − 0.983i)2-s + (−0.935 − 0.353i)4-s + (−0.683 − 0.730i)5-s + (−0.953 + 0.299i)7-s + (−0.516 + 0.856i)8-s + (−0.841 + 0.540i)10-s + (−0.548 − 0.836i)13-s + (0.123 + 0.992i)14-s + (0.749 + 0.662i)16-s + (0.466 − 0.884i)17-s + (0.985 − 0.170i)19-s + (0.380 + 0.924i)20-s + (0.580 − 0.814i)23-s + (−0.0665 + 0.997i)25-s + (−0.921 + 0.389i)26-s + ⋯ |
L(s) = 1 | + (0.179 − 0.983i)2-s + (−0.935 − 0.353i)4-s + (−0.683 − 0.730i)5-s + (−0.953 + 0.299i)7-s + (−0.516 + 0.856i)8-s + (−0.841 + 0.540i)10-s + (−0.548 − 0.836i)13-s + (0.123 + 0.992i)14-s + (0.749 + 0.662i)16-s + (0.466 − 0.884i)17-s + (0.985 − 0.170i)19-s + (0.380 + 0.924i)20-s + (0.580 − 0.814i)23-s + (−0.0665 + 0.997i)25-s + (−0.921 + 0.389i)26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.874 + 0.484i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.874 + 0.484i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(-0.3264172948 - 1.263289539i\) |
\(L(\frac12)\) |
\(\approx\) |
\(-0.3264172948 - 1.263289539i\) |
\(L(1)\) |
\(\approx\) |
\(0.5919819171 - 0.6349449605i\) |
\(L(1)\) |
\(\approx\) |
\(0.5919819171 - 0.6349449605i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 11 | \( 1 \) |
good | 2 | \( 1 + (0.179 - 0.983i)T \) |
| 5 | \( 1 + (-0.683 - 0.730i)T \) |
| 7 | \( 1 + (-0.953 + 0.299i)T \) |
| 13 | \( 1 + (-0.548 - 0.836i)T \) |
| 17 | \( 1 + (0.466 - 0.884i)T \) |
| 19 | \( 1 + (0.985 - 0.170i)T \) |
| 23 | \( 1 + (0.580 - 0.814i)T \) |
| 29 | \( 1 + (0.830 - 0.556i)T \) |
| 31 | \( 1 + (0.988 - 0.151i)T \) |
| 37 | \( 1 + (-0.254 - 0.967i)T \) |
| 41 | \( 1 + (0.851 - 0.524i)T \) |
| 43 | \( 1 + (-0.981 + 0.189i)T \) |
| 47 | \( 1 + (0.997 - 0.0760i)T \) |
| 53 | \( 1 + (0.198 + 0.980i)T \) |
| 59 | \( 1 + (0.879 - 0.475i)T \) |
| 61 | \( 1 + (0.761 - 0.647i)T \) |
| 67 | \( 1 + (0.235 + 0.971i)T \) |
| 71 | \( 1 + (0.696 - 0.717i)T \) |
| 73 | \( 1 + (-0.993 + 0.113i)T \) |
| 79 | \( 1 + (-0.820 - 0.572i)T \) |
| 83 | \( 1 + (-0.00951 + 0.999i)T \) |
| 89 | \( 1 + (-0.142 - 0.989i)T \) |
| 97 | \( 1 + (-0.683 + 0.730i)T \) |
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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
−21.99015316722715940138383167054, −21.21696834188524000668688755025, −19.830072251592338369083332647747, −19.213512791660923996317835168877, −18.66071751549787522097497274864, −17.66001951925728428702960532559, −16.858431277915813797481763574579, −16.13277569517457879547187139857, −15.52603660997032640283060837354, −14.710777596206234053255519400076, −14.027079346239533545131012575681, −13.25086498130979760796194254944, −12.27538640972135215319732733816, −11.62239432909998370039584227773, −10.26153833828634858026522808063, −9.7420489718304927285864529318, −8.67297063704568244379179749132, −7.772294504648771907605675988593, −6.97873790679492861640525713472, −6.53757096005037492007168513208, −5.47649684546728234967195987299, −4.38844634665685483802538471380, −3.567047023205253819465799142818, −2.87756420771214459769329558872, −0.97817131445754953861690592701,
0.40599982696692159382631154204, 0.87988795651916705740774296349, 2.552550932660766320794880692516, 3.12020494343819070504116705296, 4.13359418968827245774790668580, 5.04693149462086954115972876042, 5.71248083228578052514310693411, 7.09001463471835261053915505524, 8.08802918021975530242984364189, 8.96058203455410898351344726743, 9.67248645086305986150311234481, 10.37107964607121975968786056453, 11.508867760115256136236297755977, 12.15656980434132044451407092195, 12.6784484454862841806687181962, 13.44324160840505591143573268213, 14.335451606126510464706181966183, 15.41958800422984371887533233869, 15.99349449152748475357189949967, 16.976749960477469682479063524044, 17.83854234440591919640907406123, 18.818346066495254406531438002459, 19.325047809840049656628128096359, 20.1211588803217747871644705065, 20.57605237756123419666823105397