L(s) = 1 | + (0.309 + 0.951i)2-s + (−0.809 + 0.587i)4-s + (−0.978 + 0.207i)5-s + (−0.809 − 0.587i)8-s + (−0.5 − 0.866i)10-s + (−0.978 − 0.207i)13-s + (0.309 − 0.951i)16-s + (−0.978 + 0.207i)17-s + (0.913 − 0.406i)19-s + (0.669 − 0.743i)20-s + (−0.5 − 0.866i)23-s + (0.913 − 0.406i)25-s + (−0.104 − 0.994i)26-s + (0.913 + 0.406i)29-s + (0.309 + 0.951i)31-s + 32-s + ⋯ |
L(s) = 1 | + (0.309 + 0.951i)2-s + (−0.809 + 0.587i)4-s + (−0.978 + 0.207i)5-s + (−0.809 − 0.587i)8-s + (−0.5 − 0.866i)10-s + (−0.978 − 0.207i)13-s + (0.309 − 0.951i)16-s + (−0.978 + 0.207i)17-s + (0.913 − 0.406i)19-s + (0.669 − 0.743i)20-s + (−0.5 − 0.866i)23-s + (0.913 − 0.406i)25-s + (−0.104 − 0.994i)26-s + (0.913 + 0.406i)29-s + (0.309 + 0.951i)31-s + 32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.00469i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.00469i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.7742958570 + 0.001817544670i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7742958570 + 0.001817544670i\) |
\(L(1)\) |
\(\approx\) |
\(0.7560734243 + 0.3237569177i\) |
\(L(1)\) |
\(\approx\) |
\(0.7560734243 + 0.3237569177i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 \) |
| 11 | \( 1 \) |
good | 2 | \( 1 + (0.309 + 0.951i)T \) |
| 5 | \( 1 + (-0.978 + 0.207i)T \) |
| 13 | \( 1 + (-0.978 - 0.207i)T \) |
| 17 | \( 1 + (-0.978 + 0.207i)T \) |
| 19 | \( 1 + (0.913 - 0.406i)T \) |
| 23 | \( 1 + (-0.5 - 0.866i)T \) |
| 29 | \( 1 + (0.913 + 0.406i)T \) |
| 31 | \( 1 + (0.309 + 0.951i)T \) |
| 37 | \( 1 + (-0.104 - 0.994i)T \) |
| 41 | \( 1 + (0.913 - 0.406i)T \) |
| 43 | \( 1 + (-0.5 - 0.866i)T \) |
| 47 | \( 1 + (-0.809 - 0.587i)T \) |
| 53 | \( 1 + (0.669 - 0.743i)T \) |
| 59 | \( 1 + (-0.809 + 0.587i)T \) |
| 61 | \( 1 + (0.309 - 0.951i)T \) |
| 67 | \( 1 + T \) |
| 71 | \( 1 + (0.309 - 0.951i)T \) |
| 73 | \( 1 + (0.913 + 0.406i)T \) |
| 79 | \( 1 + (0.309 + 0.951i)T \) |
| 83 | \( 1 + (-0.978 + 0.207i)T \) |
| 89 | \( 1 + (-0.5 + 0.866i)T \) |
| 97 | \( 1 + (0.669 - 0.743i)T \) |
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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
−22.60472774476545653328321483816, −21.89355270146493442923073591396, −20.994949819275007526688101487763, −20.01010140220644447359273962551, −19.724971585354678713511220333381, −18.84280356032617380883165556621, −17.98940333470514174789032539532, −17.08012957663225032631569862452, −15.91855996630751102527150208382, −15.1931315498307463884805336916, −14.285317331902297322663952302318, −13.4184637235296723403238003989, −12.52070516047650677722112094463, −11.69133733947640122318679100386, −11.33296075941238080409204542914, −10.05750181213831611580150169489, −9.40576913305278391832088941312, −8.317151868937451202004532680593, −7.48184958871861235333424844005, −6.19839632685428619730775349169, −4.9384520226036346070810390678, −4.36001568253809362477186873602, −3.33136071809010407497696242654, −2.38554274393182358908943419698, −1.04966757226022147512811794096,
0.408129479863735092741632375397, 2.56895552236717434269055349159, 3.60904622365302516495353053129, 4.55191210998233018149353875859, 5.270623860315074053678835363275, 6.64320989022292010557207109455, 7.14461712256798267688432933393, 8.11354333277037494681443769525, 8.79921532928017322948369373253, 9.92135144337091823345268067410, 11.03223199537166157827424084398, 12.14660586447573468539134043982, 12.609786593488866609762916608676, 13.84328842766325697454359979116, 14.50438143221079895269333463615, 15.40119010765576054548458758671, 15.91194728411872015580244490150, 16.74862879123501278445846134807, 17.766681835869337789199593543545, 18.31893323481235089679151945770, 19.522955273911546033457354201073, 20.01227978080699001451076216537, 21.33147179707650442812048343039, 22.19088078525116115551165710310, 22.71786538954404034693309742844