L(s) = 1 | + (−0.951 + 0.309i)2-s + (0.809 − 0.587i)4-s + (0.207 + 0.978i)5-s + (0.406 − 0.913i)7-s + (−0.587 + 0.809i)8-s + (−0.5 − 0.866i)10-s + (−0.104 + 0.994i)14-s + (0.309 − 0.951i)16-s + (0.669 + 0.743i)17-s + (0.406 + 0.913i)19-s + (0.743 + 0.669i)20-s + (0.5 − 0.866i)23-s + (−0.913 + 0.406i)25-s + (−0.207 − 0.978i)28-s + (0.809 − 0.587i)29-s + ⋯ |
L(s) = 1 | + (−0.951 + 0.309i)2-s + (0.809 − 0.587i)4-s + (0.207 + 0.978i)5-s + (0.406 − 0.913i)7-s + (−0.587 + 0.809i)8-s + (−0.5 − 0.866i)10-s + (−0.104 + 0.994i)14-s + (0.309 − 0.951i)16-s + (0.669 + 0.743i)17-s + (0.406 + 0.913i)19-s + (0.743 + 0.669i)20-s + (0.5 − 0.866i)23-s + (−0.913 + 0.406i)25-s + (−0.207 − 0.978i)28-s + (0.809 − 0.587i)29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1287 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.618 + 0.785i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1287 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.618 + 0.785i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.002616848 + 0.4868836870i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.002616848 + 0.4868836870i\) |
\(L(1)\) |
\(\approx\) |
\(0.7996500995 + 0.1942558512i\) |
\(L(1)\) |
\(\approx\) |
\(0.7996500995 + 0.1942558512i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 11 | \( 1 \) |
| 13 | \( 1 \) |
good | 2 | \( 1 + (-0.951 + 0.309i)T \) |
| 5 | \( 1 + (0.207 + 0.978i)T \) |
| 7 | \( 1 + (0.406 - 0.913i)T \) |
| 17 | \( 1 + (0.669 + 0.743i)T \) |
| 19 | \( 1 + (0.406 + 0.913i)T \) |
| 23 | \( 1 + (0.5 - 0.866i)T \) |
| 29 | \( 1 + (0.809 - 0.587i)T \) |
| 31 | \( 1 + (-0.743 - 0.669i)T \) |
| 37 | \( 1 + (-0.406 + 0.913i)T \) |
| 41 | \( 1 + (0.406 + 0.913i)T \) |
| 43 | \( 1 + (-0.5 - 0.866i)T \) |
| 47 | \( 1 + (0.406 + 0.913i)T \) |
| 53 | \( 1 + (0.309 + 0.951i)T \) |
| 59 | \( 1 + (-0.587 - 0.809i)T \) |
| 61 | \( 1 + (0.978 - 0.207i)T \) |
| 67 | \( 1 + (0.866 + 0.5i)T \) |
| 71 | \( 1 + (0.743 - 0.669i)T \) |
| 73 | \( 1 + (0.587 + 0.809i)T \) |
| 79 | \( 1 + (0.978 + 0.207i)T \) |
| 83 | \( 1 + (-0.743 + 0.669i)T \) |
| 89 | \( 1 + (0.866 - 0.5i)T \) |
| 97 | \( 1 + (-0.743 - 0.669i)T \) |
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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
−20.9377795477331945409012309324, −19.97453306408023087430729569986, −19.49384191101261285137779386993, −18.490202039458129935310667705346, −17.8538785709379158719713234609, −17.31016865003221781176969667780, −16.26586790324111565201405050069, −15.904123547927352727572695036242, −14.99406523392372798582756632354, −13.888744129869123817078734162817, −12.891463720797296662045018481207, −12.18111132018294504887427513680, −11.62398147844630378892686651050, −10.75441032947313468100498763483, −9.64144243357308348579692807153, −9.12125546957207301364517268143, −8.53155539594760506174243428009, −7.645974188675131701602287578565, −6.79305286187864861830500967086, −5.51530467718414254466612917437, −5.035327856475942462378212754596, −3.61030425622089063960336913957, −2.60456948871084115740763995977, −1.698386822648272190061023092960, −0.76386359940887971859223789248,
0.979927683632109507299924186178, 1.94235529723118567050901083978, 2.99930341711009177940639253971, 3.99324365421282770613296378166, 5.31553400931666550777727831489, 6.26201564760259049938356380240, 6.8885418523975829464620194270, 7.79066311764896057287903701275, 8.23812581718174497805937814887, 9.55766308874595777417389587885, 10.18636551773756772860183118577, 10.7513666822410412582958737960, 11.45005156757578169621729985756, 12.449658715418801401960714474246, 13.74313686027608771783596183467, 14.39756520330387032088272343081, 14.94385450289967984081620034297, 15.84468475242993074176399111441, 16.92015512981494187557539345699, 17.13587212964851642114896822181, 18.17613821802753006099152238183, 18.698199671522536783364387436318, 19.36899352547013157384958628699, 20.3111036187766543037160001732, 20.87273002386958235901978042233