L(s) = 1 | + (0.309 − 0.951i)2-s − 3-s + (−0.809 − 0.587i)4-s + (0.809 + 0.587i)5-s + (−0.309 + 0.951i)6-s + (−0.809 + 0.587i)8-s + 9-s + (0.809 − 0.587i)10-s + (−0.809 + 0.587i)11-s + (0.809 + 0.587i)12-s + (−0.309 + 0.951i)13-s + (−0.809 − 0.587i)15-s + (0.309 + 0.951i)16-s + (0.809 − 0.587i)17-s + (0.309 − 0.951i)18-s + (−0.309 − 0.951i)19-s + ⋯ |
L(s) = 1 | + (0.309 − 0.951i)2-s − 3-s + (−0.809 − 0.587i)4-s + (0.809 + 0.587i)5-s + (−0.309 + 0.951i)6-s + (−0.809 + 0.587i)8-s + 9-s + (0.809 − 0.587i)10-s + (−0.809 + 0.587i)11-s + (0.809 + 0.587i)12-s + (−0.309 + 0.951i)13-s + (−0.809 − 0.587i)15-s + (0.309 + 0.951i)16-s + (0.809 − 0.587i)17-s + (0.309 − 0.951i)18-s + (−0.309 − 0.951i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.999 + 0.0317i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.999 + 0.0317i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.01039028552 - 0.6542971213i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.01039028552 - 0.6542971213i\) |
\(L(1)\) |
\(\approx\) |
\(0.7031533470 - 0.3752481439i\) |
\(L(1)\) |
\(\approx\) |
\(0.7031533470 - 0.3752481439i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 \) |
| 41 | \( 1 \) |
good | 2 | \( 1 + (0.309 - 0.951i)T \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 + (0.809 + 0.587i)T \) |
| 11 | \( 1 + (-0.809 + 0.587i)T \) |
| 13 | \( 1 + (-0.309 + 0.951i)T \) |
| 17 | \( 1 + (0.809 - 0.587i)T \) |
| 19 | \( 1 + (-0.309 - 0.951i)T \) |
| 23 | \( 1 + (0.309 - 0.951i)T \) |
| 29 | \( 1 + (-0.809 - 0.587i)T \) |
| 31 | \( 1 + (0.809 - 0.587i)T \) |
| 37 | \( 1 + (-0.809 - 0.587i)T \) |
| 43 | \( 1 + (0.309 - 0.951i)T \) |
| 47 | \( 1 + (-0.309 + 0.951i)T \) |
| 53 | \( 1 + (-0.809 - 0.587i)T \) |
| 59 | \( 1 + (-0.309 + 0.951i)T \) |
| 61 | \( 1 + (-0.309 - 0.951i)T \) |
| 67 | \( 1 + (-0.809 - 0.587i)T \) |
| 71 | \( 1 + (-0.809 + 0.587i)T \) |
| 73 | \( 1 - T \) |
| 79 | \( 1 + T \) |
| 83 | \( 1 - T \) |
| 89 | \( 1 + (-0.309 - 0.951i)T \) |
| 97 | \( 1 + (0.809 + 0.587i)T \) |
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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
−25.48608849453527161696640421733, −24.783175474512239943857649604753, −23.90440084840806580828860147738, −23.24491469581197033162557544596, −22.2925758934615117426471683652, −21.42977251890905619991245275653, −20.85557630592304500522629640788, −19.006074123976112488798312552933, −18.05617790046217375667062142743, −17.32990025709300010061022861908, −16.64417012661301948024787840895, −15.87335994241196338332403782177, −14.82829152114738445069664822646, −13.57073859834098984992096683900, −12.85262017121130922358841347058, −12.134791691903262691718513082515, −10.53791205907772616181483986538, −9.77105020472539606084440069372, −8.41449564392946453048936540706, −7.48823506605915287447515106576, −6.10691046079612161906979627766, −5.56644800576888374704057409525, −4.83103847345474124041440258274, −3.3281276106536765070716394002, −1.248394223167636001905987997010,
0.2179329550281136987093590604, 1.77021476491544096396095554309, 2.7516121948685183791292683317, 4.39367333565820476653460720116, 5.21560058986919635135543155921, 6.2336770957235632449790145853, 7.3160284018044150186119432043, 9.23088711003691585540455438657, 10.023904191100210899258024503536, 10.77542637238643474675844682087, 11.64924730571221846740784395908, 12.61715033013226884284400125960, 13.45571468745627855314740471735, 14.449656792602319580640065395258, 15.51974098782785710606980715738, 16.93743097902554298073880470158, 17.690544555022242134707271557, 18.569892561500664882436003432563, 19.11290947499132897074087336814, 20.8109841812901023495036827190, 21.16965881549702894911759634379, 22.23411398854929234640581959928, 22.73994542166473740237384056558, 23.65705841994076925835899633416, 24.53167966583141321875949185909