L(s) = 1 | + (−0.5 − 0.866i)2-s + (−0.5 + 0.866i)4-s + (−0.5 − 0.866i)5-s + 8-s + (−0.5 + 0.866i)10-s + (−0.5 + 0.866i)11-s + (−0.5 − 0.866i)16-s + (0.5 − 0.866i)17-s + (0.5 + 0.866i)19-s + 20-s + 22-s + (0.5 + 0.866i)23-s + (−0.5 + 0.866i)25-s − 29-s + (0.5 − 0.866i)31-s + (−0.5 + 0.866i)32-s + ⋯ |
L(s) = 1 | + (−0.5 − 0.866i)2-s + (−0.5 + 0.866i)4-s + (−0.5 − 0.866i)5-s + 8-s + (−0.5 + 0.866i)10-s + (−0.5 + 0.866i)11-s + (−0.5 − 0.866i)16-s + (0.5 − 0.866i)17-s + (0.5 + 0.866i)19-s + 20-s + 22-s + (0.5 + 0.866i)23-s + (−0.5 + 0.866i)25-s − 29-s + (0.5 − 0.866i)31-s + (−0.5 + 0.866i)32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.0633 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.0633 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.8019154646 - 0.7526388737i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8019154646 - 0.7526388737i\) |
\(L(1)\) |
\(\approx\) |
\(0.6814184290 - 0.3377976874i\) |
\(L(1)\) |
\(\approx\) |
\(0.6814184290 - 0.3377976874i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 \) |
| 13 | \( 1 \) |
good | 2 | \( 1 + (-0.5 - 0.866i)T \) |
| 5 | \( 1 + (-0.5 - 0.866i)T \) |
| 11 | \( 1 + (-0.5 + 0.866i)T \) |
| 17 | \( 1 + (0.5 - 0.866i)T \) |
| 19 | \( 1 + (0.5 + 0.866i)T \) |
| 23 | \( 1 + (0.5 + 0.866i)T \) |
| 29 | \( 1 - T \) |
| 31 | \( 1 + (0.5 - 0.866i)T \) |
| 37 | \( 1 + (0.5 + 0.866i)T \) |
| 41 | \( 1 + T \) |
| 43 | \( 1 + T \) |
| 47 | \( 1 + (-0.5 - 0.866i)T \) |
| 53 | \( 1 + (0.5 - 0.866i)T \) |
| 59 | \( 1 + (-0.5 + 0.866i)T \) |
| 61 | \( 1 + (-0.5 - 0.866i)T \) |
| 67 | \( 1 + (0.5 - 0.866i)T \) |
| 71 | \( 1 + T \) |
| 73 | \( 1 + (0.5 - 0.866i)T \) |
| 79 | \( 1 + (-0.5 - 0.866i)T \) |
| 83 | \( 1 + T \) |
| 89 | \( 1 + (-0.5 - 0.866i)T \) |
| 97 | \( 1 - T \) |
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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
−26.078080003221768444415913901486, −24.74814576781331566387779150295, −23.978530524587501714565525255055, −23.1630654533417346660595280669, −22.34539957678255639331100980732, −21.30342525291083307093218165867, −19.81526169688139082111008529519, −19.06139199183642708509433218845, −18.38356288667757282788986279808, −17.44109928544863121237900043710, −16.339196894279266246891752695575, −15.60480069210832195454740555372, −14.683197763982429650749009440793, −13.9423382791859893893823307868, −12.73213682479877617894361968830, −11.11194174221895400831692185051, −10.61709247133116891242508798186, −9.34190399427151491032761727255, −8.24206900209650725077477534766, −7.44324940799011500071960928689, −6.43602279762418527869326707669, −5.46891175418724339926530959499, −4.07508954892929689918102707306, −2.70455487940345036406053612416, −0.79729764721385682354658922599,
0.62177308835489028184500136325, 1.85031907332267840417706825090, 3.27050035107309866609716226308, 4.39890405635421163466579721281, 5.37644897500602522835270926160, 7.427568009122795301003527194642, 7.97526298950340937004711325195, 9.29071305770428924000668566282, 9.86110877193662523212032638074, 11.22252976086444524569541076366, 12.0245187350547464995592906045, 12.80801393261716930944585131815, 13.70541307092469434674851843665, 15.188589708672408012876755344281, 16.28440179505218972122352776726, 17.026217770136312559438389314397, 18.08551295834042195739760901235, 18.91910076213101557798604870617, 19.92561061412799268081976328408, 20.64945766151525735418880398782, 21.186081062472479637170655142050, 22.62968625699880380375046297065, 23.1687044086700281844644886926, 24.44912007457488554864646876479, 25.41715237339892208933058113949