L(s) = 1 | + (−0.848 + 0.529i)2-s + (0.438 − 0.898i)4-s + (−0.173 − 0.984i)5-s + (−0.241 − 0.970i)7-s + (0.104 + 0.994i)8-s + (0.669 + 0.743i)10-s + (0.241 + 0.970i)11-s + (0.0348 − 0.999i)13-s + (0.719 + 0.694i)14-s + (−0.615 − 0.788i)16-s + (0.809 − 0.587i)17-s + (−0.978 + 0.207i)19-s + (−0.961 − 0.275i)20-s + (−0.719 − 0.694i)22-s + (0.719 + 0.694i)23-s + ⋯ |
L(s) = 1 | + (−0.848 + 0.529i)2-s + (0.438 − 0.898i)4-s + (−0.173 − 0.984i)5-s + (−0.241 − 0.970i)7-s + (0.104 + 0.994i)8-s + (0.669 + 0.743i)10-s + (0.241 + 0.970i)11-s + (0.0348 − 0.999i)13-s + (0.719 + 0.694i)14-s + (−0.615 − 0.788i)16-s + (0.809 − 0.587i)17-s + (−0.978 + 0.207i)19-s + (−0.961 − 0.275i)20-s + (−0.719 − 0.694i)22-s + (0.719 + 0.694i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 837 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.634 + 0.773i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 837 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.634 + 0.773i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.02401482975 + 0.05077422338i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.02401482975 + 0.05077422338i\) |
\(L(1)\) |
\(\approx\) |
\(0.5989018166 - 0.07071095448i\) |
\(L(1)\) |
\(\approx\) |
\(0.5989018166 - 0.07071095448i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 31 | \( 1 \) |
good | 2 | \( 1 + (-0.848 + 0.529i)T \) |
| 5 | \( 1 + (-0.173 - 0.984i)T \) |
| 7 | \( 1 + (-0.241 - 0.970i)T \) |
| 11 | \( 1 + (0.241 + 0.970i)T \) |
| 13 | \( 1 + (0.0348 - 0.999i)T \) |
| 17 | \( 1 + (0.809 - 0.587i)T \) |
| 19 | \( 1 + (-0.978 + 0.207i)T \) |
| 23 | \( 1 + (0.719 + 0.694i)T \) |
| 29 | \( 1 + (-0.848 + 0.529i)T \) |
| 37 | \( 1 + T \) |
| 41 | \( 1 + (0.615 - 0.788i)T \) |
| 43 | \( 1 + (-0.882 - 0.469i)T \) |
| 47 | \( 1 + (-0.990 + 0.139i)T \) |
| 53 | \( 1 + (-0.913 - 0.406i)T \) |
| 59 | \( 1 + (-0.848 - 0.529i)T \) |
| 61 | \( 1 + (0.173 - 0.984i)T \) |
| 67 | \( 1 + (-0.939 - 0.342i)T \) |
| 71 | \( 1 + (0.104 + 0.994i)T \) |
| 73 | \( 1 + (-0.809 - 0.587i)T \) |
| 79 | \( 1 + (0.559 - 0.829i)T \) |
| 83 | \( 1 + (0.882 + 0.469i)T \) |
| 89 | \( 1 + (-0.913 + 0.406i)T \) |
| 97 | \( 1 + (0.961 + 0.275i)T \) |
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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
−21.48872429922752297881791259873, −21.118139081077831623522048099597, −19.65733843284515838990335511398, −19.050247526688991332049849186, −18.74379413666219022436187139410, −17.950968154297688295428333165251, −16.773344361759331578893673236540, −16.3594385821725177336359739175, −15.15799711996345698055715180403, −14.61978344566832404610379576390, −13.362069910732948193390745291952, −12.47693844476429814759650581086, −11.49700735878694821323409635603, −11.138124870388845747834435404576, −10.12956993719140814789539867018, −9.25602237843165601977233031602, −8.51648975176082532977946159554, −7.6682845216672414385334973988, −6.49396070981379385957823818033, −6.08005504112846176075145268480, −4.30389536550873433591952786959, −3.2663833472442803954260118235, −2.5942762643431558320413646460, −1.55267588476443172759636029670, −0.01960985167817519783197011428,
0.91927535447610093600007861072, 1.80854848937483095160357795684, 3.405225280247818905232689407897, 4.63285603846781668212333778895, 5.36319822268788990800117886695, 6.4765972092015021574138900487, 7.520488456652312245712674936703, 7.89795330581781766025462861644, 9.07947698983885902251249973744, 9.72002056403810679939206286896, 10.492875854610548045824533651025, 11.43859190562875648880001298866, 12.57062818925390154316720119207, 13.22689774325536687998625412380, 14.38767250418047252809892023254, 15.15492336219340789998585570748, 15.99098331559075054633644706458, 16.81633015466885673878578610587, 17.219789810426816629652341968845, 18.020983952871175348912678700761, 19.1200373097221340021599088741, 19.84311670547004886255411193441, 20.436771763841393497696693770163, 20.97195925774088686373247399905, 22.5473684236368114522832786841