L(s) = 1 | + (0.766 + 0.642i)2-s + (−0.939 + 0.342i)3-s + (0.173 + 0.984i)4-s + (−0.939 − 0.342i)6-s + (0.5 + 0.866i)7-s + (−0.5 + 0.866i)8-s + (0.766 − 0.642i)9-s + (−0.5 + 0.866i)11-s + (−0.5 − 0.866i)12-s + (−0.939 − 0.342i)13-s + (−0.173 + 0.984i)14-s + (−0.939 + 0.342i)16-s + (−0.766 − 0.642i)17-s + 18-s + (−0.766 − 0.642i)21-s + (−0.939 + 0.342i)22-s + ⋯ |
L(s) = 1 | + (0.766 + 0.642i)2-s + (−0.939 + 0.342i)3-s + (0.173 + 0.984i)4-s + (−0.939 − 0.342i)6-s + (0.5 + 0.866i)7-s + (−0.5 + 0.866i)8-s + (0.766 − 0.642i)9-s + (−0.5 + 0.866i)11-s + (−0.5 − 0.866i)12-s + (−0.939 − 0.342i)13-s + (−0.173 + 0.984i)14-s + (−0.939 + 0.342i)16-s + (−0.766 − 0.642i)17-s + 18-s + (−0.766 − 0.642i)21-s + (−0.939 + 0.342i)22-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 95 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.973 - 0.226i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 95 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.973 - 0.226i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(-0.1389703654 + 1.210225009i\) |
\(L(\frac12)\) |
\(\approx\) |
\(-0.1389703654 + 1.210225009i\) |
\(L(1)\) |
\(\approx\) |
\(0.7395184242 + 0.7453026921i\) |
\(L(1)\) |
\(\approx\) |
\(0.7395184242 + 0.7453026921i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 19 | \( 1 \) |
good | 2 | \( 1 + (0.766 + 0.642i)T \) |
| 3 | \( 1 + (-0.939 + 0.342i)T \) |
| 7 | \( 1 + (0.5 + 0.866i)T \) |
| 11 | \( 1 + (-0.5 + 0.866i)T \) |
| 13 | \( 1 + (-0.939 - 0.342i)T \) |
| 17 | \( 1 + (-0.766 - 0.642i)T \) |
| 23 | \( 1 + (-0.173 - 0.984i)T \) |
| 29 | \( 1 + (-0.766 + 0.642i)T \) |
| 31 | \( 1 + (0.5 + 0.866i)T \) |
| 37 | \( 1 + T \) |
| 41 | \( 1 + (0.939 - 0.342i)T \) |
| 43 | \( 1 + (-0.173 + 0.984i)T \) |
| 47 | \( 1 + (-0.766 + 0.642i)T \) |
| 53 | \( 1 + (0.173 + 0.984i)T \) |
| 59 | \( 1 + (-0.766 - 0.642i)T \) |
| 61 | \( 1 + (0.173 + 0.984i)T \) |
| 67 | \( 1 + (0.766 - 0.642i)T \) |
| 71 | \( 1 + (-0.173 + 0.984i)T \) |
| 73 | \( 1 + (0.939 - 0.342i)T \) |
| 79 | \( 1 + (0.939 - 0.342i)T \) |
| 83 | \( 1 + (0.5 + 0.866i)T \) |
| 89 | \( 1 + (0.939 + 0.342i)T \) |
| 97 | \( 1 + (0.766 + 0.642i)T \) |
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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
−29.5124938813841393678833375275, −28.69370175049377947998298085402, −27.58413874795955604618692629595, −26.56137471887116896506861642664, −24.46796328992775662578415308473, −24.01141804582954467832500478800, −23.089700969671510149629591187760, −21.977121179327981245500449764740, −21.22195822181831742466440380212, −19.84231137392337633136832459503, −18.88501621630869568557635351924, −17.62203717060148695920345385817, −16.53293173750450948835928820862, −15.153727122119439427068959661, −13.74447091574131715552400600216, −13.00188955750607193630405925224, −11.62031400839832196607208338615, −10.97459714124394763480354246042, −9.85533255418956091578679415287, −7.66309603397825702835466726436, −6.32665875214877419594873985141, −5.139714379606151531787640659943, −4.014332268185436940969205008073, −2.0196331188798450062024212676, −0.45871592230627734473796224831,
2.52299809327649747699812006649, 4.561009752636383692488163451, 5.19445184869830620766661679491, 6.49066143532408988218737136516, 7.718791487211997773097565622748, 9.34159366359868140047143600480, 10.9565001386911831858963454790, 12.12909051871061650388694324587, 12.8012816738334198566573807132, 14.58768524250976023863645916315, 15.380995150782514600982829733006, 16.33553969363922055258483624366, 17.59032992498896204097331917357, 18.20231455220240152649654907383, 20.34143875698203693261936712339, 21.40390778029978155327608800360, 22.26739109838360715194091848270, 23.01289770347637400279411305954, 24.22137298154220309259793008146, 24.90536916194355773584798844941, 26.316999678197422266026540214075, 27.33436001803899829953359522805, 28.463632166578355036727090401471, 29.48561136258924824553888947022, 30.70949773406556780019452913417