L(s) = 1 | + (0.866 − 0.5i)2-s + (−0.866 + 0.5i)3-s + (0.5 − 0.866i)4-s + (−0.5 + 0.866i)6-s + i·7-s − i·8-s + (0.5 − 0.866i)9-s + 11-s + i·12-s + (0.866 + 0.5i)13-s + (0.5 + 0.866i)14-s + (−0.5 − 0.866i)16-s + (0.866 − 0.5i)17-s − i·18-s + (−0.5 − 0.866i)21-s + (0.866 − 0.5i)22-s + ⋯ |
L(s) = 1 | + (0.866 − 0.5i)2-s + (−0.866 + 0.5i)3-s + (0.5 − 0.866i)4-s + (−0.5 + 0.866i)6-s + i·7-s − i·8-s + (0.5 − 0.866i)9-s + 11-s + i·12-s + (0.866 + 0.5i)13-s + (0.5 + 0.866i)14-s + (−0.5 − 0.866i)16-s + (0.866 − 0.5i)17-s − i·18-s + (−0.5 − 0.866i)21-s + (0.866 − 0.5i)22-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 95 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.960 - 0.277i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 95 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.960 - 0.277i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(2.344419438 - 0.3313005587i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.344419438 - 0.3313005587i\) |
\(L(1)\) |
\(\approx\) |
\(1.530894891 - 0.1959208863i\) |
\(L(1)\) |
\(\approx\) |
\(1.530894891 - 0.1959208863i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 19 | \( 1 \) |
good | 2 | \( 1 + (0.866 - 0.5i)T \) |
| 3 | \( 1 + (-0.866 + 0.5i)T \) |
| 7 | \( 1 + iT \) |
| 11 | \( 1 + T \) |
| 13 | \( 1 + (0.866 + 0.5i)T \) |
| 17 | \( 1 + (0.866 - 0.5i)T \) |
| 23 | \( 1 + (0.866 + 0.5i)T \) |
| 29 | \( 1 + (0.5 - 0.866i)T \) |
| 31 | \( 1 + T \) |
| 37 | \( 1 + iT \) |
| 41 | \( 1 + (-0.5 - 0.866i)T \) |
| 43 | \( 1 + (-0.866 + 0.5i)T \) |
| 47 | \( 1 + (-0.866 - 0.5i)T \) |
| 53 | \( 1 + (0.866 + 0.5i)T \) |
| 59 | \( 1 + (0.5 + 0.866i)T \) |
| 61 | \( 1 + (-0.5 + 0.866i)T \) |
| 67 | \( 1 + (-0.866 - 0.5i)T \) |
| 71 | \( 1 + (-0.5 - 0.866i)T \) |
| 73 | \( 1 + (-0.866 + 0.5i)T \) |
| 79 | \( 1 + (0.5 + 0.866i)T \) |
| 83 | \( 1 - iT \) |
| 89 | \( 1 + (0.5 - 0.866i)T \) |
| 97 | \( 1 + (0.866 - 0.5i)T \) |
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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
−30.16489505498409524583662232578, −29.3401318075638140198427602883, −27.981285507662123871351668631698, −26.80197816109177471565749516169, −25.436278628090216527021583873395, −24.593617993495818067384258889846, −23.36928303504215627055079046013, −23.0345714216326094941681301290, −21.883875808542308091170760902833, −20.69787557211250435122865080065, −19.42085836040698933478529769584, −17.8318343423190235208002282894, −16.92220743887085390905351057112, −16.20455551278781698767545901776, −14.65963158739122755866810327379, −13.56173982606388225272537685315, −12.64255833897336358220321008447, −11.51038197005386231519810905696, −10.46729554360604336909271778447, −8.24396773145668724696823650862, −7.00484947825168583593965178553, −6.17939700814652310756174467151, −4.82509138828008121652765113998, −3.527293208268228799076109774867, −1.22867737961521653048319483706,
1.28284886304653415904873776766, 3.25373417644507285848385261437, 4.58029910905898443431811046602, 5.74515258895035219731240087968, 6.64765891097233932435973609947, 9.067276273419899540744220282655, 10.17900312670447485889456157719, 11.74383352604032963593422917842, 11.81684461026534567204074578014, 13.430549687604880533960092006620, 14.809850487772893375517730551136, 15.68545338970137126307737387232, 16.786888512355121841634831465058, 18.3197527112816743213111982456, 19.29504059205155820790800581571, 20.90022383506525959187994462266, 21.47211979970008955392067509761, 22.53240176337206722193823406589, 23.21034820959238071699953561556, 24.43187529166938330237808476934, 25.44258953699070675360190227513, 27.26809814722584989004736062668, 28.06857867975249960194055445259, 28.82778208500985338769309299782, 29.86606534623725855144953085660