L(s) = 1 | + (0.866 − 0.5i)3-s + (0.866 − 0.5i)5-s − 7-s + (0.5 − 0.866i)9-s + i·11-s + (−0.866 − 0.5i)13-s + (0.5 − 0.866i)15-s + (−0.5 − 0.866i)17-s + (−0.866 + 0.5i)21-s + (0.5 − 0.866i)23-s + (0.5 − 0.866i)25-s − i·27-s + (−0.866 − 0.5i)29-s − 31-s + (0.5 + 0.866i)33-s + ⋯ |
L(s) = 1 | + (0.866 − 0.5i)3-s + (0.866 − 0.5i)5-s − 7-s + (0.5 − 0.866i)9-s + i·11-s + (−0.866 − 0.5i)13-s + (0.5 − 0.866i)15-s + (−0.5 − 0.866i)17-s + (−0.866 + 0.5i)21-s + (0.5 − 0.866i)23-s + (0.5 − 0.866i)25-s − i·27-s + (−0.866 − 0.5i)29-s − 31-s + (0.5 + 0.866i)33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.730 - 0.683i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.730 - 0.683i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.7061596716 - 1.787710552i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7061596716 - 1.787710552i\) |
\(L(1)\) |
\(\approx\) |
\(1.188330689 - 0.5491493068i\) |
\(L(1)\) |
\(\approx\) |
\(1.188330689 - 0.5491493068i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 19 | \( 1 \) |
good | 3 | \( 1 + (0.866 - 0.5i)T \) |
| 5 | \( 1 + (0.866 - 0.5i)T \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 + iT \) |
| 13 | \( 1 + (-0.866 - 0.5i)T \) |
| 17 | \( 1 + (-0.5 - 0.866i)T \) |
| 23 | \( 1 + (0.5 - 0.866i)T \) |
| 29 | \( 1 + (-0.866 - 0.5i)T \) |
| 31 | \( 1 - T \) |
| 37 | \( 1 - iT \) |
| 41 | \( 1 + (-0.5 - 0.866i)T \) |
| 43 | \( 1 + (0.866 - 0.5i)T \) |
| 47 | \( 1 + (-0.5 + 0.866i)T \) |
| 53 | \( 1 + (0.866 + 0.5i)T \) |
| 59 | \( 1 + (-0.866 + 0.5i)T \) |
| 61 | \( 1 + (0.866 + 0.5i)T \) |
| 67 | \( 1 + (-0.866 - 0.5i)T \) |
| 71 | \( 1 + (-0.5 - 0.866i)T \) |
| 73 | \( 1 + (0.5 + 0.866i)T \) |
| 79 | \( 1 + (0.5 + 0.866i)T \) |
| 83 | \( 1 - iT \) |
| 89 | \( 1 + (-0.5 + 0.866i)T \) |
| 97 | \( 1 + (0.5 + 0.866i)T \) |
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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
−25.605091852467401671680417153301, −24.7859011381972363737037068029, −23.79802671145311677148913682373, −22.26723835160755190865806118837, −21.89278916295860926793940068539, −21.1564898597710837432042970736, −19.89599311961116655974400302430, −19.24310213872861863386296776899, −18.46658845750231355531491180143, −17.04162648828635637089657649615, −16.38091543500041584489748501824, −15.20424699303922668775264581305, −14.478385392657118234659151434641, −13.47064906724924975526638309277, −12.950074326819739943018506235643, −11.24875073123144259264758433601, −10.26136011698323517050583023575, −9.486100883791160886290616055798, −8.76716131632485617380189045024, −7.360770029673710125049223120, −6.35779898181080468835961584047, −5.22714785524714124986791912335, −3.70560448747802418550735451727, −2.92699727089971148544965009746, −1.79806275160313932640148096596,
0.459984253521653519338379643797, 2.04408929246005791845657781746, 2.77725862262860058988103818946, 4.24952374453328368590324856708, 5.54606844280068978081201981830, 6.79874403017899606496717483183, 7.50352032365875657677262695417, 9.035907349206777193523336260474, 9.44015711768853093684415787238, 10.39882215282225911730669696034, 12.29560612489209454013820871290, 12.77419423074482015479597066141, 13.57319298575228685438555492562, 14.57031039783126485161475093884, 15.475756660599975756340018709055, 16.62747538122260418317578470601, 17.62441474031421603394237092707, 18.4216286308354594906654491411, 19.50438603302987213705058523214, 20.268928273476979309736385254961, 20.85909142961018540669390960754, 22.1594262611669641498550047423, 22.85570194245001172220234481564, 24.220357231398535234828597365687, 24.88187358114918004159999794827