L(s) = 1 | + (−0.425 − 0.904i)2-s + (0.929 + 0.368i)3-s + (−0.637 + 0.770i)4-s + (−0.0627 − 0.998i)6-s + (−0.809 + 0.587i)7-s + (0.968 + 0.248i)8-s + (0.728 + 0.684i)9-s + (−0.876 + 0.481i)12-s + (0.728 + 0.684i)13-s + (0.876 + 0.481i)14-s + (−0.187 − 0.982i)16-s + (−0.929 + 0.368i)17-s + (0.309 − 0.951i)18-s + (−0.0627 − 0.998i)19-s + (−0.968 + 0.248i)21-s + ⋯ |
L(s) = 1 | + (−0.425 − 0.904i)2-s + (0.929 + 0.368i)3-s + (−0.637 + 0.770i)4-s + (−0.0627 − 0.998i)6-s + (−0.809 + 0.587i)7-s + (0.968 + 0.248i)8-s + (0.728 + 0.684i)9-s + (−0.876 + 0.481i)12-s + (0.728 + 0.684i)13-s + (0.876 + 0.481i)14-s + (−0.187 − 0.982i)16-s + (−0.929 + 0.368i)17-s + (0.309 − 0.951i)18-s + (−0.0627 − 0.998i)19-s + (−0.968 + 0.248i)21-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1375 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.106 - 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1375 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.106 - 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.9559885348 - 1.063523647i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9559885348 - 1.063523647i\) |
\(L(1)\) |
\(\approx\) |
\(0.9571436532 - 0.2240036917i\) |
\(L(1)\) |
\(\approx\) |
\(0.9571436532 - 0.2240036917i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 11 | \( 1 \) |
good | 2 | \( 1 + (-0.425 - 0.904i)T \) |
| 3 | \( 1 + (0.929 + 0.368i)T \) |
| 7 | \( 1 + (-0.809 + 0.587i)T \) |
| 13 | \( 1 + (0.728 + 0.684i)T \) |
| 17 | \( 1 + (-0.929 + 0.368i)T \) |
| 19 | \( 1 + (-0.0627 - 0.998i)T \) |
| 23 | \( 1 + (0.187 - 0.982i)T \) |
| 29 | \( 1 + (-0.535 - 0.844i)T \) |
| 31 | \( 1 + (0.968 + 0.248i)T \) |
| 37 | \( 1 + (-0.728 - 0.684i)T \) |
| 41 | \( 1 + (-0.728 - 0.684i)T \) |
| 43 | \( 1 + (-0.809 - 0.587i)T \) |
| 47 | \( 1 + (-0.0627 + 0.998i)T \) |
| 53 | \( 1 + (-0.0627 + 0.998i)T \) |
| 59 | \( 1 + (-0.992 - 0.125i)T \) |
| 61 | \( 1 + (0.992 - 0.125i)T \) |
| 67 | \( 1 + (0.929 - 0.368i)T \) |
| 71 | \( 1 + (-0.929 - 0.368i)T \) |
| 73 | \( 1 + (-0.425 - 0.904i)T \) |
| 79 | \( 1 + (-0.968 + 0.248i)T \) |
| 83 | \( 1 + (0.968 + 0.248i)T \) |
| 89 | \( 1 + (0.876 + 0.481i)T \) |
| 97 | \( 1 + (0.637 - 0.770i)T \) |
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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
−20.39773002769405040255023711331, −20.08912591471170591459595640408, −19.18476673330330351208094722452, −18.61792150053650086344680714896, −17.87500553175565828381974641836, −17.08033189845781862965244898622, −16.12837456854994146643124156090, −15.5947129769883206086395763497, −14.862142691192927278727572122, −14.00444330862590695554723202656, −13.245717652472279837097378145169, −13.02145672561368199086904859247, −11.57621390389795502209587211708, −10.2941803230842882228241770520, −9.89498578591397351833090557393, −8.9257115502234239653964180765, −8.313592234356135308254849597549, −7.50839123020027972049266928054, −6.76715283836501632016672278070, −6.1493014679788666687372690907, −4.98414846114558077357972720174, −3.835464209092397592634689542971, −3.2101198785899640866044651775, −1.75777014099599893868215556782, −0.87194006563952076336908100234,
0.32976444724805260751086391828, 1.790468146795492387689278183202, 2.47286207841711294238932342052, 3.276434469319161781549917794686, 4.10649521521270381491146451455, 4.85343497582086040547575480603, 6.33444783488128181274121764705, 7.208768891828927902362112569164, 8.41357508360692553604948689490, 8.86388026086908140600080890753, 9.41143801610518451773109855866, 10.32443633275414877990788780565, 10.99458467760834967703303659135, 11.95577227830592867438444708763, 12.85325344111689194626647935982, 13.441653937823738770957897411071, 14.07447120364854931356936377656, 15.291463067434259101952400863249, 15.80420339555183816471091166119, 16.676879301750376406212078698579, 17.58286642943029428911567926554, 18.60936452824582176264599682296, 19.04313001443653479562114817881, 19.643944757824191821446429089554, 20.44444686575571368486835507837