L(s) = 1 | + (−0.309 + 0.951i)3-s + 7-s + (−0.809 − 0.587i)9-s + (0.809 − 0.587i)11-s + (0.809 + 0.587i)13-s + (0.309 + 0.951i)17-s + (−0.309 − 0.951i)19-s + (−0.309 + 0.951i)21-s + (−0.809 + 0.587i)23-s + (0.809 − 0.587i)27-s + (−0.309 + 0.951i)29-s + (0.309 + 0.951i)31-s + (0.309 + 0.951i)33-s + (0.809 + 0.587i)37-s + (−0.809 + 0.587i)39-s + ⋯ |
L(s) = 1 | + (−0.309 + 0.951i)3-s + 7-s + (−0.809 − 0.587i)9-s + (0.809 − 0.587i)11-s + (0.809 + 0.587i)13-s + (0.309 + 0.951i)17-s + (−0.309 − 0.951i)19-s + (−0.309 + 0.951i)21-s + (−0.809 + 0.587i)23-s + (0.809 − 0.587i)27-s + (−0.309 + 0.951i)29-s + (0.309 + 0.951i)31-s + (0.309 + 0.951i)33-s + (0.809 + 0.587i)37-s + (−0.809 + 0.587i)39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 200 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.535 + 0.844i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 200 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.535 + 0.844i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.040693357 + 0.5721260147i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.040693357 + 0.5721260147i\) |
\(L(1)\) |
\(\approx\) |
\(1.032624294 + 0.3319573545i\) |
\(L(1)\) |
\(\approx\) |
\(1.032624294 + 0.3319573545i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
good | 3 | \( 1 + (-0.309 + 0.951i)T \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 + (0.809 - 0.587i)T \) |
| 13 | \( 1 + (0.809 + 0.587i)T \) |
| 17 | \( 1 + (0.309 + 0.951i)T \) |
| 19 | \( 1 + (-0.309 - 0.951i)T \) |
| 23 | \( 1 + (-0.809 + 0.587i)T \) |
| 29 | \( 1 + (-0.309 + 0.951i)T \) |
| 31 | \( 1 + (0.309 + 0.951i)T \) |
| 37 | \( 1 + (0.809 + 0.587i)T \) |
| 41 | \( 1 + (-0.809 - 0.587i)T \) |
| 43 | \( 1 - T \) |
| 47 | \( 1 + (0.309 - 0.951i)T \) |
| 53 | \( 1 + (-0.309 + 0.951i)T \) |
| 59 | \( 1 + (0.809 + 0.587i)T \) |
| 61 | \( 1 + (0.809 - 0.587i)T \) |
| 67 | \( 1 + (-0.309 - 0.951i)T \) |
| 71 | \( 1 + (0.309 - 0.951i)T \) |
| 73 | \( 1 + (-0.809 + 0.587i)T \) |
| 79 | \( 1 + (0.309 - 0.951i)T \) |
| 83 | \( 1 + (-0.309 - 0.951i)T \) |
| 89 | \( 1 + (-0.809 + 0.587i)T \) |
| 97 | \( 1 + (0.309 - 0.951i)T \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−26.88877084334063813598467113632, −25.37009877295466429648652677538, −24.94998306851510643653632825282, −23.94730394736278481492288779724, −23.02904895817732906152786642958, −22.32162199326546611890079520802, −20.81074083548657058008753405540, −20.18261053979356186310876355564, −18.86802489067200219529053873949, −18.12694528527520289282473158893, −17.352760247028595610641327830618, −16.3599073387843128993688030911, −14.864947711313681533897428250826, −14.08060849347568135755753762639, −13.022889194788427487500870169461, −11.90187025148506168750763510395, −11.30802136675975427035154910783, −9.93853993881460957367553307906, −8.396610716358763270866166752734, −7.71404037427088751884778188494, −6.47266908409354110707482063383, −5.46778875770818886817227743445, −4.10181513124177438500224428307, −2.31006237250729210363208505240, −1.16678896088193981694731306773,
1.51913789710526858107221771566, 3.4508369809728700803769304941, 4.38736573001378531112502530712, 5.53166484364844090124808636854, 6.62409005725283799438506788200, 8.35762918882993024295670316128, 9.017827924832673812419031925172, 10.390064341480197906218815357206, 11.243920251523204464556676901985, 11.95857912684032758683550202463, 13.65940039807263016811371895021, 14.5514829134894910930326577621, 15.4267173826685565637825413485, 16.53129423109257270860148445048, 17.273854273843435924379402119545, 18.26935404298679345531803455744, 19.608846609683568961856634849881, 20.54416147113324436426753536110, 21.72745084588767567378441814007, 21.80057543319904379808371179324, 23.46240934071449897012221980182, 23.92438971402613971126642870101, 25.327939929239203759111454250542, 26.26319024656322280711311972661, 27.16648536375092136918154233581