L(s) = 1 | + 3-s + (0.5 − 0.866i)5-s + 9-s + (0.5 − 0.866i)11-s + (0.5 − 0.866i)13-s + (0.5 − 0.866i)15-s − 17-s − 23-s + (−0.5 − 0.866i)25-s + 27-s + (−0.5 + 0.866i)29-s + (−0.5 + 0.866i)31-s + (0.5 − 0.866i)33-s + (−0.5 − 0.866i)37-s + (0.5 − 0.866i)39-s + ⋯ |
L(s) = 1 | + 3-s + (0.5 − 0.866i)5-s + 9-s + (0.5 − 0.866i)11-s + (0.5 − 0.866i)13-s + (0.5 − 0.866i)15-s − 17-s − 23-s + (−0.5 − 0.866i)25-s + 27-s + (−0.5 + 0.866i)29-s + (−0.5 + 0.866i)31-s + (0.5 − 0.866i)33-s + (−0.5 − 0.866i)37-s + (0.5 − 0.866i)39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 532 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.572 - 0.819i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 532 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.572 - 0.819i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.914010058 - 0.9977863777i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.914010058 - 0.9977863777i\) |
\(L(1)\) |
\(\approx\) |
\(1.552660415 - 0.3938392219i\) |
\(L(1)\) |
\(\approx\) |
\(1.552660415 - 0.3938392219i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
| 19 | \( 1 \) |
good | 3 | \( 1 + T \) |
| 5 | \( 1 + (0.5 - 0.866i)T \) |
| 11 | \( 1 + (0.5 - 0.866i)T \) |
| 13 | \( 1 + (0.5 - 0.866i)T \) |
| 17 | \( 1 - T \) |
| 23 | \( 1 - T \) |
| 29 | \( 1 + (-0.5 + 0.866i)T \) |
| 31 | \( 1 + (-0.5 + 0.866i)T \) |
| 37 | \( 1 + (-0.5 - 0.866i)T \) |
| 41 | \( 1 + (0.5 + 0.866i)T \) |
| 43 | \( 1 + (0.5 + 0.866i)T \) |
| 47 | \( 1 + T \) |
| 53 | \( 1 + (-0.5 - 0.866i)T \) |
| 59 | \( 1 + T \) |
| 61 | \( 1 - T \) |
| 67 | \( 1 + (0.5 + 0.866i)T \) |
| 71 | \( 1 + (0.5 + 0.866i)T \) |
| 73 | \( 1 - T \) |
| 79 | \( 1 + (0.5 - 0.866i)T \) |
| 83 | \( 1 + T \) |
| 89 | \( 1 - T \) |
| 97 | \( 1 + (0.5 + 0.866i)T \) |
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
−23.75065895844098592771450888372, −22.46316557042260976791245051320, −22.01439554242397058802775388375, −20.94481977584059974215902035135, −20.32058954673606548721779243372, −19.34154798758501452009629410944, −18.64126166218656538784905519927, −17.87362011731015683970083661402, −16.91936962087893561213283338730, −15.56582166131662453235558251374, −15.11013275559824602541471474373, −13.99277898517120483322654032140, −13.7395995813507299720534571867, −12.55420705372693676038074659543, −11.440593298636328332146549098950, −10.41321451784590514367154268073, −9.53095148705032234449575400522, −8.91605038235908660865981360867, −7.64468229827437510078159412616, −6.89162996949280217924384216204, −6.03365721136701099625415836289, −4.37726702453869955132556878537, −3.69841137145733227548195077813, −2.31582124487479001918520691214, −1.83673676169111869694516896831,
1.08571587321343820702347258740, 2.141001890643082638041462639645, 3.35333128351311160198963352126, 4.26117016476697980648259992112, 5.45185569341183083381504993860, 6.44007662349350304022216010085, 7.719348989576244207889188200578, 8.65553066208761440346278247492, 9.05845095353274494844297756025, 10.13265894361932247230569657651, 11.12737599997387350038896400893, 12.49260311264231780631611189075, 13.119846060675615053932385162019, 13.89630852248456627214507410666, 14.63308524072704714793585616326, 15.88105687324898726647581327363, 16.25966539642394767789758304474, 17.586744178886155201128379149407, 18.21726848860793334160391242505, 19.425366087094034076623160701154, 20.0325077091280125411264420219, 20.680376800201468948562233275604, 21.59016587817091488189564288425, 22.201657128190592652046577028283, 23.616166244187212638301697780537