L(s) = 1 | + (0.5 + 0.866i)2-s + (−0.499 + 0.866i)4-s + (−0.5 − 0.866i)5-s + i·7-s − 0.999·8-s + (0.499 − 0.866i)10-s + (0.866 − 1.5i)11-s + 1.73·13-s + (−0.866 + 0.5i)14-s + (−0.5 − 0.866i)16-s + (−0.5 − 0.866i)19-s + 0.999·20-s + 1.73·22-s + (0.5 + 0.866i)23-s + (−0.499 + 0.866i)25-s + (0.866 + 1.49i)26-s + ⋯ |
L(s) = 1 | + (0.5 + 0.866i)2-s + (−0.499 + 0.866i)4-s + (−0.5 − 0.866i)5-s + i·7-s − 0.999·8-s + (0.499 − 0.866i)10-s + (0.866 − 1.5i)11-s + 1.73·13-s + (−0.866 + 0.5i)14-s + (−0.5 − 0.866i)16-s + (−0.5 − 0.866i)19-s + 0.999·20-s + 1.73·22-s + (0.5 + 0.866i)23-s + (−0.499 + 0.866i)25-s + (0.866 + 1.49i)26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.553 - 0.832i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.553 - 0.832i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.473304953\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.473304953\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.5 - 0.866i)T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (0.5 + 0.866i)T \) |
| 7 | \( 1 - iT \) |
good | 11 | \( 1 + (-0.866 + 1.5i)T + (-0.5 - 0.866i)T^{2} \) |
| 13 | \( 1 - 1.73T + T^{2} \) |
| 17 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 19 | \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 23 | \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 29 | \( 1 - T^{2} \) |
| 31 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 37 | \( 1 + (-0.866 - 1.5i)T + (-0.5 + 0.866i)T^{2} \) |
| 41 | \( 1 - 1.73T + T^{2} \) |
| 43 | \( 1 - T^{2} \) |
| 47 | \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 53 | \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 59 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 61 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 67 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 79 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 83 | \( 1 - T^{2} \) |
| 89 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 97 | \( 1 - T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.993071937707239449845893165373, −8.421416973948489865647296127332, −7.916207959102967853406624966557, −6.65161810939165058954022621953, −6.04110300891148399010946195427, −5.50598818165184724551454028923, −4.52525704264673422429047343810, −3.70988321191806048983105925157, −2.98476059432350568431861698592, −1.13118150223226981639620625182,
1.17761286650172589734516317596, 2.25734861373653594131652818678, 3.48608916360494848136469909798, 4.05914706977121436057489958542, 4.52800755991690738777534013797, 6.02996015974499335661973024975, 6.51814815217135803081316912572, 7.34921385882895836404119004755, 8.229132130625812306224005482521, 9.234694323088628406512584630478