L(s) = 1 | − i·2-s − 4-s + (0.866 − 0.5i)5-s − 1.73i·7-s + i·8-s + (−0.5 − 0.866i)10-s − 1.73·11-s − 1.73·14-s + 16-s + (−0.866 + 0.5i)20-s + 1.73i·22-s + (0.499 − 0.866i)25-s + 1.73i·28-s + 31-s − i·32-s + ⋯ |
L(s) = 1 | − i·2-s − 4-s + (0.866 − 0.5i)5-s − 1.73i·7-s + i·8-s + (−0.5 − 0.866i)10-s − 1.73·11-s − 1.73·14-s + 16-s + (−0.866 + 0.5i)20-s + 1.73i·22-s + (0.499 − 0.866i)25-s + 1.73i·28-s + 31-s − i·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1080 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.866 + 0.5i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1080 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.866 + 0.5i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.9271061268\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9271061268\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + iT \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (-0.866 + 0.5i)T \) |
good | 7 | \( 1 + 1.73iT - T^{2} \) |
| 11 | \( 1 + 1.73T + T^{2} \) |
| 13 | \( 1 + T^{2} \) |
| 17 | \( 1 + T^{2} \) |
| 19 | \( 1 - T^{2} \) |
| 23 | \( 1 + T^{2} \) |
| 29 | \( 1 + T^{2} \) |
| 31 | \( 1 - T + T^{2} \) |
| 37 | \( 1 + T^{2} \) |
| 41 | \( 1 - T^{2} \) |
| 43 | \( 1 + T^{2} \) |
| 47 | \( 1 + T^{2} \) |
| 53 | \( 1 - iT - T^{2} \) |
| 59 | \( 1 + T^{2} \) |
| 61 | \( 1 - T^{2} \) |
| 67 | \( 1 + T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 + 1.73iT - T^{2} \) |
| 79 | \( 1 - 2T + T^{2} \) |
| 83 | \( 1 + iT - T^{2} \) |
| 89 | \( 1 - T^{2} \) |
| 97 | \( 1 - 1.73iT - T^{2} \) |
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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
−10.10054753173865492609566976555, −9.180057933832893529515346655209, −8.132031209679243081392231242504, −7.51778571151360962519223862736, −6.19186171240020255060742843792, −5.06209797470060113091615043386, −4.51904233837962069829048957274, −3.32062298491310001628278791586, −2.20834013717454747780806594918, −0.869459263362982107627160513250,
2.24268083273325330541403018118, 3.09842598395990234858796801035, 4.86994752018897555580602482898, 5.49588552536215980422463874785, 6.05106501151413431306618060982, 6.97021515484389462163631641657, 8.054150553360396994732463364170, 8.597242644480923531800374545078, 9.543364995404761822417107962717, 10.08322887491663591233966540071