L(s) = 1 | − 3·2-s + 5·4-s + 5.19i·5-s − 3·8-s − 15.5i·10-s − 15·11-s + 13.8i·13-s − 11·16-s − 10.3i·17-s + 10.3i·19-s + 25.9i·20-s + 45·22-s − 2·25-s − 41.5i·26-s + 9·29-s + ⋯ |
L(s) = 1 | − 1.5·2-s + 1.25·4-s + 1.03i·5-s − 0.375·8-s − 1.55i·10-s − 1.36·11-s + 1.06i·13-s − 0.687·16-s − 0.611i·17-s + 0.546i·19-s + 1.29i·20-s + 2.04·22-s − 0.0800·25-s − 1.59i·26-s + 0.310·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.755 + 0.654i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.755 + 0.654i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.00937502 - 0.0251459i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.00937502 - 0.0251459i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 \) |
good | 2 | \( 1 + 3T + 4T^{2} \) |
| 5 | \( 1 - 5.19iT - 25T^{2} \) |
| 11 | \( 1 + 15T + 121T^{2} \) |
| 13 | \( 1 - 13.8iT - 169T^{2} \) |
| 17 | \( 1 + 10.3iT - 289T^{2} \) |
| 19 | \( 1 - 10.3iT - 361T^{2} \) |
| 23 | \( 1 + 529T^{2} \) |
| 29 | \( 1 - 9T + 841T^{2} \) |
| 31 | \( 1 + 12.1iT - 961T^{2} \) |
| 37 | \( 1 - 10T + 1.36e3T^{2} \) |
| 41 | \( 1 + 10.3iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 74T + 1.84e3T^{2} \) |
| 47 | \( 1 - 2.20e3T^{2} \) |
| 53 | \( 1 + 33T + 2.80e3T^{2} \) |
| 59 | \( 1 + 15.5iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 90.0iT - 3.72e3T^{2} \) |
| 67 | \( 1 + 76T + 4.48e3T^{2} \) |
| 71 | \( 1 + 84T + 5.04e3T^{2} \) |
| 73 | \( 1 + 62.3iT - 5.32e3T^{2} \) |
| 79 | \( 1 + 43T + 6.24e3T^{2} \) |
| 83 | \( 1 + 119. iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 72.7iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 185. iT - 9.40e3T^{2} \) |
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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
−11.09888261283272589951389788521, −10.37047761456781050939185621851, −9.777712475648281033585369991811, −8.783643832403680375643234695473, −7.83427574755241016975782640010, −7.16996899487590012274196676380, −6.28293011160590821135575855321, −4.75970162257730277722623794836, −3.03941847789391547335797718041, −1.91251358320683494224378694168,
0.01970016574074470454491590490, 1.25806336519482208263484974215, 2.77479981456033149304625145546, 4.65301231819552776298261789673, 5.59119126720464350910997810112, 7.02310348503872889219807850794, 8.113715754154657924090906621988, 8.376740728640365535845007469597, 9.381372013227864316781795347582, 10.29045715910638252764443605169