L(s) = 1 | − 2i·2-s + 0.607i·3-s − 4·4-s + 1.21·6-s − 13.9i·7-s + 8i·8-s + 8.63·9-s − 2.42i·12-s − 27.9·14-s + 16·16-s − 17.2i·18-s + 8.50·21-s − 26.3i·23-s − 4.85·24-s + 10.7i·27-s + 55.9i·28-s + ⋯ |
L(s) = 1 | − i·2-s + 0.202i·3-s − 4-s + 0.202·6-s − 1.99i·7-s + i·8-s + 0.959·9-s − 0.202i·12-s − 1.99·14-s + 16-s − 0.959i·18-s + 0.404·21-s − 1.14i·23-s − 0.202·24-s + 0.396i·27-s + 1.99i·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 500 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 500 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(1.215467890\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.215467890\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + 2iT \) |
| 5 | \( 1 \) |
good | 3 | \( 1 - 0.607iT - 9T^{2} \) |
| 7 | \( 1 + 13.9iT - 49T^{2} \) |
| 11 | \( 1 - 121T^{2} \) |
| 13 | \( 1 + 169T^{2} \) |
| 17 | \( 1 + 289T^{2} \) |
| 19 | \( 1 - 361T^{2} \) |
| 23 | \( 1 + 26.3iT - 529T^{2} \) |
| 29 | \( 1 + 49.3T + 841T^{2} \) |
| 31 | \( 1 - 961T^{2} \) |
| 37 | \( 1 + 1.36e3T^{2} \) |
| 41 | \( 1 + 81.7T + 1.68e3T^{2} \) |
| 43 | \( 1 + 37.8iT - 1.84e3T^{2} \) |
| 47 | \( 1 + 51.9iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 2.80e3T^{2} \) |
| 59 | \( 1 - 3.48e3T^{2} \) |
| 61 | \( 1 - 84.1T + 3.72e3T^{2} \) |
| 67 | \( 1 - 116iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 5.04e3T^{2} \) |
| 73 | \( 1 + 5.32e3T^{2} \) |
| 79 | \( 1 - 6.24e3T^{2} \) |
| 83 | \( 1 + 116. iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 145.T + 7.92e3T^{2} \) |
| 97 | \( 1 + 9.40e3T^{2} \) |
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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
−10.27902775101789974718327818098, −9.881435320591175180838865748820, −8.667518042382559103997687622631, −7.56278356828340772452628492746, −6.80079565508311725573857568497, −5.08882626810620976862146656137, −4.13919772089289814673325982429, −3.57056718518344498225706053467, −1.75999807839968682707927850669, −0.50445031221110938526151080075,
1.80858659372502410299368499589, 3.45738829345319500934003592276, 4.89909510541930449502252193530, 5.65761871377742053123691175302, 6.50947038963453959044579070490, 7.55259934777509491419800559722, 8.387199009910882075674100525729, 9.311227520576869055327632109594, 9.779340340958023855397508150149, 11.30706867602629231639460976195