L(s) = 1 | − 2.82i·5-s − i·7-s − 1.41·11-s − 2·13-s − 7.07·23-s − 3.00·25-s − 9.89i·29-s + 4i·31-s − 2.82·35-s + 2.82i·41-s + 4i·43-s − 2.82·47-s − 49-s − 4.24i·53-s + 4.00i·55-s + ⋯ |
L(s) = 1 | − 1.26i·5-s − 0.377i·7-s − 0.426·11-s − 0.554·13-s − 1.47·23-s − 0.600·25-s − 1.83i·29-s + 0.718i·31-s − 0.478·35-s + 0.441i·41-s + 0.609i·43-s − 0.412·47-s − 0.142·49-s − 0.582i·53-s + 0.539i·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2016 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.985 - 0.169i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2016 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.985 - 0.169i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.5385346126\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5385346126\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + iT \) |
good | 5 | \( 1 + 2.82iT - 5T^{2} \) |
| 11 | \( 1 + 1.41T + 11T^{2} \) |
| 13 | \( 1 + 2T + 13T^{2} \) |
| 17 | \( 1 - 17T^{2} \) |
| 19 | \( 1 - 19T^{2} \) |
| 23 | \( 1 + 7.07T + 23T^{2} \) |
| 29 | \( 1 + 9.89iT - 29T^{2} \) |
| 31 | \( 1 - 4iT - 31T^{2} \) |
| 37 | \( 1 + 37T^{2} \) |
| 41 | \( 1 - 2.82iT - 41T^{2} \) |
| 43 | \( 1 - 4iT - 43T^{2} \) |
| 47 | \( 1 + 2.82T + 47T^{2} \) |
| 53 | \( 1 + 4.24iT - 53T^{2} \) |
| 59 | \( 1 - 2.82T + 59T^{2} \) |
| 61 | \( 1 + 14T + 61T^{2} \) |
| 67 | \( 1 - 10iT - 67T^{2} \) |
| 71 | \( 1 - 7.07T + 71T^{2} \) |
| 73 | \( 1 + 6T + 73T^{2} \) |
| 79 | \( 1 - 14iT - 79T^{2} \) |
| 83 | \( 1 + 11.3T + 83T^{2} \) |
| 89 | \( 1 - 8.48iT - 89T^{2} \) |
| 97 | \( 1 + 14T + 97T^{2} \) |
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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
−8.562388125035950299082070839179, −8.114324781043737924608546505940, −7.33741453208251919975745664479, −6.25456939991234159985501383387, −5.45134145812226616470449838336, −4.61929811004212363961206708831, −4.00924234978420940382886947876, −2.64997537838759852476012929644, −1.47874488893559954389493611207, −0.18336990947785650624021473923,
1.91814148541747434332632901686, 2.80212292510036994662328863548, 3.59958093013743889585048963707, 4.74802571157142289038999913584, 5.70958883213699101794291930542, 6.43441876828900572970375733728, 7.25827716433049915597682272227, 7.82957903175270038131401809991, 8.806786070206760093486480569353, 9.637055315887257287363720956340