L(s) = 1 | + i·3-s − 4.02i·5-s + 4.61·7-s − 9-s + 1.53i·11-s − 5.57i·13-s + 4.02·15-s − 1.29·17-s − 4.35i·19-s + 4.61i·21-s + 4·23-s − 11.2·25-s − i·27-s − 1.86i·29-s + 2.77·31-s + ⋯ |
L(s) = 1 | + 0.577i·3-s − 1.80i·5-s + 1.74·7-s − 0.333·9-s + 0.461i·11-s − 1.54i·13-s + 1.03·15-s − 0.314·17-s − 1.00i·19-s + 1.00i·21-s + 0.834·23-s − 2.24·25-s − 0.192i·27-s − 0.345i·29-s + 0.498·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3072 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3072 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.090216231\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.090216231\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 - iT \) |
good | 5 | \( 1 + 4.02iT - 5T^{2} \) |
| 7 | \( 1 - 4.61T + 7T^{2} \) |
| 11 | \( 1 - 1.53iT - 11T^{2} \) |
| 13 | \( 1 + 5.57iT - 13T^{2} \) |
| 17 | \( 1 + 1.29T + 17T^{2} \) |
| 19 | \( 1 + 4.35iT - 19T^{2} \) |
| 23 | \( 1 - 4T + 23T^{2} \) |
| 29 | \( 1 + 1.86iT - 29T^{2} \) |
| 31 | \( 1 - 2.77T + 31T^{2} \) |
| 37 | \( 1 + 3.97iT - 37T^{2} \) |
| 41 | \( 1 - 3.03T + 41T^{2} \) |
| 43 | \( 1 - 3.03iT - 43T^{2} \) |
| 47 | \( 1 + 9.65T + 47T^{2} \) |
| 53 | \( 1 - 8.58iT - 53T^{2} \) |
| 59 | \( 1 - 5.73iT - 59T^{2} \) |
| 61 | \( 1 + 7.64iT - 61T^{2} \) |
| 67 | \( 1 - 8.79iT - 67T^{2} \) |
| 71 | \( 1 + 6.88T + 71T^{2} \) |
| 73 | \( 1 - 3.50T + 73T^{2} \) |
| 79 | \( 1 + 8.18T + 79T^{2} \) |
| 83 | \( 1 + 4.12iT - 83T^{2} \) |
| 89 | \( 1 + 7.98T + 89T^{2} \) |
| 97 | \( 1 + 1.40T + 97T^{2} \) |
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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
−8.512018011676067161129745501510, −8.037016171355534161244797076034, −7.33420544292949811709677037534, −5.81381819645855784350676724306, −5.20039012055988813167929955756, −4.69286546205909603914202477685, −4.24286711194051877991461619850, −2.77834836673844326691152587051, −1.57141648205243704570208329740, −0.66898198359746370396300153024,
1.53221452820273476656633193571, 2.17295713753005569606904762939, 3.18323738678047936114388647731, 4.14591767081921907193213743997, 5.08481973565345530742191899197, 6.11049888916896407253744158683, 6.74329704012571931593851661016, 7.30720093502269936346482660003, 8.052712899192797221154871495716, 8.637649185951870048587870875884