L(s) = 1 | − i·3-s + i·5-s − i·7-s − 9-s + i·11-s − 13-s + 15-s − 19-s − 21-s − i·23-s − 25-s + i·27-s + i·29-s + i·31-s + 33-s + ⋯ |
L(s) = 1 | − i·3-s + i·5-s − i·7-s − 9-s + i·11-s − 13-s + 15-s − 19-s − 21-s − i·23-s − 25-s + i·27-s + i·29-s + i·31-s + 33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 136 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.615 + 0.788i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 136 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.615 + 0.788i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.1244085814 + 0.2549730460i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.1244085814 + 0.2549730460i\) |
\(L(1)\) |
\(\approx\) |
\(0.7562392846 - 0.09309731024i\) |
\(L(1)\) |
\(\approx\) |
\(0.7562392846 - 0.09309731024i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 17 | \( 1 \) |
good | 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| 7 | \( 1 - iT \) |
| 11 | \( 1 \) |
| 13 | \( 1 + iT \) |
| 19 | \( 1 - iT \) |
| 23 | \( 1 \) |
| 29 | \( 1 - T \) |
| 31 | \( 1 \) |
| 37 | \( 1 + iT \) |
| 41 | \( 1 \) |
| 43 | \( 1 - T \) |
| 47 | \( 1 \) |
| 53 | \( 1 + T \) |
| 59 | \( 1 \) |
| 61 | \( 1 \) |
| 67 | \( 1 \) |
| 71 | \( 1 - T \) |
| 73 | \( 1 \) |
| 79 | \( 1 - T \) |
| 83 | \( 1 \) |
| 89 | \( 1 - iT \) |
| 97 | \( 1 \) |
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\(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−27.82504338723185267207222564188, −27.193822419109076730640981265444, −25.98512353975013284860802099677, −24.91740120256258420739250296296, −24.11389004717864291094513537351, −22.76630963246001423230377628276, −21.47271057968734680001685849438, −21.35073432623792008854955937332, −19.92241414184827828293630217699, −19.072490053002757122939496225051, −17.42607737772377709945380944377, −16.644407438116307485888610784281, −15.66989486813702223080374997402, −14.84024408056652347995717538485, −13.469803705937317740161242268056, −12.18280093889166845585878679334, −11.28740536958546263643245160091, −9.812818860013439778864366977729, −8.97953508914503090503183097847, −8.10043650763573390395062648611, −5.92096161389584698528753326987, −5.14210601404982872516384870184, −3.90238058266905516886037157021, −2.35445857209522629260005694542, −0.10372809833489118836417972618,
1.784015543364541407517245955370, 3.04805211923289649127034345846, 4.71833646969700179773710723961, 6.6053588888178704636890472471, 7.05258467856932466483888982366, 8.16744448133686566713741015148, 9.94182714268389593654001028000, 10.8683501482153879505123690854, 12.14303732091477948746194391775, 13.116938921949845919190250858904, 14.32599569840794610875084201913, 14.89373687984412042205469295243, 16.75208098295581665954050539984, 17.60543688332156693772491301777, 18.474010895365753997386107376749, 19.57040846179680500423416212598, 20.23660712108612491568848843037, 21.83575209075905887093265807413, 22.97650114860515592637497043705, 23.4342613348185567419911073280, 24.672720736643936846803644111256, 25.70790317199763028338367592667, 26.44398363855975214857106611054, 27.569647433710978931284273505463, 28.97782180986109345333196202712