L(s) = 1 | − 3-s + i·7-s + 9-s − i·11-s − 13-s − i·17-s − i·19-s − i·21-s − 27-s + i·29-s + 31-s + i·33-s + 37-s + 39-s − 41-s + ⋯ |
L(s) = 1 | − 3-s + i·7-s + 9-s − i·11-s − 13-s − i·17-s − i·19-s − i·21-s − 27-s + i·29-s + 31-s + i·33-s + 37-s + 39-s − 41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.987 - 0.160i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.987 - 0.160i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.006999551401 - 0.08683060331i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.006999551401 - 0.08683060331i\) |
\(L(1)\) |
\(\approx\) |
\(0.6355812797 + 0.007182954180i\) |
\(L(1)\) |
\(\approx\) |
\(0.6355812797 + 0.007182954180i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 23 | \( 1 \) |
good | 3 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 \) |
| 13 | \( 1 \) |
| 17 | \( 1 \) |
| 19 | \( 1 + iT \) |
| 29 | \( 1 + T \) |
| 31 | \( 1 \) |
| 37 | \( 1 - iT \) |
| 41 | \( 1 \) |
| 43 | \( 1 - T \) |
| 47 | \( 1 \) |
| 53 | \( 1 \) |
| 59 | \( 1 \) |
| 61 | \( 1 - iT \) |
| 67 | \( 1 \) |
| 71 | \( 1 - iT \) |
| 73 | \( 1 \) |
| 79 | \( 1 - iT \) |
| 83 | \( 1 \) |
| 89 | \( 1 \) |
| 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
−20.54511141894472152448219159487, −19.72188147353854824289532046978, −19.03304238545376555765465338470, −18.09670311177589056680174975763, −17.37989309669862186039312348486, −16.94957476507718852590965389621, −16.37927255274444406536859055509, −15.230464744228227181376982475802, −14.81862924594637325687097218213, −13.68362527292227009676357685696, −12.975817877867260920671734908808, −12.19100289663656167141007345976, −11.69634730779897271319283125024, −10.58194339822427792497514472972, −10.12363534113638167764733063881, −9.60997316514525716475300206758, −8.08824988577477669755069425609, −7.51317466325060911142607335250, −6.68310830300749262419764139977, −6.04491347673372261550199655462, −4.91027030041788032506404669042, −4.417377789981816366921058080200, −3.56954577282562261530506145971, −2.0999654191934409353288692834, −1.28449675195145926922553678656,
0.03855291416984383199808609938, 1.22221420583401865994870335778, 2.48930835442507304424434145865, 3.2051480098844311097993430737, 4.69806786919686085538988712095, 5.04749329767493722084054649611, 5.94998513519671953532195200035, 6.64943855629324212030545972626, 7.47400004383816069036137983869, 8.500842666436399565939483640948, 9.34142349170526329619508950250, 10.00768531143102532757969200236, 11.06556192602808058433659026391, 11.569439411721746526497064422435, 12.19102691882794142018605371103, 12.97306758316709537608872795555, 13.77741584298183806215362803501, 14.77439060398195234338935225209, 15.53904613261093790326420370034, 16.175088444145450700894144385155, 16.816403380886034643143136524484, 17.676073037992835350118265638394, 18.24889708270137681754201749686, 18.95288053959290860936124546840, 19.58918876809158745709325570149