L(s) = 1 | + (0.953 − 0.299i)2-s + (0.820 − 0.572i)4-s + (−0.851 + 0.524i)5-s + (−0.532 − 0.846i)7-s + (0.610 − 0.791i)8-s + (−0.654 + 0.755i)10-s + (0.797 + 0.603i)13-s + (−0.761 − 0.647i)14-s + (0.345 − 0.938i)16-s + (−0.254 + 0.967i)17-s + (−0.362 + 0.931i)19-s + (−0.398 + 0.917i)20-s + (0.928 − 0.371i)23-s + (0.449 − 0.893i)25-s + (0.941 + 0.336i)26-s + ⋯ |
L(s) = 1 | + (0.953 − 0.299i)2-s + (0.820 − 0.572i)4-s + (−0.851 + 0.524i)5-s + (−0.532 − 0.846i)7-s + (0.610 − 0.791i)8-s + (−0.654 + 0.755i)10-s + (0.797 + 0.603i)13-s + (−0.761 − 0.647i)14-s + (0.345 − 0.938i)16-s + (−0.254 + 0.967i)17-s + (−0.362 + 0.931i)19-s + (−0.398 + 0.917i)20-s + (0.928 − 0.371i)23-s + (0.449 − 0.893i)25-s + (0.941 + 0.336i)26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.573 - 0.819i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.573 - 0.819i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(2.087037662 - 1.086634356i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.087037662 - 1.086634356i\) |
\(L(1)\) |
\(\approx\) |
\(1.593687175 - 0.4184607132i\) |
\(L(1)\) |
\(\approx\) |
\(1.593687175 - 0.4184607132i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 11 | \( 1 \) |
good | 2 | \( 1 + (0.953 - 0.299i)T \) |
| 5 | \( 1 + (-0.851 + 0.524i)T \) |
| 7 | \( 1 + (-0.532 - 0.846i)T \) |
| 13 | \( 1 + (0.797 + 0.603i)T \) |
| 17 | \( 1 + (-0.254 + 0.967i)T \) |
| 19 | \( 1 + (-0.362 + 0.931i)T \) |
| 23 | \( 1 + (0.928 - 0.371i)T \) |
| 29 | \( 1 + (0.548 - 0.836i)T \) |
| 31 | \( 1 + (0.483 - 0.875i)T \) |
| 37 | \( 1 + (0.974 - 0.226i)T \) |
| 41 | \( 1 + (0.749 - 0.662i)T \) |
| 43 | \( 1 + (0.235 - 0.971i)T \) |
| 47 | \( 1 + (0.861 - 0.508i)T \) |
| 53 | \( 1 + (-0.985 - 0.170i)T \) |
| 59 | \( 1 + (-0.948 + 0.318i)T \) |
| 61 | \( 1 + (-0.217 - 0.976i)T \) |
| 67 | \( 1 + (-0.995 + 0.0950i)T \) |
| 71 | \( 1 + (0.774 + 0.633i)T \) |
| 73 | \( 1 + (0.696 - 0.717i)T \) |
| 79 | \( 1 + (-0.432 - 0.901i)T \) |
| 83 | \( 1 + (-0.0665 + 0.997i)T \) |
| 89 | \( 1 + (0.841 + 0.540i)T \) |
| 97 | \( 1 + (-0.851 - 0.524i)T \) |
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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
−21.57396839948476200284151221781, −20.95702548102486609961900779520, −19.99373662133972497020674758580, −19.56925217379854865316376586320, −18.47965639705507945590150707507, −17.56204119706051677423827714752, −16.49621117096546248282585154806, −15.81155973823555048886103061285, −15.50595852935564545729765223771, −14.644379735993503733601548036001, −13.52873256850388877024585637726, −12.86789592527572246359057123956, −12.3057425555509872676835007534, −11.39622023641326761989815843631, −10.866414353353143851057918434867, −9.30076413427878003063830754528, −8.62203941396030418917648052399, −7.732663036323709657838444330274, −6.83299571211976253413958435864, −6.00445563059092048836707126973, −5.01984498108469250350963376471, −4.45315148302379245013596952466, −3.15651877634986022436398505943, −2.81019365479197869246511104519, −1.13606341225004118022444245154,
0.83481690158960671448376214914, 2.15056017405755469969426118245, 3.25326227066399329876522769147, 4.01700792885017947045778692620, 4.407132501786952762981400700751, 5.99833784685134145417708839456, 6.493875268793956081717652045640, 7.373222507605246139868463534532, 8.229164532993133295245386993579, 9.558348176908415308528331028639, 10.714629231946741651991744113640, 10.82908216068082371354777144649, 11.926314913140799233348771918930, 12.64839136354036651146868467741, 13.48664537859646284196217395691, 14.177055411201661423968296979567, 15.01107901156331534867887882994, 15.668512887431691315883372681191, 16.45748579871991445169329793075, 17.18298302244380069428839425974, 18.74205945737193635958349794883, 19.05473958589425717451931681518, 19.84246078766958071853253988043, 20.617589527749305957278087783171, 21.28773001983886308297700712928