L(s) = 1 | + (0.897 + 0.441i)2-s + (−0.809 + 0.587i)3-s + (0.610 + 0.791i)4-s + (−0.736 + 0.676i)5-s + (−0.985 + 0.170i)6-s + (−0.998 − 0.0570i)7-s + (0.198 + 0.980i)8-s + (0.309 − 0.951i)9-s + (−0.959 + 0.281i)10-s + (−0.959 − 0.281i)12-s + (−0.564 + 0.825i)13-s + (−0.870 − 0.491i)14-s + (0.198 − 0.980i)15-s + (−0.254 + 0.967i)16-s + (−0.921 − 0.389i)17-s + (0.696 − 0.717i)18-s + ⋯ |
L(s) = 1 | + (0.897 + 0.441i)2-s + (−0.809 + 0.587i)3-s + (0.610 + 0.791i)4-s + (−0.736 + 0.676i)5-s + (−0.985 + 0.170i)6-s + (−0.998 − 0.0570i)7-s + (0.198 + 0.980i)8-s + (0.309 − 0.951i)9-s + (−0.959 + 0.281i)10-s + (−0.959 − 0.281i)12-s + (−0.564 + 0.825i)13-s + (−0.870 − 0.491i)14-s + (0.198 − 0.980i)15-s + (−0.254 + 0.967i)16-s + (−0.921 − 0.389i)17-s + (0.696 − 0.717i)18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 121 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.885 + 0.464i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 121 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.885 + 0.464i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.2262759857 + 0.9188274751i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.2262759857 + 0.9188274751i\) |
\(L(1)\) |
\(\approx\) |
\(0.7642515785 + 0.6916035976i\) |
\(L(1)\) |
\(\approx\) |
\(0.7642515785 + 0.6916035976i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 11 | \( 1 \) |
good | 2 | \( 1 + (0.897 + 0.441i)T \) |
| 3 | \( 1 + (-0.809 + 0.587i)T \) |
| 5 | \( 1 + (-0.736 + 0.676i)T \) |
| 7 | \( 1 + (-0.998 - 0.0570i)T \) |
| 13 | \( 1 + (-0.564 + 0.825i)T \) |
| 17 | \( 1 + (-0.921 - 0.389i)T \) |
| 19 | \( 1 + (0.974 + 0.226i)T \) |
| 23 | \( 1 + (0.841 + 0.540i)T \) |
| 29 | \( 1 + (0.0855 + 0.996i)T \) |
| 31 | \( 1 + (-0.0285 + 0.999i)T \) |
| 37 | \( 1 + (0.941 + 0.336i)T \) |
| 41 | \( 1 + (-0.466 - 0.884i)T \) |
| 43 | \( 1 + (0.415 - 0.909i)T \) |
| 47 | \( 1 + (0.696 + 0.717i)T \) |
| 53 | \( 1 + (-0.254 - 0.967i)T \) |
| 59 | \( 1 + (-0.466 + 0.884i)T \) |
| 61 | \( 1 + (0.897 - 0.441i)T \) |
| 67 | \( 1 + (-0.142 + 0.989i)T \) |
| 71 | \( 1 + (0.516 - 0.856i)T \) |
| 73 | \( 1 + (-0.362 - 0.931i)T \) |
| 79 | \( 1 + (0.993 - 0.113i)T \) |
| 83 | \( 1 + (0.774 + 0.633i)T \) |
| 89 | \( 1 + (-0.654 + 0.755i)T \) |
| 97 | \( 1 + (-0.736 - 0.676i)T \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−28.667068855721634431395158503856, −28.21792842532882445679102296040, −26.841359185930289777623549033506, −24.946778750951052928964664760040, −24.45757490336038356825071918805, −23.31995244180138620016755850844, −22.67223086794427608356848090958, −21.865854917630507428976170707344, −20.276995734249080533733449543962, −19.59732022691600744794074368951, −18.63684196343928830691081888912, −17.06282320884958352401575168404, −16.02970963004695287230160552687, −15.18421027511061503588892296041, −13.33046068171785149187036051422, −12.82774289082870371557904748934, −11.91025854749127079305593290553, −10.94857353474449480422158660257, −9.60586052069537535346348387770, −7.69089319011884400674259952096, −6.51351295755044763938665245592, −5.384562879195695334718981009207, −4.258451968072923654449063027715, −2.67846890623578467324239256557, −0.73506328149887974957117021699,
3.005523386719594284350523844353, 4.01410505154144700137617994056, 5.20831581288316969454298615784, 6.62925695731973513168525435643, 7.17918513846010950808774303168, 9.19212370162404441240785948930, 10.6697986659018041073947611754, 11.6537308744785520141525266062, 12.48004049674675096784594192932, 13.91155896175783861197024350251, 15.12396893288243785330282030204, 15.92255766795029089580420445926, 16.60825185612323380103731916457, 17.91523097919121151884634345549, 19.35704445093847266231589144133, 20.52416652006584327211785269110, 22.01385741607515204217749137483, 22.26864260221097362469755064169, 23.29164443056725056712854018682, 23.992235307727595377101563221022, 25.45980984474875890368781486145, 26.58250044330944581111611544278, 27.09027522941628443889819968096, 28.913263183009468515345209838958, 29.26599219105291046205109729835