L(s) = 1 | + (0.587 − 0.809i)2-s + (−0.951 + 0.309i)3-s + (−0.309 − 0.951i)4-s + (−0.309 + 0.951i)6-s + i·7-s + (−0.951 − 0.309i)8-s + (0.809 − 0.587i)9-s + (0.587 + 0.809i)12-s + (0.587 + 0.809i)13-s + (0.809 + 0.587i)14-s + (−0.809 + 0.587i)16-s + (0.951 + 0.309i)17-s − i·18-s + (0.309 − 0.951i)19-s + (−0.309 − 0.951i)21-s + ⋯ |
L(s) = 1 | + (0.587 − 0.809i)2-s + (−0.951 + 0.309i)3-s + (−0.309 − 0.951i)4-s + (−0.309 + 0.951i)6-s + i·7-s + (−0.951 − 0.309i)8-s + (0.809 − 0.587i)9-s + (0.587 + 0.809i)12-s + (0.587 + 0.809i)13-s + (0.809 + 0.587i)14-s + (−0.809 + 0.587i)16-s + (0.951 + 0.309i)17-s − i·18-s + (0.309 − 0.951i)19-s + (−0.309 − 0.951i)21-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 275 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.770 - 0.637i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 275 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.770 - 0.637i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.148409277 - 0.4134527807i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.148409277 - 0.4134527807i\) |
\(L(1)\) |
\(\approx\) |
\(1.041386717 - 0.3200707264i\) |
\(L(1)\) |
\(\approx\) |
\(1.041386717 - 0.3200707264i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 11 | \( 1 \) |
good | 2 | \( 1 + (0.587 - 0.809i)T \) |
| 3 | \( 1 + (-0.951 + 0.309i)T \) |
| 7 | \( 1 + iT \) |
| 13 | \( 1 + (0.587 + 0.809i)T \) |
| 17 | \( 1 + (0.951 + 0.309i)T \) |
| 19 | \( 1 + (0.309 - 0.951i)T \) |
| 23 | \( 1 + (0.587 - 0.809i)T \) |
| 29 | \( 1 + (0.309 + 0.951i)T \) |
| 31 | \( 1 + (0.309 - 0.951i)T \) |
| 37 | \( 1 + (0.587 + 0.809i)T \) |
| 41 | \( 1 + (0.809 - 0.587i)T \) |
| 43 | \( 1 - iT \) |
| 47 | \( 1 + (0.951 - 0.309i)T \) |
| 53 | \( 1 + (-0.951 + 0.309i)T \) |
| 59 | \( 1 + (0.809 - 0.587i)T \) |
| 61 | \( 1 + (0.809 + 0.587i)T \) |
| 67 | \( 1 + (-0.951 - 0.309i)T \) |
| 71 | \( 1 + (0.309 + 0.951i)T \) |
| 73 | \( 1 + (-0.587 + 0.809i)T \) |
| 79 | \( 1 + (0.309 + 0.951i)T \) |
| 83 | \( 1 + (-0.951 - 0.309i)T \) |
| 89 | \( 1 + (0.809 + 0.587i)T \) |
| 97 | \( 1 + (0.951 - 0.309i)T \) |
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
−25.414906403460189950549584729070, −24.85574088336645154384490945881, −23.69377768446475566246212169598, −23.08924133646821608039015481751, −22.70078796214034658054960695544, −21.397641235650180110196089399234, −20.675897890978609566850140699572, −19.21591561818781005499672129585, −17.9938645259655169913834517758, −17.42439655925094284398792983200, −16.44472776548095874330292100877, −15.90687871307018011478080597383, −14.57159841413481318601270289667, −13.59408189273528608182197368087, −12.84556252709930259088689569739, −11.88122680484844995225747849554, −10.83064099600502530843315675807, −9.725621767503116288088351816376, −7.96079033722463647641426065842, −7.42394506730216540778855376375, −6.25428666225254407514105052177, −5.477000210100481069489225155391, −4.377350151430047906651198378650, −3.25915666879125707042796474944, −1.053789919500835836477975205522,
1.140538205640531771963812389541, 2.60791576325726221259873365855, 3.942986358206453647908769982723, 5.00658171331710240863718960018, 5.82685142496476759152096318067, 6.77476838196457211766843506652, 8.77600897969363193770589916201, 9.643506850653920569596141612450, 10.73647362876799248578786017861, 11.549477386715309389396935034655, 12.25202606367280128014146547432, 13.12699665580437026311976656866, 14.41844083344911668073836594252, 15.35164998224199210528042434628, 16.20196044380470928243569234645, 17.43158978312517863852053212946, 18.59212318140821802964378538859, 18.92142099373267876895780319226, 20.45110858134068685246992435515, 21.27275245107130098218929721776, 21.933991253877543820869782533022, 22.629472608505384066431570686216, 23.664704056465029382610692032557, 24.19892685844702184893573121219, 25.50712401579202053485021906614