L(s) = 1 | + 2-s + (−0.5 − 0.866i)3-s + 4-s + (−0.5 − 0.866i)5-s + (−0.5 − 0.866i)6-s + (−0.5 + 0.866i)7-s + 8-s + (−0.5 + 0.866i)9-s + (−0.5 − 0.866i)10-s + 11-s + (−0.5 − 0.866i)12-s + (−0.5 + 0.866i)13-s + (−0.5 + 0.866i)14-s + (−0.5 + 0.866i)15-s + 16-s + (−0.5 + 0.866i)17-s + ⋯ |
L(s) = 1 | + 2-s + (−0.5 − 0.866i)3-s + 4-s + (−0.5 − 0.866i)5-s + (−0.5 − 0.866i)6-s + (−0.5 + 0.866i)7-s + 8-s + (−0.5 + 0.866i)9-s + (−0.5 − 0.866i)10-s + 11-s + (−0.5 − 0.866i)12-s + (−0.5 + 0.866i)13-s + (−0.5 + 0.866i)14-s + (−0.5 + 0.866i)15-s + 16-s + (−0.5 + 0.866i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.736 - 0.675i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.736 - 0.675i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.062973404 - 0.4136849104i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.062973404 - 0.4136849104i\) |
\(L(1)\) |
\(\approx\) |
\(1.281795694 - 0.3423057538i\) |
\(L(1)\) |
\(\approx\) |
\(1.281795694 - 0.3423057538i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 43 | \( 1 \) |
good | 2 | \( 1 + T \) |
| 3 | \( 1 + (-0.5 - 0.866i)T \) |
| 5 | \( 1 + (-0.5 - 0.866i)T \) |
| 7 | \( 1 + (-0.5 + 0.866i)T \) |
| 11 | \( 1 + T \) |
| 13 | \( 1 + (-0.5 + 0.866i)T \) |
| 17 | \( 1 + (-0.5 + 0.866i)T \) |
| 19 | \( 1 + (-0.5 - 0.866i)T \) |
| 23 | \( 1 + (-0.5 - 0.866i)T \) |
| 29 | \( 1 + (-0.5 + 0.866i)T \) |
| 31 | \( 1 + (-0.5 - 0.866i)T \) |
| 37 | \( 1 + (-0.5 - 0.866i)T \) |
| 41 | \( 1 + T \) |
| 47 | \( 1 + T \) |
| 53 | \( 1 + (-0.5 - 0.866i)T \) |
| 59 | \( 1 + T \) |
| 61 | \( 1 + (-0.5 + 0.866i)T \) |
| 67 | \( 1 + (-0.5 - 0.866i)T \) |
| 71 | \( 1 + (-0.5 + 0.866i)T \) |
| 73 | \( 1 + (-0.5 + 0.866i)T \) |
| 79 | \( 1 + (-0.5 + 0.866i)T \) |
| 83 | \( 1 + (-0.5 - 0.866i)T \) |
| 89 | \( 1 + (-0.5 - 0.866i)T \) |
| 97 | \( 1 + T \) |
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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
−34.35123470531189341937925306703, −33.4637713257218731271959355403, −32.52541454916462303596608896802, −31.50609852754867947086485653936, −29.97627395655533468200699306087, −29.37771016219550144226377426941, −27.60630021380698140569717651773, −26.61154235954059879888363960619, −25.27212788247672027739344399431, −23.53722891701169444596997070874, −22.64000907782534067761048444312, −22.10511635612365401816307500844, −20.504326387867857747071701170656, −19.52794369926892615390476237257, −17.340269016533038665614393300016, −16.1119927419996294217674625963, −15.055023760673684597183648077760, −13.98033703190551588163770949528, −12.15769709370406409158855491779, −11.03135786332682597247967294348, −9.982617212940609794290545511925, −7.3036800779324419824399179562, −6.05798551481637015006045765322, −4.26946752696245929219109306146, −3.28094717712705603898194001140,
2.04207198603197446195261156079, 4.297617203734646786314182174756, 5.827306809281636210352851808801, 7.00734058174388255700433727443, 8.80248953995798203098224767936, 11.3101142053880267348294074073, 12.28507179147785800679626880372, 12.99190428180432611567132771767, 14.611452038236319763941221613116, 16.122120451661042992355717557479, 17.11687792680087581462770218955, 19.14803993896234736559989673004, 19.89264453192944600924673808152, 21.69959556284112712599834078935, 22.58471435844809456801338095961, 24.04514573055887535627535659919, 24.421364197778380215681194164244, 25.7092158583536843989574562930, 28.07162598900808192462247008189, 28.72033461081603112326729915245, 29.97767583802619608376648309752, 31.09649627625923147597892870936, 31.96130395439532110985251468270, 33.18101304050135073666329332679, 34.737198939494816839854463184393