L(s) = 1 | + (−0.5 − 0.866i)2-s + (−0.5 + 0.866i)4-s + 5-s + (−0.5 + 0.866i)7-s + 8-s + (−0.5 − 0.866i)10-s + (−0.5 + 0.866i)11-s + (−0.5 + 0.866i)13-s + 14-s + (−0.5 − 0.866i)16-s + (−0.5 + 0.866i)17-s + (−0.5 + 0.866i)20-s + 22-s + (−0.5 + 0.866i)23-s + 25-s + 26-s + ⋯ |
L(s) = 1 | + (−0.5 − 0.866i)2-s + (−0.5 + 0.866i)4-s + 5-s + (−0.5 + 0.866i)7-s + 8-s + (−0.5 − 0.866i)10-s + (−0.5 + 0.866i)11-s + (−0.5 + 0.866i)13-s + 14-s + (−0.5 − 0.866i)16-s + (−0.5 + 0.866i)17-s + (−0.5 + 0.866i)20-s + 22-s + (−0.5 + 0.866i)23-s + 25-s + 26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 171 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.884 + 0.466i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 171 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.884 + 0.466i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.7940758964 + 0.1965463349i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7940758964 + 0.1965463349i\) |
\(L(1)\) |
\(\approx\) |
\(0.8247709625 - 0.04178634678i\) |
\(L(1)\) |
\(\approx\) |
\(0.8247709625 - 0.04178634678i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 19 | \( 1 \) |
good | 2 | \( 1 + (-0.5 - 0.866i)T \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 + (-0.5 + 0.866i)T \) |
| 11 | \( 1 + (-0.5 + 0.866i)T \) |
| 13 | \( 1 + (-0.5 + 0.866i)T \) |
| 17 | \( 1 + (-0.5 + 0.866i)T \) |
| 23 | \( 1 + (-0.5 + 0.866i)T \) |
| 29 | \( 1 + T \) |
| 31 | \( 1 + (-0.5 - 0.866i)T \) |
| 37 | \( 1 + T \) |
| 41 | \( 1 + T \) |
| 43 | \( 1 + (-0.5 - 0.866i)T \) |
| 47 | \( 1 + T \) |
| 53 | \( 1 + (-0.5 - 0.866i)T \) |
| 59 | \( 1 + T \) |
| 61 | \( 1 + T \) |
| 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 | \( 1 + (-0.5 - 0.866i)T \) |
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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
−26.97101664464088163985947143696, −26.61239123030564162168660465632, −25.47169893869266802509519244615, −24.80148771767372009641423247610, −23.80775229510852654928940981989, −22.76407597668567460331942353834, −21.89585914184838645336157093853, −20.47850789799266570458328401372, −19.56738813026676646919899960780, −18.302146628800390729684599278047, −17.64474391573910030392606457173, −16.59004289544250631477564934951, −15.93074573200182314845706242482, −14.476497331192118437691588347682, −13.71459365789445855044941928214, −12.877129954201354049937606167042, −10.77802092090153048184256355619, −10.11883550199837155387312717516, −9.09612157187156911214603250459, −7.8785311206235006517795624729, −6.73852261284606628573727087091, −5.80745176077873103133310246204, −4.66937375262808915568822359522, −2.74796530842060482906118190553, −0.799099629488144993301257873812,
1.87543394674714542032188569319, 2.58309455923726472989835628030, 4.27639722629058417679730363533, 5.662985195152691382104514477394, 7.05196273820200281534881683886, 8.55167555783675412621677347975, 9.54832741843866734930824260963, 10.121773114017500001852999854491, 11.50759208019432722978113047820, 12.584316355547443386871300712013, 13.27206222382788873727296818366, 14.57194846329136322060737937643, 15.95374793213344535459356856640, 17.18308489236734060693694619475, 17.90375795595758315355087427682, 18.82658108401116663418617321511, 19.76772997629231798755534678206, 20.89058544647307116467604307520, 21.771958035662910799645697374130, 22.23542049117036050103902333157, 23.692555305737783243704719540133, 25.160770711172915295147022275568, 25.782947493610688622509462541053, 26.58454414427503955566749948045, 27.97161115028109986427147664770