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 + 8-s + (−0.5 − 0.866i)9-s + (0.5 − 0.866i)10-s + (−0.5 + 0.866i)11-s + (0.5 − 0.866i)12-s + (−0.5 − 0.866i)15-s + 16-s − 17-s + (−0.5 − 0.866i)18-s + (0.5 + 0.866i)19-s + (0.5 − 0.866i)20-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 + 8-s + (−0.5 − 0.866i)9-s + (0.5 − 0.866i)10-s + (−0.5 + 0.866i)11-s + (0.5 − 0.866i)12-s + (−0.5 − 0.866i)15-s + 16-s − 17-s + (−0.5 − 0.866i)18-s + (0.5 + 0.866i)19-s + (0.5 − 0.866i)20-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.372 - 0.927i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.372 - 0.927i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(3.185169817 - 2.152677575i\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.185169817 - 2.152677575i\) |
\(L(1)\) |
\(\approx\) |
\(2.201849028 - 0.8897386131i\) |
\(L(1)\) |
\(\approx\) |
\(2.201849028 - 0.8897386131i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 \) |
| 13 | \( 1 \) |
good | 2 | \( 1 + T \) |
| 3 | \( 1 + (0.5 - 0.866i)T \) |
| 5 | \( 1 + (0.5 - 0.866i)T \) |
| 11 | \( 1 + (-0.5 + 0.866i)T \) |
| 17 | \( 1 - T \) |
| 19 | \( 1 + (0.5 + 0.866i)T \) |
| 23 | \( 1 + T \) |
| 29 | \( 1 + (-0.5 - 0.866i)T \) |
| 31 | \( 1 + (0.5 + 0.866i)T \) |
| 37 | \( 1 + T \) |
| 41 | \( 1 + (0.5 + 0.866i)T \) |
| 43 | \( 1 + (-0.5 + 0.866i)T \) |
| 47 | \( 1 + (0.5 - 0.866i)T \) |
| 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 | \( 1 - T \) |
| 89 | \( 1 - T \) |
| 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
−30.64398159855833291761241882882, −29.4729312121865532616338879916, −28.529415066925859400314164943802, −26.864447276891699556983964372963, −26.12519653283424559777193100263, −25.13816280095355668222057229669, −23.97185255193866820158581324015, −22.53444257613548027950542883059, −21.924189974655709448110963742645, −21.08862375806505150856754290570, −20.01433102596633927265335909602, −18.811102248824328253918665568039, −17.09901653871938785204110978337, −15.796881003737518329710841990615, −15.04980143752467422099538047690, −13.94594931374376557755212672838, −13.24596072516263036371477128258, −11.21024330242407472520948579449, −10.683656621625631311265756363557, −9.17796425479932213727581884040, −7.47131727576953797727931400384, −6.044245047082554872105438471456, −4.82289267221164018911124261548, −3.33940627049138566889065664762, −2.4517720946516190585517525120,
1.45192039226743102128059683980, 2.65973404623887319990438868594, 4.43968367265399756029011062582, 5.76150655215480074085753722798, 7.02488342849972649888469566433, 8.25471272790176430742713164309, 9.784429838982925856811192568208, 11.58554123658822047014260413099, 12.761119515408858404935414234993, 13.24098635216284888275285419524, 14.408110051760002344323523894415, 15.54507744765648408278361935759, 16.93286383364867861598435199314, 18.10359443200615760068803225816, 19.65083428305208075228529234089, 20.46660400366812620087533464274, 21.25131893580809716462670937615, 22.79389418205148003838985170998, 23.67071212076722136881394906350, 24.80107067538935635709829793928, 25.14348193379650195796396479640, 26.435751059244444170344574875881, 28.512329725298044463014254907321, 29.00836885419353172956241744854, 30.14323409487851962605475455765