L(s) = 1 | + (−0.973 + 0.227i)3-s + (0.812 + 0.582i)5-s + (0.896 − 0.442i)9-s + (0.0327 + 0.999i)11-s + (0.995 − 0.0980i)13-s + (−0.923 − 0.382i)15-s + (−0.793 − 0.608i)17-s + (0.935 − 0.352i)19-s + (0.946 + 0.321i)23-s + (0.321 + 0.946i)25-s + (−0.773 + 0.634i)27-s + (−0.471 − 0.881i)29-s + (−0.258 + 0.965i)31-s + (−0.258 − 0.965i)33-s + (0.986 − 0.162i)37-s + ⋯ |
L(s) = 1 | + (−0.973 + 0.227i)3-s + (0.812 + 0.582i)5-s + (0.896 − 0.442i)9-s + (0.0327 + 0.999i)11-s + (0.995 − 0.0980i)13-s + (−0.923 − 0.382i)15-s + (−0.793 − 0.608i)17-s + (0.935 − 0.352i)19-s + (0.946 + 0.321i)23-s + (0.321 + 0.946i)25-s + (−0.773 + 0.634i)27-s + (−0.471 − 0.881i)29-s + (−0.258 + 0.965i)31-s + (−0.258 − 0.965i)33-s + (0.986 − 0.162i)37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.643 + 0.765i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.643 + 0.765i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(2.049662851 + 0.9539996121i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.049662851 + 0.9539996121i\) |
\(L(1)\) |
\(\approx\) |
\(1.051223419 + 0.2356387358i\) |
\(L(1)\) |
\(\approx\) |
\(1.051223419 + 0.2356387358i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
good | 3 | \( 1 + (-0.973 + 0.227i)T \) |
| 5 | \( 1 + (0.812 + 0.582i)T \) |
| 11 | \( 1 + (0.0327 + 0.999i)T \) |
| 13 | \( 1 + (0.995 - 0.0980i)T \) |
| 17 | \( 1 + (-0.793 - 0.608i)T \) |
| 19 | \( 1 + (0.935 - 0.352i)T \) |
| 23 | \( 1 + (0.946 + 0.321i)T \) |
| 29 | \( 1 + (-0.471 - 0.881i)T \) |
| 31 | \( 1 + (-0.258 + 0.965i)T \) |
| 37 | \( 1 + (0.986 - 0.162i)T \) |
| 41 | \( 1 + (0.980 - 0.195i)T \) |
| 43 | \( 1 + (0.290 + 0.956i)T \) |
| 47 | \( 1 + (-0.991 - 0.130i)T \) |
| 53 | \( 1 + (0.999 - 0.0327i)T \) |
| 59 | \( 1 + (-0.412 - 0.910i)T \) |
| 61 | \( 1 + (0.729 - 0.683i)T \) |
| 67 | \( 1 + (0.227 + 0.973i)T \) |
| 71 | \( 1 + (0.831 - 0.555i)T \) |
| 73 | \( 1 + (-0.997 - 0.0654i)T \) |
| 79 | \( 1 + (0.608 + 0.793i)T \) |
| 83 | \( 1 + (0.634 - 0.773i)T \) |
| 89 | \( 1 + (0.751 + 0.659i)T \) |
| 97 | \( 1 + (0.707 + 0.707i)T \) |
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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
−19.975511043430828374833627443185, −18.94494981254195731565945120058, −18.297767021463930371356451449261, −17.780496929390273314045923221727, −16.814423216324848955466333412306, −16.49528777522556826089069453255, −15.79282717897788774334028898836, −14.71365917565230449389845016516, −13.631734021755058768736116209260, −13.23161457705850270787113972217, −12.59134831283339710196659397009, −11.53690764100780814978875078861, −11.00642472285508647934760252966, −10.29876345860254138165959689749, −9.27065058356943672122165891006, −8.670971717575539026481501584495, −7.66769314997025469113488317554, −6.60380772885648156613615389111, −5.9565654784421366027575414527, −5.46474284594338590861984073371, −4.52065707482381637824218512953, −3.5773057169411392204721584169, −2.27310405275331493954330219068, −1.24802360822138024004820336740, −0.690852356081347225877572407873,
0.764003839471029914693362319472, 1.6862266725350661263884961169, 2.72959667921030730319855662495, 3.79237299461416017964629073175, 4.80545190931765557160135726223, 5.43518065513176619630105904259, 6.33856169036329262563369864011, 6.891317396956169537471520563884, 7.6684101579116945147604602352, 9.19119652468523329353015184991, 9.548329756955842101889759173122, 10.43022394573304518971081492161, 11.1861057954252405261703941610, 11.598750647288318645604782278287, 12.852255066346546764787480190688, 13.22404496187471685729156708946, 14.20263916246154461904271483355, 15.09086321445079857842532207843, 15.73526798297190625036699484251, 16.457188384902980903092089973223, 17.4123199309200668502558843961, 17.93983739531747239544913983767, 18.23276477590952718298602332623, 19.25241566663011215963966470081, 20.34849101153507725157612705358