L(s) = 1 | + (−0.951 − 0.309i)2-s + (0.707 + 0.707i)3-s + (0.809 + 0.587i)4-s + (0.587 − 0.809i)5-s + (−0.453 − 0.891i)6-s + (−0.453 + 0.891i)7-s + (−0.587 − 0.809i)8-s + i·9-s + (−0.809 + 0.587i)10-s + (0.987 + 0.156i)11-s + (0.156 + 0.987i)12-s + (0.891 − 0.453i)13-s + (0.707 − 0.707i)14-s + (0.987 − 0.156i)15-s + (0.309 + 0.951i)16-s + (−0.156 + 0.987i)17-s + ⋯ |
L(s) = 1 | + (−0.951 − 0.309i)2-s + (0.707 + 0.707i)3-s + (0.809 + 0.587i)4-s + (0.587 − 0.809i)5-s + (−0.453 − 0.891i)6-s + (−0.453 + 0.891i)7-s + (−0.587 − 0.809i)8-s + i·9-s + (−0.809 + 0.587i)10-s + (0.987 + 0.156i)11-s + (0.156 + 0.987i)12-s + (0.891 − 0.453i)13-s + (0.707 − 0.707i)14-s + (0.987 − 0.156i)15-s + (0.309 + 0.951i)16-s + (−0.156 + 0.987i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 41 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.848 + 0.529i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 41 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.848 + 0.529i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.290662765 + 0.3701519624i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.290662765 + 0.3701519624i\) |
\(L(1)\) |
\(\approx\) |
\(1.012857173 + 0.1422301524i\) |
\(L(1)\) |
\(\approx\) |
\(1.012857173 + 0.1422301524i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 41 | \( 1 \) |
good | 2 | \( 1 + (-0.951 - 0.309i)T \) |
| 3 | \( 1 + (0.707 + 0.707i)T \) |
| 5 | \( 1 + (0.587 - 0.809i)T \) |
| 7 | \( 1 + (-0.453 + 0.891i)T \) |
| 11 | \( 1 + (0.987 + 0.156i)T \) |
| 13 | \( 1 + (0.891 - 0.453i)T \) |
| 17 | \( 1 + (-0.156 + 0.987i)T \) |
| 19 | \( 1 + (0.891 + 0.453i)T \) |
| 23 | \( 1 + (-0.309 + 0.951i)T \) |
| 29 | \( 1 + (-0.156 - 0.987i)T \) |
| 31 | \( 1 + (0.809 - 0.587i)T \) |
| 37 | \( 1 + (-0.809 - 0.587i)T \) |
| 43 | \( 1 + (-0.951 - 0.309i)T \) |
| 47 | \( 1 + (-0.453 - 0.891i)T \) |
| 53 | \( 1 + (0.156 + 0.987i)T \) |
| 59 | \( 1 + (0.309 - 0.951i)T \) |
| 61 | \( 1 + (0.951 - 0.309i)T \) |
| 67 | \( 1 + (-0.987 + 0.156i)T \) |
| 71 | \( 1 + (-0.987 - 0.156i)T \) |
| 73 | \( 1 - iT \) |
| 79 | \( 1 + (-0.707 - 0.707i)T \) |
| 83 | \( 1 + T \) |
| 89 | \( 1 + (-0.453 + 0.891i)T \) |
| 97 | \( 1 + (0.987 - 0.156i)T \) |
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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
−34.88209541341057386016046961266, −33.406744505503294891739428832817, −32.57043249536803759301541043769, −30.58271310939852356773442304456, −29.765372238410249712060042020826, −28.8382248671874333482123579156, −26.95811455858225637514130320366, −26.160742140059396429139163590662, −25.2936763501737332318363206736, −24.16747329122293174061373553719, −22.72659398109181290088627582376, −20.702541779249651629529360199696, −19.66051911547850862767150655000, −18.57471289465196283960179581962, −17.64592451811580798214800780017, −16.182075520493339104463611001855, −14.46010629549467334223453887126, −13.67017727872615654664489330563, −11.511014200418459766631518851960, −9.97954939783475929784352693553, −8.820203933389740198737255571132, −7.09219651136961143817733150443, −6.48646759412420371961568334669, −3.09815583481187596222012844934, −1.26894083228531502524611001975,
1.80188429269940173261359849970, 3.61032993913576862797806004036, 5.92404118713321510093341496873, 8.24100364395326289344355177591, 9.16316079866680465358531085353, 10.053922552048582697745725065856, 11.86518001632856217848663568366, 13.36314981160954994677077047186, 15.28297206918018735515132071371, 16.29019247424876725496171629331, 17.528310374078353587244749186453, 19.09390058413052254190522190904, 20.14887540479150465917403652356, 21.15512074413777948663577058459, 22.13715886980792255520170861309, 24.76910259127585905367857608026, 25.30722625520054629005852998599, 26.410109986367595022836196426512, 27.970053666244520938101695922133, 28.26539694758653802301110025990, 29.90094082546060090528705390120, 31.22358903506587516317489531311, 32.54318920974087497670573418962, 33.432371681972833646364191817376, 35.16953826672849825713142245023