L(s) = 1 | + (−0.963 + 0.266i)2-s + (−0.691 + 0.722i)3-s + (0.858 − 0.512i)4-s + (0.983 − 0.178i)5-s + (0.473 − 0.880i)6-s + (−0.691 + 0.722i)8-s + (−0.0448 − 0.998i)9-s + (−0.900 + 0.433i)10-s + (−0.222 + 0.974i)12-s + (−0.963 + 0.266i)13-s + (−0.550 + 0.834i)15-s + (0.473 − 0.880i)16-s + (−0.393 − 0.919i)17-s + (0.309 + 0.951i)18-s + (0.309 − 0.951i)19-s + (0.753 − 0.657i)20-s + ⋯ |
L(s) = 1 | + (−0.963 + 0.266i)2-s + (−0.691 + 0.722i)3-s + (0.858 − 0.512i)4-s + (0.983 − 0.178i)5-s + (0.473 − 0.880i)6-s + (−0.691 + 0.722i)8-s + (−0.0448 − 0.998i)9-s + (−0.900 + 0.433i)10-s + (−0.222 + 0.974i)12-s + (−0.963 + 0.266i)13-s + (−0.550 + 0.834i)15-s + (0.473 − 0.880i)16-s + (−0.393 − 0.919i)17-s + (0.309 + 0.951i)18-s + (0.309 − 0.951i)19-s + (0.753 − 0.657i)20-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 539 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0183 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 539 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0183 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.2597833982 - 0.2645958969i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.2597833982 - 0.2645958969i\) |
\(L(1)\) |
\(\approx\) |
\(0.5326980898 + 0.03888171439i\) |
\(L(1)\) |
\(\approx\) |
\(0.5326980898 + 0.03888171439i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 \) |
| 11 | \( 1 \) |
good | 2 | \( 1 + (-0.963 + 0.266i)T \) |
| 3 | \( 1 + (-0.691 + 0.722i)T \) |
| 5 | \( 1 + (0.983 - 0.178i)T \) |
| 13 | \( 1 + (-0.963 + 0.266i)T \) |
| 17 | \( 1 + (-0.393 - 0.919i)T \) |
| 19 | \( 1 + (0.309 - 0.951i)T \) |
| 23 | \( 1 + (-0.222 - 0.974i)T \) |
| 29 | \( 1 + (-0.995 + 0.0896i)T \) |
| 31 | \( 1 + (-0.809 - 0.587i)T \) |
| 37 | \( 1 + (-0.995 + 0.0896i)T \) |
| 41 | \( 1 + (-0.691 + 0.722i)T \) |
| 43 | \( 1 + (-0.900 + 0.433i)T \) |
| 47 | \( 1 + (-0.550 - 0.834i)T \) |
| 53 | \( 1 + (-0.393 + 0.919i)T \) |
| 59 | \( 1 + (-0.691 - 0.722i)T \) |
| 61 | \( 1 + (-0.393 - 0.919i)T \) |
| 67 | \( 1 + T \) |
| 71 | \( 1 + (0.753 + 0.657i)T \) |
| 73 | \( 1 + (-0.550 + 0.834i)T \) |
| 79 | \( 1 + (-0.809 - 0.587i)T \) |
| 83 | \( 1 + (-0.963 - 0.266i)T \) |
| 89 | \( 1 + (0.623 + 0.781i)T \) |
| 97 | \( 1 + (-0.809 - 0.587i)T \) |
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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
−23.98017317514794041282090673960, −22.57932392898230119710637194994, −21.93656685995138070243940102741, −21.15005904160179311351157404262, −20.04934632415957597939284989643, −19.2519453870640482814329892677, −18.45144227786267230777257714753, −17.73644668823612116093510424465, −17.14240518991540295300399463793, −16.546487337777741055743066147647, −15.304766103180148237649412228, −14.17068090788137600008445400141, −13.047060422164662514290633491672, −12.40948793339536990139833077214, −11.47221662923516387659140535667, −10.5040520615122632325861610843, −9.944903737738001755911372647493, −8.84578510109040851460026399228, −7.73108228492165915016007682163, −6.995579268855219866419046528478, −6.04506967123242851503692846463, −5.26461477977891635669180515045, −3.40461481263986740502951125933, −2.03112031308675924321119392490, −1.554363507401654736595659765192,
0.27469285854989556456614037522, 1.81982642668937617266820465405, 2.944293975094166634739857815, 4.76063926968110059857428420370, 5.38579427321299219486243245587, 6.47820229440798418783126369633, 7.14455191286183883823515323910, 8.65479125382455437932580905898, 9.49949680976424328627159362518, 9.897245170236596614442757974461, 10.93674611006258471029828542717, 11.65538513982527163336824095873, 12.75017882973340885888706213524, 14.096455903071125270146988060394, 14.94505487438231320669679717849, 15.83045095214694475835105245421, 16.778026606062158669857084786231, 17.1034713250951011826503241625, 18.063625085094757435233561536882, 18.61137966368178474634668309136, 20.134725866420064253909677015, 20.46866080600910747064948436957, 21.65102368624242399668291835092, 22.12422830998362575440332950334, 23.26823673867083823290040947045