L(s) = 1 | + (−0.766 + 0.642i)5-s + (0.939 − 0.342i)7-s + (0.766 + 0.642i)11-s + (0.173 + 0.984i)13-s + (0.5 + 0.866i)17-s + (0.5 − 0.866i)19-s + (−0.939 − 0.342i)23-s + (0.173 − 0.984i)25-s + (−0.173 + 0.984i)29-s + (0.939 + 0.342i)31-s + (−0.5 + 0.866i)35-s + (−0.5 − 0.866i)37-s + (−0.173 − 0.984i)41-s + (−0.766 − 0.642i)43-s + (−0.939 + 0.342i)47-s + ⋯ |
L(s) = 1 | + (−0.766 + 0.642i)5-s + (0.939 − 0.342i)7-s + (0.766 + 0.642i)11-s + (0.173 + 0.984i)13-s + (0.5 + 0.866i)17-s + (0.5 − 0.866i)19-s + (−0.939 − 0.342i)23-s + (0.173 − 0.984i)25-s + (−0.173 + 0.984i)29-s + (0.939 + 0.342i)31-s + (−0.5 + 0.866i)35-s + (−0.5 − 0.866i)37-s + (−0.173 − 0.984i)41-s + (−0.766 − 0.642i)43-s + (−0.939 + 0.342i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.835 + 0.549i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.835 + 0.549i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.9781863630 + 0.2928497729i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9781863630 + 0.2928497729i\) |
\(L(1)\) |
\(\approx\) |
\(1.025005483 + 0.1592095066i\) |
\(L(1)\) |
\(\approx\) |
\(1.025005483 + 0.1592095066i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + (-0.766 + 0.642i)T \) |
| 7 | \( 1 + (0.939 - 0.342i)T \) |
| 11 | \( 1 + (0.766 + 0.642i)T \) |
| 13 | \( 1 + (0.173 + 0.984i)T \) |
| 17 | \( 1 + (0.5 + 0.866i)T \) |
| 19 | \( 1 + (0.5 - 0.866i)T \) |
| 23 | \( 1 + (-0.939 - 0.342i)T \) |
| 29 | \( 1 + (-0.173 + 0.984i)T \) |
| 31 | \( 1 + (0.939 + 0.342i)T \) |
| 37 | \( 1 + (-0.5 - 0.866i)T \) |
| 41 | \( 1 + (-0.173 - 0.984i)T \) |
| 43 | \( 1 + (-0.766 - 0.642i)T \) |
| 47 | \( 1 + (-0.939 + 0.342i)T \) |
| 53 | \( 1 - T \) |
| 59 | \( 1 + (0.766 - 0.642i)T \) |
| 61 | \( 1 + (-0.939 + 0.342i)T \) |
| 67 | \( 1 + (-0.173 - 0.984i)T \) |
| 71 | \( 1 + (-0.5 - 0.866i)T \) |
| 73 | \( 1 + (-0.5 + 0.866i)T \) |
| 79 | \( 1 + (-0.173 + 0.984i)T \) |
| 83 | \( 1 + (0.173 - 0.984i)T \) |
| 89 | \( 1 + (0.5 - 0.866i)T \) |
| 97 | \( 1 + (0.766 + 0.642i)T \) |
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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
−29.62279171888400572099173645639, −28.13628295138077508893888006133, −27.54292939421717311246029719458, −26.74479581567678296068338620819, −24.9704376304836701847227387807, −24.559973745404836896183413159476, −23.361891040704692026545303762200, −22.34511957349440518819194234530, −20.9904292860498972934327451993, −20.25459518484623184732394144378, −19.08504268629091930721769525902, −17.97395724159464233640015625744, −16.77975290495532004701541605328, −15.76805559360681256388212120855, −14.68693284193660419381135588300, −13.49981968148518016116211094578, −11.969715349260240286350523334217, −11.515222848116827289893963218868, −9.84310932826649050421255709548, −8.39231639576046258993232151842, −7.78401711334032457969058111793, −5.88605527699139420812814896228, −4.6996792813107099634099913695, −3.32819997943692241393208766460, −1.249769475689516605608505998837,
1.760824527843204233887955414541, 3.6576536444983275868107787277, 4.69647296061458142007018171589, 6.57761289210904084081951771237, 7.55111390298205282364817430699, 8.77007212577008655132246458754, 10.325580970182756982295893078262, 11.40700028095423065824732017378, 12.22231507751378645721011780220, 14.04034171315031165829410442265, 14.674179593788947058137493079247, 15.858913714850679535663833203952, 17.15676916838782564071850111305, 18.15631018912069960544404195038, 19.31826599738831847881742790100, 20.19975500986039200667140076880, 21.46220996185484522384344531136, 22.49213361177083335262626924130, 23.62312323181905468157736124069, 24.27357030743845203575702864751, 25.803695909608090658834926606, 26.62519287666737866555622432874, 27.63174101446545766676872881189, 28.415131632699486066197216579806, 30.12070512759085710523731113850