L(s) = 1 | + (−0.342 − 0.939i)5-s + (−0.173 − 0.984i)7-s + (0.342 − 0.939i)11-s + (−0.642 + 0.766i)13-s + (0.5 − 0.866i)17-s + (0.866 − 0.5i)19-s + (0.173 − 0.984i)23-s + (−0.766 + 0.642i)25-s + (−0.642 − 0.766i)29-s + (0.173 − 0.984i)31-s + (−0.866 + 0.5i)35-s + (0.866 + 0.5i)37-s + (0.766 + 0.642i)41-s + (−0.342 + 0.939i)43-s + (−0.173 − 0.984i)47-s + ⋯ |
L(s) = 1 | + (−0.342 − 0.939i)5-s + (−0.173 − 0.984i)7-s + (0.342 − 0.939i)11-s + (−0.642 + 0.766i)13-s + (0.5 − 0.866i)17-s + (0.866 − 0.5i)19-s + (0.173 − 0.984i)23-s + (−0.766 + 0.642i)25-s + (−0.642 − 0.766i)29-s + (0.173 − 0.984i)31-s + (−0.866 + 0.5i)35-s + (0.866 + 0.5i)37-s + (0.766 + 0.642i)41-s + (−0.342 + 0.939i)43-s + (−0.173 − 0.984i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.999 + 0.0145i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.999 + 0.0145i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.008322708376 - 1.144434378i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.008322708376 - 1.144434378i\) |
\(L(1)\) |
\(\approx\) |
\(0.8193535601 - 0.4379966177i\) |
\(L(1)\) |
\(\approx\) |
\(0.8193535601 - 0.4379966177i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + (-0.342 - 0.939i)T \) |
| 7 | \( 1 + (-0.173 - 0.984i)T \) |
| 11 | \( 1 + (0.342 - 0.939i)T \) |
| 13 | \( 1 + (-0.642 + 0.766i)T \) |
| 17 | \( 1 + (0.5 - 0.866i)T \) |
| 19 | \( 1 + (0.866 - 0.5i)T \) |
| 23 | \( 1 + (0.173 - 0.984i)T \) |
| 29 | \( 1 + (-0.642 - 0.766i)T \) |
| 31 | \( 1 + (0.173 - 0.984i)T \) |
| 37 | \( 1 + (0.866 + 0.5i)T \) |
| 41 | \( 1 + (0.766 + 0.642i)T \) |
| 43 | \( 1 + (-0.342 + 0.939i)T \) |
| 47 | \( 1 + (-0.173 - 0.984i)T \) |
| 53 | \( 1 - iT \) |
| 59 | \( 1 + (-0.342 - 0.939i)T \) |
| 61 | \( 1 + (-0.984 + 0.173i)T \) |
| 67 | \( 1 + (-0.642 + 0.766i)T \) |
| 71 | \( 1 + (-0.5 + 0.866i)T \) |
| 73 | \( 1 + (0.5 + 0.866i)T \) |
| 79 | \( 1 + (0.766 - 0.642i)T \) |
| 83 | \( 1 + (-0.642 - 0.766i)T \) |
| 89 | \( 1 + (-0.5 - 0.866i)T \) |
| 97 | \( 1 + (-0.939 - 0.342i)T \) |
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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
−24.39965168121095500429514752525, −23.31320112033138803458664596710, −22.51349290831582726656600647604, −22.010347385443992489196013488215, −21.01473502585655957503045790754, −19.77511890454709510327260775619, −19.28729563908181179240226116433, −18.16390029920831045809421154677, −17.72625969087957508054822305531, −16.4421276345374286548889269584, −15.27923849149266437301635848198, −14.98168278658264631432802421320, −14.02084532031850558018807966824, −12.552760255488878263219502247998, −12.14147974201584503973738409828, −11.00504473373716831493165902854, −10.0394477167027005619255054251, −9.245414960198286053024297636322, −7.89557072240606297675542760754, −7.22296926432303540152582955547, −6.0506230683752553677063441319, −5.17041273768860769221491202347, −3.700321445871199235184168121960, −2.84873702866061542692608528985, −1.66803540326502751199141964732,
0.3352609007288726730021059775, 1.16426060759691485735740801812, 2.86650778015394311083002132922, 4.09858925948278877962143315379, 4.81082791860655366403725109188, 6.06524157007989340552035265004, 7.246640951855952821208566019915, 8.02437155750789891245774955464, 9.20725242803146047600694516110, 9.834137757403387322012490574410, 11.27636269640470435587442222512, 11.79312876973038699088896149746, 13.019511889600551315500549022950, 13.71098595979879409552923439073, 14.58233140145228868806637176278, 15.89672673745004748388489137852, 16.69922011129735134694110857027, 16.95721941622488371942792045291, 18.39974362466742675674010925384, 19.31053000823583557799692057676, 20.09401122235584094469173345119, 20.73971665559730546901744344390, 21.72416224811784932401414216505, 22.73541310975017736460372347162, 23.55769171281038995772729662611