L(s) = 1 | + (−0.866 − 0.5i)7-s + (−0.5 − 0.866i)11-s + (−0.642 + 0.766i)13-s + (0.984 − 0.173i)17-s + (−0.342 − 0.939i)23-s + (−0.173 + 0.984i)29-s + (−0.5 + 0.866i)31-s − i·37-s + (−0.766 + 0.642i)41-s + (0.342 − 0.939i)43-s + (−0.984 − 0.173i)47-s + (0.5 + 0.866i)49-s + (0.342 + 0.939i)53-s + (−0.173 − 0.984i)59-s + (0.939 − 0.342i)61-s + ⋯ |
L(s) = 1 | + (−0.866 − 0.5i)7-s + (−0.5 − 0.866i)11-s + (−0.642 + 0.766i)13-s + (0.984 − 0.173i)17-s + (−0.342 − 0.939i)23-s + (−0.173 + 0.984i)29-s + (−0.5 + 0.866i)31-s − i·37-s + (−0.766 + 0.642i)41-s + (0.342 − 0.939i)43-s + (−0.984 − 0.173i)47-s + (0.5 + 0.866i)49-s + (0.342 + 0.939i)53-s + (−0.173 − 0.984i)59-s + (0.939 − 0.342i)61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2280 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.384 + 0.923i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2280 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.384 + 0.923i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.2398827381 + 0.3595915865i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.2398827381 + 0.3595915865i\) |
\(L(1)\) |
\(\approx\) |
\(0.7771932031 + 0.02513134759i\) |
\(L(1)\) |
\(\approx\) |
\(0.7771932031 + 0.02513134759i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 19 | \( 1 \) |
good | 7 | \( 1 + (-0.866 - 0.5i)T \) |
| 11 | \( 1 + (-0.5 - 0.866i)T \) |
| 13 | \( 1 + (-0.642 + 0.766i)T \) |
| 17 | \( 1 + (0.984 - 0.173i)T \) |
| 23 | \( 1 + (-0.342 - 0.939i)T \) |
| 29 | \( 1 + (-0.173 + 0.984i)T \) |
| 31 | \( 1 + (-0.5 + 0.866i)T \) |
| 37 | \( 1 - iT \) |
| 41 | \( 1 + (-0.766 + 0.642i)T \) |
| 43 | \( 1 + (0.342 - 0.939i)T \) |
| 47 | \( 1 + (-0.984 - 0.173i)T \) |
| 53 | \( 1 + (0.342 + 0.939i)T \) |
| 59 | \( 1 + (-0.173 - 0.984i)T \) |
| 61 | \( 1 + (0.939 - 0.342i)T \) |
| 67 | \( 1 + (-0.984 - 0.173i)T \) |
| 71 | \( 1 + (0.939 + 0.342i)T \) |
| 73 | \( 1 + (-0.642 - 0.766i)T \) |
| 79 | \( 1 + (-0.766 + 0.642i)T \) |
| 83 | \( 1 + (0.866 + 0.5i)T \) |
| 89 | \( 1 + (0.766 + 0.642i)T \) |
| 97 | \( 1 + (-0.984 + 0.173i)T \) |
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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
−19.37948485256181309759875084566, −18.80536699387854156066327445454, −17.98041882106243928152988080151, −17.33224373095365979981462574933, −16.5424487411880244875270151554, −15.76584455140781498286145597104, −15.11187057291605250221469497179, −14.626116300012772445693834846312, −13.33939257761097720662888331411, −13.01966524168876565210402944484, −12.12736612088444232471286370089, −11.65408197728414733267389325822, −10.36407697654966730311665119173, −9.884120420602080829549560478412, −9.36754070393992974179580961152, −8.16304053979869625350326869583, −7.62110171574698973113328136406, −6.779790658348921596831980395986, −5.77210855083242110069942673856, −5.31679357522464513691207931111, −4.235460744117870851406735807904, −3.2689557831977201504392159124, −2.58954972902310406548943618793, −1.62558208589678424687434956623, −0.15664255944933712926760847462,
1.00687168453239907126770677384, 2.22855394640200237893867093343, 3.18001560407684056262592590252, 3.76800729535668110381846385148, 4.86022653623197948540972753354, 5.61684841229443529475633494483, 6.551939580029844871157833440790, 7.14932181285243894760028347050, 8.0114312164464796950096233979, 8.89567442937929777195092764293, 9.62143597421389314487007623073, 10.403535849139139066170453306443, 10.95778600113626127459412642131, 12.05341712210580151118588562733, 12.56737815378141570233124293944, 13.40145363178553739865920170829, 14.148735180976702855940537381959, 14.636444068405004477027364713837, 15.7975336243736323286820092824, 16.50689675315746065834409320086, 16.63221854252170129386252060932, 17.76290939744546298588662224312, 18.66735594686446162253772187933, 19.05103602231279140259417170556, 19.87462230706024714746223343278