L(s) = 1 | + (0.866 + 0.5i)7-s + (0.5 + 0.866i)11-s + (−0.984 − 0.173i)13-s + (−0.342 + 0.939i)17-s + (0.642 − 0.766i)23-s + (−0.939 + 0.342i)29-s + (−0.5 + 0.866i)31-s − i·37-s + (0.173 + 0.984i)41-s + (−0.642 − 0.766i)43-s + (−0.342 − 0.939i)47-s + (0.5 + 0.866i)49-s + (−0.642 + 0.766i)53-s + (0.939 + 0.342i)59-s + (−0.766 − 0.642i)61-s + ⋯ |
L(s) = 1 | + (0.866 + 0.5i)7-s + (0.5 + 0.866i)11-s + (−0.984 − 0.173i)13-s + (−0.342 + 0.939i)17-s + (0.642 − 0.766i)23-s + (−0.939 + 0.342i)29-s + (−0.5 + 0.866i)31-s − i·37-s + (0.173 + 0.984i)41-s + (−0.642 − 0.766i)43-s + (−0.342 − 0.939i)47-s + (0.5 + 0.866i)49-s + (−0.642 + 0.766i)53-s + (0.939 + 0.342i)59-s + (−0.766 − 0.642i)61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2280 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.505 + 0.862i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2280 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.505 + 0.862i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.6085409368 + 1.062359132i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6085409368 + 1.062359132i\) |
\(L(1)\) |
\(\approx\) |
\(0.9996016536 + 0.2678134161i\) |
\(L(1)\) |
\(\approx\) |
\(0.9996016536 + 0.2678134161i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
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 | \( 1 + (-0.984 - 0.173i)T \) |
| 17 | \( 1 + (-0.342 + 0.939i)T \) |
| 23 | \( 1 + (0.642 - 0.766i)T \) |
| 29 | \( 1 + (-0.939 + 0.342i)T \) |
| 31 | \( 1 + (-0.5 + 0.866i)T \) |
| 37 | \( 1 - iT \) |
| 41 | \( 1 + (0.173 + 0.984i)T \) |
| 43 | \( 1 + (-0.642 - 0.766i)T \) |
| 47 | \( 1 + (-0.342 - 0.939i)T \) |
| 53 | \( 1 + (-0.642 + 0.766i)T \) |
| 59 | \( 1 + (0.939 + 0.342i)T \) |
| 61 | \( 1 + (-0.766 - 0.642i)T \) |
| 67 | \( 1 + (0.342 + 0.939i)T \) |
| 71 | \( 1 + (-0.766 + 0.642i)T \) |
| 73 | \( 1 + (0.984 - 0.173i)T \) |
| 79 | \( 1 + (-0.173 - 0.984i)T \) |
| 83 | \( 1 + (-0.866 - 0.5i)T \) |
| 89 | \( 1 + (-0.173 + 0.984i)T \) |
| 97 | \( 1 + (-0.342 + 0.939i)T \) |
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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
−19.484120947114017190337465557102, −18.723354839593343433380348814493, −17.917081514929106952724124685610, −17.197186360858452121986094937820, −16.70971664019944984753447592562, −15.878022638337015239872188849546, −14.92732470802135870727041253602, −14.37992767362364632538144609234, −13.717059512295719614173265351044, −12.99893398735302718226940278224, −11.9833577865314434069349632585, −11.246750472237189076235915716300, −10.95457898488191451972235580736, −9.66755316576659496319354794658, −9.24632612195013117600575585276, −8.236545964225876595717156048704, −7.47402955733447711512067737933, −6.930736934960445264893566171960, −5.78957487796922679319937913810, −5.086009927407195146803719704940, −4.2764284716282449527621948072, −3.44941508203396483009194495016, −2.40499084813861077725126875562, −1.50675946544778245029079356532, −0.38859536321133494323110661423,
1.39918348770257097753918917672, 2.038072962401010987822332434325, 2.98240058522745322533985369310, 4.13279874721176592024885582468, 4.83759308772780374094256397905, 5.47440596599808024864871736003, 6.5897489357626611326168658433, 7.217477807387054415921946144622, 8.13327108093832160459468078844, 8.78607677697392179483795793716, 9.59405597720366532045218925380, 10.40911563012694585369017052236, 11.15632582201872491970343031811, 12.00325435323458303431625510624, 12.51401133135201751628618670260, 13.28663027275983387474935232187, 14.42758168961831311279830138466, 14.880016679950520338480369350005, 15.2526642397989811081292430672, 16.45127349094051552293910714868, 17.17751352781078995358913723991, 17.634401983445666263114329257680, 18.45660888724872009439235326231, 19.12075936489803933990700541838, 20.12129634431800637874596239811