L(s) = 1 | + (0.707 + 0.707i)3-s + (0.707 − 0.707i)5-s + i·7-s + i·9-s + (0.707 − 0.707i)11-s + (0.707 + 0.707i)13-s + 15-s − 17-s + (−0.707 − 0.707i)19-s + (−0.707 + 0.707i)21-s − i·23-s − i·25-s + (−0.707 + 0.707i)27-s + (−0.707 − 0.707i)29-s − 31-s + ⋯ |
L(s) = 1 | + (0.707 + 0.707i)3-s + (0.707 − 0.707i)5-s + i·7-s + i·9-s + (0.707 − 0.707i)11-s + (0.707 + 0.707i)13-s + 15-s − 17-s + (−0.707 − 0.707i)19-s + (−0.707 + 0.707i)21-s − i·23-s − i·25-s + (−0.707 + 0.707i)27-s + (−0.707 − 0.707i)29-s − 31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 32 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.831 + 0.555i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 32 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.831 + 0.555i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.808198019 + 0.5485108726i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.808198019 + 0.5485108726i\) |
\(L(1)\) |
\(\approx\) |
\(1.423341664 + 0.2831202600i\) |
\(L(1)\) |
\(\approx\) |
\(1.423341664 + 0.2831202600i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
good | 3 | \( 1 + (0.707 + 0.707i)T \) |
| 5 | \( 1 + (0.707 - 0.707i)T \) |
| 7 | \( 1 + iT \) |
| 11 | \( 1 + (0.707 - 0.707i)T \) |
| 13 | \( 1 + (0.707 + 0.707i)T \) |
| 17 | \( 1 - T \) |
| 19 | \( 1 + (-0.707 - 0.707i)T \) |
| 23 | \( 1 - iT \) |
| 29 | \( 1 + (-0.707 - 0.707i)T \) |
| 31 | \( 1 - T \) |
| 37 | \( 1 + (0.707 - 0.707i)T \) |
| 41 | \( 1 + iT \) |
| 43 | \( 1 + (0.707 - 0.707i)T \) |
| 47 | \( 1 + T \) |
| 53 | \( 1 + (-0.707 + 0.707i)T \) |
| 59 | \( 1 + (-0.707 + 0.707i)T \) |
| 61 | \( 1 + (-0.707 - 0.707i)T \) |
| 67 | \( 1 + (0.707 + 0.707i)T \) |
| 71 | \( 1 + iT \) |
| 73 | \( 1 + iT \) |
| 79 | \( 1 + T \) |
| 83 | \( 1 + (-0.707 - 0.707i)T \) |
| 89 | \( 1 - iT \) |
| 97 | \( 1 + T \) |
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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
−36.191648219598528379062050628373, −35.26702705750753035320270543181, −33.55058988425371830514389971908, −32.6668073754382393468188834826, −30.97942908874554956522548170408, −30.057899216613783309087949166571, −29.28612035597456373320287123621, −27.30201585136737243001613384566, −25.92210250981586179255059257165, −25.26204011432168976313088623105, −23.70308022984907141220013318767, −22.47027247730529576759225056746, −20.72569155984747047415154155781, −19.70766320514182973203812787312, −18.234247045831694711022588586676, −17.26187013802039644735136478376, −15.04019301381373658472390381402, −13.92789197325934873393141496216, −12.89527440790653569427679489166, −10.84912514697350119980684825974, −9.33658997625249447827700621129, −7.52319196293555974166849458647, −6.35900187234094306006619502021, −3.640656916871261635983272091893, −1.72096909693814813092436307440,
2.24335513554469438713146152642, 4.33888362925581802874904501208, 6.032042852509231921086069214510, 8.71610959747903301666137937837, 9.193735552677570762148667626089, 11.12231938416884050823641586706, 13.028204813562807670242051480390, 14.29611150719206453655236933967, 15.72094109915403842745920173932, 16.89273313282052864794273354121, 18.68296180803459633844611346848, 20.09281129587004903401079664786, 21.368882435449109982458532901573, 22.038890501729185440827949094937, 24.31756691525862003410131205072, 25.21821206629074999356042207656, 26.38109685862421843718963220524, 27.85937665149967744962289924955, 28.73683384048286617104121249941, 30.50889164782938318855510252930, 31.77901428437357431018605218581, 32.58925639882058419378699304750, 33.69864573992814775165743979532, 35.31581570894988921655710203111, 36.64985030842321296223373644535