L(s) = 1 | + (−0.988 − 0.149i)2-s + (0.955 + 0.294i)4-s + (−0.900 − 0.433i)5-s + (−0.900 − 0.433i)8-s + (0.826 + 0.563i)10-s + (0.623 − 0.781i)11-s + (−0.988 − 0.149i)13-s + (0.826 + 0.563i)16-s + (0.955 − 0.294i)17-s + (−0.5 + 0.866i)19-s + (−0.733 − 0.680i)20-s + (−0.733 + 0.680i)22-s + (−0.222 − 0.974i)23-s + (0.623 + 0.781i)25-s + (0.955 + 0.294i)26-s + ⋯ |
L(s) = 1 | + (−0.988 − 0.149i)2-s + (0.955 + 0.294i)4-s + (−0.900 − 0.433i)5-s + (−0.900 − 0.433i)8-s + (0.826 + 0.563i)10-s + (0.623 − 0.781i)11-s + (−0.988 − 0.149i)13-s + (0.826 + 0.563i)16-s + (0.955 − 0.294i)17-s + (−0.5 + 0.866i)19-s + (−0.733 − 0.680i)20-s + (−0.733 + 0.680i)22-s + (−0.222 − 0.974i)23-s + (0.623 + 0.781i)25-s + (0.955 + 0.294i)26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.944 - 0.328i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.944 - 0.328i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.04745373722 - 0.2808066450i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.04745373722 - 0.2808066450i\) |
\(L(1)\) |
\(\approx\) |
\(0.4793672100 - 0.1414886859i\) |
\(L(1)\) |
\(\approx\) |
\(0.4793672100 - 0.1414886859i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 \) |
good | 2 | \( 1 + (-0.988 - 0.149i)T \) |
| 5 | \( 1 + (-0.900 - 0.433i)T \) |
| 11 | \( 1 + (0.623 - 0.781i)T \) |
| 13 | \( 1 + (-0.988 - 0.149i)T \) |
| 17 | \( 1 + (0.955 - 0.294i)T \) |
| 19 | \( 1 + (-0.5 + 0.866i)T \) |
| 23 | \( 1 + (-0.222 - 0.974i)T \) |
| 29 | \( 1 + (-0.733 - 0.680i)T \) |
| 31 | \( 1 + (-0.5 + 0.866i)T \) |
| 37 | \( 1 + (-0.733 - 0.680i)T \) |
| 41 | \( 1 + (0.0747 + 0.997i)T \) |
| 43 | \( 1 + (0.0747 - 0.997i)T \) |
| 47 | \( 1 + (-0.988 - 0.149i)T \) |
| 53 | \( 1 + (-0.733 + 0.680i)T \) |
| 59 | \( 1 + (0.0747 - 0.997i)T \) |
| 61 | \( 1 + (0.955 - 0.294i)T \) |
| 67 | \( 1 + (-0.5 + 0.866i)T \) |
| 71 | \( 1 + (-0.222 - 0.974i)T \) |
| 73 | \( 1 + (0.365 - 0.930i)T \) |
| 79 | \( 1 + (-0.5 - 0.866i)T \) |
| 83 | \( 1 + (-0.988 + 0.149i)T \) |
| 89 | \( 1 + (-0.988 + 0.149i)T \) |
| 97 | \( 1 + (-0.5 + 0.866i)T \) |
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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
−24.35760643228993523221782284334, −23.86605648353064851749288893398, −22.81473318849430039705114401067, −21.883779946292572194069449324865, −20.74423284196152302585996297119, −19.72805711676113773049760933261, −19.403609029052873289810553660351, −18.47719424957170675646578781823, −17.471759022665749604743579313023, −16.827719232594079790167632444905, −15.7957035778888622372166652570, −14.94586919054153881833653638651, −14.47501285253101099544680599777, −12.68901429423241352488416442951, −11.83475882594341428211833531399, −11.15283489574639719999021953476, −10.04837141705778431348841872732, −9.32862638148061156133185327166, −8.17007130484121293125444416270, −7.31668976401916484947857298870, −6.766212325978196415178353245313, −5.35688238122113427702908940281, −3.98771631239866435477099998805, −2.79484078170745703218910835128, −1.55259665310733025358229632797,
0.226478204665773083664332328433, 1.57795649631299633453736377182, 3.03742221114446672215858288217, 3.986828271384747757132914505498, 5.444790467485266746530466438080, 6.67190772744381331767052887980, 7.687565785447277092213182432541, 8.350242944332857498347307592922, 9.268687243522395488928862123572, 10.256698326265912341877886737770, 11.23360878850897224682914801709, 12.10326084203571000115793277870, 12.62832655919554475018487608138, 14.327968504854425407139630939, 15.07786975493916168167587421338, 16.34441454684644902637682857736, 16.560237406960438739092561219518, 17.56295402260594058130515538375, 18.86603567718612194156639824478, 19.1522056374930921083647067752, 20.111739449403670659637364928757, 20.813256975954006113397835828431, 21.79924768665563346610160966455, 22.87274523662188370726686668119, 23.93384051175900444816840880608