L(s) = 1 | + (−0.5 + 0.866i)5-s + (0.5 + 0.866i)11-s − 13-s + (0.5 + 0.866i)17-s + (−0.5 + 0.866i)19-s + (−0.5 + 0.866i)23-s + (−0.5 − 0.866i)25-s + 29-s + (0.5 + 0.866i)31-s + (0.5 − 0.866i)37-s − 41-s + 43-s + (−0.5 + 0.866i)47-s + (−0.5 − 0.866i)53-s − 55-s + ⋯ |
L(s) = 1 | + (−0.5 + 0.866i)5-s + (0.5 + 0.866i)11-s − 13-s + (0.5 + 0.866i)17-s + (−0.5 + 0.866i)19-s + (−0.5 + 0.866i)23-s + (−0.5 − 0.866i)25-s + 29-s + (0.5 + 0.866i)31-s + (0.5 − 0.866i)37-s − 41-s + 43-s + (−0.5 + 0.866i)47-s + (−0.5 − 0.866i)53-s − 55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 168 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0633 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 168 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0633 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.6594850301 + 0.6189605914i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6594850301 + 0.6189605914i\) |
\(L(1)\) |
\(\approx\) |
\(0.8716241994 + 0.2985236742i\) |
\(L(1)\) |
\(\approx\) |
\(0.8716241994 + 0.2985236742i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
good | 5 | \( 1 + (-0.5 + 0.866i)T \) |
| 11 | \( 1 + (0.5 + 0.866i)T \) |
| 13 | \( 1 - T \) |
| 17 | \( 1 + (0.5 + 0.866i)T \) |
| 19 | \( 1 + (-0.5 + 0.866i)T \) |
| 23 | \( 1 + (-0.5 + 0.866i)T \) |
| 29 | \( 1 + T \) |
| 31 | \( 1 + (0.5 + 0.866i)T \) |
| 37 | \( 1 + (0.5 - 0.866i)T \) |
| 41 | \( 1 - T \) |
| 43 | \( 1 + T \) |
| 47 | \( 1 + (-0.5 + 0.866i)T \) |
| 53 | \( 1 + (-0.5 - 0.866i)T \) |
| 59 | \( 1 + (0.5 + 0.866i)T \) |
| 61 | \( 1 + (0.5 - 0.866i)T \) |
| 67 | \( 1 + (-0.5 - 0.866i)T \) |
| 71 | \( 1 + T \) |
| 73 | \( 1 + (-0.5 - 0.866i)T \) |
| 79 | \( 1 + (0.5 - 0.866i)T \) |
| 83 | \( 1 - T \) |
| 89 | \( 1 + (0.5 - 0.866i)T \) |
| 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
−27.34700688259878567187517607700, −26.6648742311191016204928827575, −25.25687991290486817500973168041, −24.41108289003133558338003765706, −23.73744283523970808013198762198, −22.5057034702252131457620774040, −21.56075309977618843677903791891, −20.45644065794825396745867001994, −19.62169175512878376237783243282, −18.74499923549795703552503073219, −17.29426786032352734251009339887, −16.5650449225101578929055999661, −15.60799531913482570788285553162, −14.4107046560596876841045709776, −13.32346539877746665139187313868, −12.17272723825973815685141154873, −11.45747734168602885349992403133, −9.965549114177424954717760193479, −8.86357391466434323668214883211, −7.93965432516971138731017093209, −6.622226261300865042144629423383, −5.18374247721784007210610071364, −4.206002317234120291872075684085, −2.68398606859280669200205477895, −0.765022630274107827981426045658,
1.937719625148746198100849801715, 3.38454857982509925054733587356, 4.53012870822821549176745456523, 6.12790912704331308957196393807, 7.207284212889681243449308012524, 8.136951101287144091583342251706, 9.73271867576872175345243366376, 10.51475388578296421685025063306, 11.85011142498226725210133216560, 12.54863767233249745226305130989, 14.23104888695860575554762028034, 14.795566233704683528762688785545, 15.834602462728824579088342317790, 17.15392342383461899089905913584, 17.97419858102378620324152964180, 19.286038229553340021082027091990, 19.7075293042252042604721878241, 21.16846134479338321200707722561, 22.1275488244425774054433836315, 23.00521330147949642862584461298, 23.80220626256366309116986686475, 25.13525183256493216434491188577, 25.875418760781783189512583384, 27.022370118350250746329929231167, 27.60171774759677379615679387122