L(s) = 1 | + (0.5 − 0.866i)2-s + (−0.5 − 0.866i)4-s + (0.866 − 0.5i)5-s + (0.866 + 0.5i)7-s − 8-s − i·10-s + (0.866 + 0.5i)11-s + (−0.5 − 0.866i)13-s + (0.866 − 0.5i)14-s + (−0.5 + 0.866i)16-s − 19-s + (−0.866 − 0.5i)20-s + (0.866 − 0.5i)22-s + (−0.866 + 0.5i)23-s + (0.5 − 0.866i)25-s − 26-s + ⋯ |
L(s) = 1 | + (0.5 − 0.866i)2-s + (−0.5 − 0.866i)4-s + (0.866 − 0.5i)5-s + (0.866 + 0.5i)7-s − 8-s − i·10-s + (0.866 + 0.5i)11-s + (−0.5 − 0.866i)13-s + (0.866 − 0.5i)14-s + (−0.5 + 0.866i)16-s − 19-s + (−0.866 − 0.5i)20-s + (0.866 − 0.5i)22-s + (−0.866 + 0.5i)23-s + (0.5 − 0.866i)25-s − 26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 153 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0352 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 153 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0352 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.146451506 - 1.106765027i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.146451506 - 1.106765027i\) |
\(L(1)\) |
\(\approx\) |
\(1.248495463 - 0.7634644349i\) |
\(L(1)\) |
\(\approx\) |
\(1.248495463 - 0.7634644349i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 17 | \( 1 \) |
good | 2 | \( 1 + (0.5 - 0.866i)T \) |
| 5 | \( 1 + (0.866 - 0.5i)T \) |
| 7 | \( 1 + (0.866 + 0.5i)T \) |
| 11 | \( 1 + (0.866 + 0.5i)T \) |
| 13 | \( 1 + (-0.5 - 0.866i)T \) |
| 19 | \( 1 - T \) |
| 23 | \( 1 + (-0.866 + 0.5i)T \) |
| 29 | \( 1 + (-0.866 - 0.5i)T \) |
| 31 | \( 1 + (0.866 - 0.5i)T \) |
| 37 | \( 1 + iT \) |
| 41 | \( 1 + (-0.866 + 0.5i)T \) |
| 43 | \( 1 + (0.5 - 0.866i)T \) |
| 47 | \( 1 + (-0.5 + 0.866i)T \) |
| 53 | \( 1 - T \) |
| 59 | \( 1 + (0.5 + 0.866i)T \) |
| 61 | \( 1 + (0.866 + 0.5i)T \) |
| 67 | \( 1 + (-0.5 - 0.866i)T \) |
| 71 | \( 1 + iT \) |
| 73 | \( 1 + iT \) |
| 79 | \( 1 + (0.866 + 0.5i)T \) |
| 83 | \( 1 + (0.5 - 0.866i)T \) |
| 89 | \( 1 + T \) |
| 97 | \( 1 + (-0.866 - 0.5i)T \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−28.05133134008901546408291480662, −26.85954065815011878028193405496, −26.29568318200590670729497249711, −25.15900753008376681032993512973, −24.37851561452780860332816345099, −23.53641507287821003644736900671, −22.29207491039483970185468179097, −21.62810503585009632485668841029, −20.76219673312109306748304671915, −19.14177297234515689615339062545, −17.92045964119579219188233643811, −17.173630824895821294324129125490, −16.39457534229773697850409824294, −14.7244416740526992658375500937, −14.33278609853987775345796905817, −13.45194968787029037931086441565, −12.07652302503669711922595705975, −10.878282653618593374579467843808, −9.439539652013933735913880103175, −8.33492796696626468342559379231, −6.999490860293043982952534934532, −6.21598384672766413716125674036, −4.902320097969029936810435215152, −3.7654748110103168191030400737, −2.021926386824463290221516275962,
1.48270202494914654235208049577, 2.4491321036410351159268244072, 4.231409766398614024604502950525, 5.23225584954524587599288918230, 6.23231227301024504500122238819, 8.24279994072022052935556775471, 9.39535702277331714438770003667, 10.2570951527880535005904288257, 11.5874865300104499818153918831, 12.43210163383354500247912016461, 13.436699765821814541794060865429, 14.50612777331179551358019291056, 15.273639964119230087535139459351, 17.194063920183487940479092079889, 17.7698105008625828842044141448, 18.98794502182364563187543779777, 20.183485067918127245513203793476, 20.853957727777886219020741041670, 21.83428912352490160037559999483, 22.49486539332064810511578140246, 23.91285357551771229150831357160, 24.66134291583498511166127540500, 25.59474350954172091978338040373, 27.33686330505112688579999953593, 27.877742710550094792906356340051