L(s) = 1 | + i·2-s + (−0.5 − 0.866i)3-s − 4-s + (0.866 − 0.5i)5-s + (0.866 − 0.5i)6-s − i·8-s + (−0.5 + 0.866i)9-s + (0.5 + 0.866i)10-s + (−0.866 + 0.5i)11-s + (0.5 + 0.866i)12-s + (−0.866 − 0.5i)15-s + 16-s − 17-s + (−0.866 − 0.5i)18-s + (−0.866 − 0.5i)19-s + (−0.866 + 0.5i)20-s + ⋯ |
L(s) = 1 | + i·2-s + (−0.5 − 0.866i)3-s − 4-s + (0.866 − 0.5i)5-s + (0.866 − 0.5i)6-s − i·8-s + (−0.5 + 0.866i)9-s + (0.5 + 0.866i)10-s + (−0.866 + 0.5i)11-s + (0.5 + 0.866i)12-s + (−0.866 − 0.5i)15-s + 16-s − 17-s + (−0.866 − 0.5i)18-s + (−0.866 − 0.5i)19-s + (−0.866 + 0.5i)20-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.810 - 0.585i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.810 - 0.585i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.05869203956 - 0.1814782838i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.05869203956 - 0.1814782838i\) |
\(L(1)\) |
\(\approx\) |
\(0.6552341662 + 0.06705931931i\) |
\(L(1)\) |
\(\approx\) |
\(0.6552341662 + 0.06705931931i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 \) |
| 13 | \( 1 \) |
good | 2 | \( 1 + iT \) |
| 3 | \( 1 + (-0.5 - 0.866i)T \) |
| 5 | \( 1 + (0.866 - 0.5i)T \) |
| 11 | \( 1 + (-0.866 + 0.5i)T \) |
| 17 | \( 1 - T \) |
| 19 | \( 1 + (-0.866 - 0.5i)T \) |
| 23 | \( 1 - T \) |
| 29 | \( 1 + (-0.5 + 0.866i)T \) |
| 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 | \( 1 + (-0.866 + 0.5i)T \) |
| 53 | \( 1 + (-0.5 + 0.866i)T \) |
| 59 | \( 1 - iT \) |
| 61 | \( 1 + (-0.5 + 0.866i)T \) |
| 67 | \( 1 + (0.866 - 0.5i)T \) |
| 71 | \( 1 + (0.866 - 0.5i)T \) |
| 73 | \( 1 + (0.866 + 0.5i)T \) |
| 79 | \( 1 + (-0.5 - 0.866i)T \) |
| 83 | \( 1 + iT \) |
| 89 | \( 1 - iT \) |
| 97 | \( 1 + (0.866 - 0.5i)T \) |
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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
−30.35093798493584504007095100887, −29.21047303594523537215034942023, −28.76830144049808549723599947962, −27.56155922974310473354160123328, −26.58307110271856631600923534644, −25.82998607314284064833006008858, −23.8862776695565057166722207507, −22.71883557518867341918835442235, −21.86340195896712282081490980938, −21.20718199061186132797905261265, −20.219970823577848967164268175882, −18.67532682793129242572873138396, −17.810968694118304425513075943203, −16.78854153185745716936344464356, −15.24443013529347885648972960838, −14.020118843125880498019830394739, −12.91397073834300565326596748443, −11.42177822544112101657810296874, −10.54346636805817758192957172497, −9.787314891306898123400394928779, −8.51323396461322154630579650079, −6.17165255383881319088257178473, −5.03959730997008992405959367818, −3.60730460450412587632035526026, −2.18681694169503596206561855607,
0.08712184384855611536598193592, 2.01213560899659944580508452267, 4.70638813813313373026618446944, 5.74634702209193737058700465388, 6.77600960462081145046362209745, 8.00402540986125018962074280204, 9.22191261188131321313625370440, 10.69510645988850152503539754187, 12.651236569064651225632927768971, 13.17237052012638565192073204464, 14.3022111714943631248655503216, 15.801635171865661254602348346173, 16.917482136411043545920082130367, 17.778428967390603566427179701438, 18.42406566094387493101729601493, 19.95487759091627201811067902319, 21.58458934238405624784995089900, 22.56620869298784662680725506336, 23.79422837334263951790178345786, 24.32011218899206658899617206333, 25.46788011848396167382537691262, 26.10550068778038237472174923686, 27.83332369365840110289963198663, 28.56989324654164602628192441104, 29.66689929626299795471060387404