L(s) = 1 | + (0.5 − 0.866i)2-s + (−0.5 − 0.866i)4-s + 5-s + (−0.5 − 0.866i)7-s − 8-s + (0.5 − 0.866i)10-s + (−0.5 − 0.866i)11-s + (0.5 + 0.866i)13-s − 14-s + (−0.5 + 0.866i)16-s + (−0.5 − 0.866i)17-s + (−0.5 − 0.866i)20-s − 22-s + (−0.5 − 0.866i)23-s + 25-s + 26-s + ⋯ |
L(s) = 1 | + (0.5 − 0.866i)2-s + (−0.5 − 0.866i)4-s + 5-s + (−0.5 − 0.866i)7-s − 8-s + (0.5 − 0.866i)10-s + (−0.5 − 0.866i)11-s + (0.5 + 0.866i)13-s − 14-s + (−0.5 + 0.866i)16-s + (−0.5 − 0.866i)17-s + (−0.5 − 0.866i)20-s − 22-s + (−0.5 − 0.866i)23-s + 25-s + 26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 171 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.997 + 0.0765i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 171 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.997 + 0.0765i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(-0.06709699604 - 1.750094286i\) |
\(L(\frac12)\) |
\(\approx\) |
\(-0.06709699604 - 1.750094286i\) |
\(L(1)\) |
\(\approx\) |
\(0.8944334228 - 0.9033628432i\) |
\(L(1)\) |
\(\approx\) |
\(0.8944334228 - 0.9033628432i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 19 | \( 1 \) |
good | 2 | \( 1 + (0.5 - 0.866i)T \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 + (-0.5 - 0.866i)T \) |
| 11 | \( 1 + (-0.5 - 0.866i)T \) |
| 13 | \( 1 + (0.5 + 0.866i)T \) |
| 17 | \( 1 + (-0.5 - 0.866i)T \) |
| 23 | \( 1 + (-0.5 - 0.866i)T \) |
| 29 | \( 1 - T \) |
| 31 | \( 1 + (0.5 - 0.866i)T \) |
| 37 | \( 1 - T \) |
| 41 | \( 1 - T \) |
| 43 | \( 1 + (-0.5 + 0.866i)T \) |
| 47 | \( 1 + T \) |
| 53 | \( 1 + (0.5 - 0.866i)T \) |
| 59 | \( 1 - T \) |
| 61 | \( 1 + T \) |
| 67 | \( 1 + (0.5 + 0.866i)T \) |
| 71 | \( 1 + (0.5 + 0.866i)T \) |
| 73 | \( 1 + (-0.5 - 0.866i)T \) |
| 79 | \( 1 + (0.5 - 0.866i)T \) |
| 83 | \( 1 + (-0.5 - 0.866i)T \) |
| 89 | \( 1 + (0.5 - 0.866i)T \) |
| 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
−27.85910590968082727935709932538, −26.355140751607763713331193918019, −25.582924720104071381934767988691, −25.143616140251543525567108596674, −24.04156943341056918160357332225, −22.91637616308359973968175471549, −22.094144338886617016713382085661, −21.35875522873052971632305819094, −20.27262269128305951832249203011, −18.59196613918984780073594026118, −17.80602849617710485237281544837, −17.025985334186635097198183212100, −15.605609783988243154885361093355, −15.18721419647709204236668368124, −13.80134637046592971199559211585, −12.9846135078315301408467235892, −12.22159079234832991829898399126, −10.40338715407853728251804832342, −9.285282959345058722789234853890, −8.295046421674184885624754346150, −6.8964999465729207764800272782, −5.85190542658668669453190244298, −5.15787981251878808921833779659, −3.47992597698358830008471598107, −2.1333093069018148031464879702,
0.50632220318080663162842049240, 1.98708455662157581241693677509, 3.24328469273615684692783923908, 4.50819009508613856055845712452, 5.79761436429870870713279796094, 6.76359616241001727820901640055, 8.73336208855129396265668248226, 9.76146953829849019276377581007, 10.61538191910235539055031300622, 11.58406917913687177805192633827, 13.093826749400466595590026307703, 13.59545020555238543831819071567, 14.35556525201916674793699482628, 15.95937785928201941434812660091, 17.00358501221861753600209570946, 18.31069581960153438732505316101, 18.97890237856827321494080246950, 20.32707483631677815951899449874, 20.89750350801099322711372162109, 21.932202118889449692800876016964, 22.69787691687973600256099780546, 23.81202229448080525426729016540, 24.60776217322529910249137443956, 26.12888796445927024285554111538, 26.69145787367010648056913163486