L(s) = 1 | + (−0.5 + 0.866i)3-s + (0.866 − 0.5i)5-s + i·7-s + (−0.5 + 0.866i)11-s + 0.999i·15-s + (−0.5 + 0.866i)17-s + (−0.5 − 0.866i)19-s + (−0.866 − 0.5i)21-s + (0.866 − 0.5i)23-s − 27-s + (0.866 + 0.5i)31-s + (−0.499 − 0.866i)33-s + (0.5 + 0.866i)35-s + (−0.866 + 0.5i)37-s + (0.866 − 0.5i)47-s + ⋯ |
L(s) = 1 | + (−0.5 + 0.866i)3-s + (0.866 − 0.5i)5-s + i·7-s + (−0.5 + 0.866i)11-s + 0.999i·15-s + (−0.5 + 0.866i)17-s + (−0.5 − 0.866i)19-s + (−0.866 − 0.5i)21-s + (0.866 − 0.5i)23-s − 27-s + (0.866 + 0.5i)31-s + (−0.499 − 0.866i)33-s + (0.5 + 0.866i)35-s + (−0.866 + 0.5i)37-s + (0.866 − 0.5i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.197 - 0.980i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.197 - 0.980i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.023148036\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.023148036\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 - iT \) |
good | 3 | \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 5 | \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \) |
| 11 | \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 13 | \( 1 - T^{2} \) |
| 17 | \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 19 | \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 23 | \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \) |
| 29 | \( 1 - T^{2} \) |
| 31 | \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \) |
| 37 | \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \) |
| 41 | \( 1 + T^{2} \) |
| 43 | \( 1 + T^{2} \) |
| 47 | \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \) |
| 53 | \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \) |
| 59 | \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 61 | \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \) |
| 67 | \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 79 | \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \) |
| 83 | \( 1 + T^{2} \) |
| 89 | \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 97 | \( 1 + T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.855539640884952660908453764488, −8.883526481227635267557586338812, −8.539733206121923304721844218461, −7.19735059357150107154717962147, −6.30392643777435479660454004803, −5.39721941077021707979494147490, −4.98049159816657324743026059224, −4.18667464773596365279267402766, −2.67374484209718768372937777335, −1.81741518880511112825319749343,
0.826320707957913952098483638792, 2.05427606741120708158007354215, 3.16956897222994540849375548485, 4.27928939984757762591806973471, 5.51459625276506014789135165387, 6.11022700008236262197360786205, 6.89742545502721203061821063778, 7.38747680532847967741619811925, 8.339186966133633638538282795514, 9.345582491746437012333586800738