L(s) = 1 | + i·2-s − 4-s + (0.866 + 0.5i)5-s − i·8-s + (−0.5 + 0.866i)10-s + 16-s + 1.73·17-s − 1.73i·19-s + (−0.866 − 0.5i)20-s + (0.866 − 1.5i)23-s + (0.499 + 0.866i)25-s + (−0.5 + 0.866i)31-s + i·32-s + 1.73i·34-s + 1.73·38-s + ⋯ |
L(s) = 1 | + i·2-s − 4-s + (0.866 + 0.5i)5-s − i·8-s + (−0.5 + 0.866i)10-s + 16-s + 1.73·17-s − 1.73i·19-s + (−0.866 − 0.5i)20-s + (0.866 − 1.5i)23-s + (0.499 + 0.866i)25-s + (−0.5 + 0.866i)31-s + i·32-s + 1.73i·34-s + 1.73·38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.342 - 0.939i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.342 - 0.939i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.404545203\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.404545203\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - iT \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (-0.866 - 0.5i)T \) |
good | 7 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 11 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 13 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 17 | \( 1 - 1.73T + T^{2} \) |
| 19 | \( 1 + 1.73iT - T^{2} \) |
| 23 | \( 1 + (-0.866 + 1.5i)T + (-0.5 - 0.866i)T^{2} \) |
| 29 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 31 | \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 37 | \( 1 + T^{2} \) |
| 41 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 43 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 47 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 53 | \( 1 + iT - T^{2} \) |
| 59 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 61 | \( 1 + (1.5 - 0.866i)T + (0.5 - 0.866i)T^{2} \) |
| 67 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 - T^{2} \) |
| 79 | \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 83 | \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \) |
| 89 | \( 1 - T^{2} \) |
| 97 | \( 1 + (0.5 - 0.866i)T^{2} \) |
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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
−9.060088547771569945979543680887, −8.148618518599986364943368090415, −7.30411786431986310336092726877, −6.74400315288030076761473052846, −6.09473027846842103943806862673, −5.22042247191546266071876571888, −4.74455852737514805773542366627, −3.44127360169184307980857327107, −2.66708967435778655997564908732, −1.10769528209305279792276205124,
1.22515391252203919489691505681, 1.86352375353004977644902132480, 3.12749700554314818024171032123, 3.74998324164523930779071634949, 4.86607119607262878619061869337, 5.58945419719353814302194953542, 6.01111114201036442244783694400, 7.50758576045833287609677709094, 8.050309572068750927354834773217, 8.939206964550131587024051135227