L(s) = 1 | + i·2-s + (−1.06 + 1.36i)3-s − 4-s + (0.866 + 0.5i)5-s + (−1.36 − 1.06i)6-s + (0.0693 + 0.120i)7-s − i·8-s + (−0.715 − 2.91i)9-s + (−0.5 + 0.866i)10-s + (1.29 − 2.24i)11-s + (1.06 − 1.36i)12-s + (−1.11 − 0.646i)13-s + (−0.120 + 0.0693i)14-s + (−1.60 + 0.646i)15-s + 16-s + (−3.25 − 5.63i)17-s + ⋯ |
L(s) = 1 | + 0.707i·2-s + (−0.617 + 0.786i)3-s − 0.5·4-s + (0.387 + 0.223i)5-s + (−0.556 − 0.436i)6-s + (0.0262 + 0.0454i)7-s − 0.353i·8-s + (−0.238 − 0.971i)9-s + (−0.158 + 0.273i)10-s + (0.390 − 0.675i)11-s + (0.308 − 0.393i)12-s + (−0.310 − 0.179i)13-s + (−0.0321 + 0.0185i)14-s + (−0.414 + 0.166i)15-s + 0.250·16-s + (−0.789 − 1.36i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.851 + 0.524i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.851 + 0.524i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.682974 - 0.193600i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.682974 - 0.193600i\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (1.06 - 1.36i)T \) |
| 5 | \( 1 + (-0.866 - 0.5i)T \) |
| 31 | \( 1 + (-0.735 + 5.51i)T \) |
good | 7 | \( 1 + (-0.0693 - 0.120i)T + (-3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (-1.29 + 2.24i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (1.11 + 0.646i)T + (6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + (3.25 + 5.63i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (2.87 + 4.98i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + 4.55T + 23T^{2} \) |
| 29 | \( 1 + 3.95T + 29T^{2} \) |
| 37 | \( 1 + (-3.55 + 2.05i)T + (18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (0.0318 + 0.0183i)T + (20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-0.173 + 0.100i)T + (21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 - 8.23iT - 47T^{2} \) |
| 53 | \( 1 + (-1.48 + 2.57i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (11.6 - 6.74i)T + (29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + 12.4iT - 61T^{2} \) |
| 67 | \( 1 + (5.56 - 9.64i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (-9.28 - 5.36i)T + (35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (-7.32 - 4.22i)T + (36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-11.0 + 6.35i)T + (39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (5.41 - 9.37i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + 6.31T + 89T^{2} \) |
| 97 | \( 1 + 0.338T + 97T^{2} \) |
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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
−9.688980732845854542491010385143, −9.354319313021524810251834362369, −8.436172538806469429800511462581, −7.23381220560731425356651427828, −6.42608808321179185735645758462, −5.71918644505123787808097513591, −4.81826417729255623917643693369, −3.98570507454524592242615302408, −2.64006223860506929447986707718, −0.35904718670506661089152877625,
1.54747634284548040736432419157, 2.17565069899258758504068748111, 3.88706857350908453368339092651, 4.78912419320742795926934746189, 5.91891062828116616044901959563, 6.53611651320944218009542109301, 7.70305544571830341794877350511, 8.466896066672011055993596265043, 9.441024509368610683824118956118, 10.41662470198563955738548119460