L(s) = 1 | + (−0.866 + 0.5i)2-s + (−0.5 − 0.866i)3-s + (0.499 − 0.866i)4-s + (0.866 + 0.499i)6-s + (−2.32 − 1.34i)7-s + 0.999i·8-s + (−0.499 + 0.866i)9-s + (4.61 − 2.66i)11-s − 0.999·12-s + (3.42 + 1.12i)13-s + 2.68·14-s + (−0.5 − 0.866i)16-s + (−2.18 + 3.78i)17-s − 0.999i·18-s + (2.70 + 1.56i)19-s + ⋯ |
L(s) = 1 | + (−0.612 + 0.353i)2-s + (−0.288 − 0.499i)3-s + (0.249 − 0.433i)4-s + (0.353 + 0.204i)6-s + (−0.880 − 0.508i)7-s + 0.353i·8-s + (−0.166 + 0.288i)9-s + (1.39 − 0.802i)11-s − 0.288·12-s + (0.950 + 0.310i)13-s + 0.718·14-s + (−0.125 − 0.216i)16-s + (−0.529 + 0.917i)17-s − 0.235i·18-s + (0.620 + 0.358i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.946 + 0.322i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.946 + 0.322i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.120500001\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.120500001\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.866 - 0.5i)T \) |
| 3 | \( 1 + (0.5 + 0.866i)T \) |
| 5 | \( 1 \) |
| 13 | \( 1 + (-3.42 - 1.12i)T \) |
good | 7 | \( 1 + (2.32 + 1.34i)T + (3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (-4.61 + 2.66i)T + (5.5 - 9.52i)T^{2} \) |
| 17 | \( 1 + (2.18 - 3.78i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-2.70 - 1.56i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-0.478 - 0.828i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (1.59 + 2.76i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 - 6.98iT - 31T^{2} \) |
| 37 | \( 1 + (4.20 - 2.42i)T + (18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-3.78 + 2.18i)T + (20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (-0.620 + 1.07i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + 2.61iT - 47T^{2} \) |
| 53 | \( 1 - 6.98T + 53T^{2} \) |
| 59 | \( 1 + (-5.03 - 2.90i)T + (29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (1.42 - 2.47i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (1.34 - 0.778i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (-8.89 - 5.13i)T + (35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + 0.569iT - 73T^{2} \) |
| 79 | \( 1 - 17.1T + 79T^{2} \) |
| 83 | \( 1 + 13.3iT - 83T^{2} \) |
| 89 | \( 1 + (0.243 - 0.140i)T + (44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-10.4 - 6.01i)T + (48.5 + 84.0i)T^{2} \) |
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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
−8.879244889773573622526749330505, −8.581025590485838322875959716287, −7.46572394885184229410412104579, −6.64531380196855725131501692576, −6.32256787263226780342490159447, −5.50603901268422451611322906064, −4.02411761186798127487763331688, −3.36238653419728930171547152155, −1.72791385987071500488502959019, −0.791243778620755397797294788379,
0.837322630507555505173873403674, 2.26785443255854547174115553494, 3.36428926307057081846912389276, 4.08563922530813073627465641762, 5.18853244918337742001549188951, 6.26240835341330415670440137870, 6.76411754185779721428477573486, 7.71682182880125365459367892465, 8.891443720485389815264195339927, 9.312990111774903269121362186371