L(s) = 1 | + (−1.98 + 1.14i)2-s + (−1.71 − 0.259i)3-s + (1.62 − 2.81i)4-s + (1.57 − 0.911i)5-s + (3.69 − 1.44i)6-s − i·7-s + 2.86i·8-s + (2.86 + 0.888i)9-s + (−2.08 + 3.61i)10-s + (5.48 − 3.16i)11-s + (−3.51 + 4.39i)12-s + (−3.12 + 1.79i)13-s + (1.14 + 1.98i)14-s + (−2.94 + 1.15i)15-s + (−0.0323 − 0.0560i)16-s + (0.400 + 0.694i)17-s + ⋯ |
L(s) = 1 | + (−1.40 + 0.810i)2-s + (−0.988 − 0.149i)3-s + (0.812 − 1.40i)4-s + (0.706 − 0.407i)5-s + (1.50 − 0.590i)6-s − 0.377i·7-s + 1.01i·8-s + (0.955 + 0.296i)9-s + (−0.660 + 1.14i)10-s + (1.65 − 0.953i)11-s + (−1.01 + 1.26i)12-s + (−0.867 + 0.497i)13-s + (0.306 + 0.530i)14-s + (−0.759 + 0.297i)15-s + (−0.00809 − 0.0140i)16-s + (0.0972 + 0.168i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.893 + 0.448i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.893 + 0.448i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.619871 - 0.146771i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.619871 - 0.146771i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (1.71 + 0.259i)T \) |
| 7 | \( 1 + iT \) |
| 13 | \( 1 + (3.12 - 1.79i)T \) |
good | 2 | \( 1 + (1.98 - 1.14i)T + (1 - 1.73i)T^{2} \) |
| 5 | \( 1 + (-1.57 + 0.911i)T + (2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (-5.48 + 3.16i)T + (5.5 - 9.52i)T^{2} \) |
| 17 | \( 1 + (-0.400 - 0.694i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-6.09 + 3.52i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 - 1.19T + 23T^{2} \) |
| 29 | \( 1 + (2.42 + 4.19i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (7.58 - 4.37i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-2.63 - 1.52i)T + (18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + 12.7iT - 41T^{2} \) |
| 43 | \( 1 - 1.74T + 43T^{2} \) |
| 47 | \( 1 + (-1.65 - 0.956i)T + (23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 - 6.89T + 53T^{2} \) |
| 59 | \( 1 + (-6.40 - 3.69i)T + (29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + 1.86T + 61T^{2} \) |
| 67 | \( 1 - 4.93iT - 67T^{2} \) |
| 71 | \( 1 + (-0.0906 + 0.0523i)T + (35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + 1.03iT - 73T^{2} \) |
| 79 | \( 1 + (2.62 - 4.55i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-10.8 - 6.27i)T + (41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (3.60 + 2.08i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + 10.2iT - 97T^{2} \) |
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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
−9.782448639235061949314582243340, −9.383095739802034372430414371906, −8.675796346661378799837029886107, −7.20231533447064178540326891060, −7.09000017956231252473239700878, −5.94759578578354961215229990319, −5.38227954533996684415730881191, −3.94632931086306509981955889288, −1.64493847486476143571659132993, −0.70360614838155385107779201855,
1.18803418065448372213334409027, 2.17442967020439478182448062319, 3.60927405436804423380226812215, 5.06036141780937468047677167239, 6.04276742520729201254223852271, 7.06782194856674519967409512184, 7.70978129503244464780579856204, 9.227900253697155523232622330912, 9.595972097255051473566670456609, 10.08184055707309753886470537427