| L(s) = 1 | + (0.141 − 1.72i)3-s + (0.298 − 1.69i)5-s + (0.0848 − 0.0711i)7-s + (−2.96 − 0.488i)9-s + (−0.107 − 0.612i)11-s + (4.44 − 1.61i)13-s + (−2.87 − 0.754i)15-s + (0.373 − 0.646i)17-s + (−1.50 − 2.60i)19-s + (−0.110 − 0.156i)21-s + (−5.03 − 4.22i)23-s + (1.92 + 0.701i)25-s + (−1.26 + 5.04i)27-s + (−0.518 − 0.188i)29-s + (−6.23 − 5.22i)31-s + ⋯ |
| L(s) = 1 | + (0.0816 − 0.996i)3-s + (0.133 − 0.756i)5-s + (0.0320 − 0.0268i)7-s + (−0.986 − 0.162i)9-s + (−0.0325 − 0.184i)11-s + (1.23 − 0.449i)13-s + (−0.743 − 0.194i)15-s + (0.0905 − 0.156i)17-s + (−0.345 − 0.598i)19-s + (−0.0241 − 0.0341i)21-s + (−1.04 − 0.880i)23-s + (0.385 + 0.140i)25-s + (−0.242 + 0.970i)27-s + (−0.0962 − 0.0350i)29-s + (−1.11 − 0.939i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.773 + 0.633i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.773 + 0.633i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.468985 - 1.31389i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.468985 - 1.31389i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (-0.141 + 1.72i)T \) |
| good | 5 | \( 1 + (-0.298 + 1.69i)T + (-4.69 - 1.71i)T^{2} \) |
| 7 | \( 1 + (-0.0848 + 0.0711i)T + (1.21 - 6.89i)T^{2} \) |
| 11 | \( 1 + (0.107 + 0.612i)T + (-10.3 + 3.76i)T^{2} \) |
| 13 | \( 1 + (-4.44 + 1.61i)T + (9.95 - 8.35i)T^{2} \) |
| 17 | \( 1 + (-0.373 + 0.646i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (1.50 + 2.60i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (5.03 + 4.22i)T + (3.99 + 22.6i)T^{2} \) |
| 29 | \( 1 + (0.518 + 0.188i)T + (22.2 + 18.6i)T^{2} \) |
| 31 | \( 1 + (6.23 + 5.22i)T + (5.38 + 30.5i)T^{2} \) |
| 37 | \( 1 + (-1.23 + 2.13i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (3.38 - 1.23i)T + (31.4 - 26.3i)T^{2} \) |
| 43 | \( 1 + (-1.06 - 6.06i)T + (-40.4 + 14.7i)T^{2} \) |
| 47 | \( 1 + (1.70 - 1.43i)T + (8.16 - 46.2i)T^{2} \) |
| 53 | \( 1 + 1.81T + 53T^{2} \) |
| 59 | \( 1 + (1.00 - 5.68i)T + (-55.4 - 20.1i)T^{2} \) |
| 61 | \( 1 + (-5.68 + 4.76i)T + (10.5 - 60.0i)T^{2} \) |
| 67 | \( 1 + (-2.65 + 0.968i)T + (51.3 - 43.0i)T^{2} \) |
| 71 | \( 1 + (-5.81 + 10.0i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (-4.32 - 7.48i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (3.14 + 1.14i)T + (60.5 + 50.7i)T^{2} \) |
| 83 | \( 1 + (9.62 + 3.50i)T + (63.5 + 53.3i)T^{2} \) |
| 89 | \( 1 + (7.11 + 12.3i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-2.11 - 11.9i)T + (-91.1 + 33.1i)T^{2} \) |
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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
−9.664878568829424175855493328116, −8.719660557190601717886348781791, −8.268902445822079341409913031147, −7.34394531043755700005361176984, −6.26373648900412021561052872748, −5.68966763902496673189454798971, −4.47430671036802842171131982660, −3.19381281194987815184039909080, −1.89435450041182591131078151059, −0.66970312263141394798505596792,
1.96644973747368490596112531703, 3.39436154442239036515466516616, 3.94351394033662020828281104641, 5.23706345828939854932272869347, 6.07239615515141410808887283663, 6.96580117181556694466664401494, 8.192261879844407787485778903286, 8.838192236867021199853403662245, 9.832458195727725551531260278287, 10.42890417295915464759465322639
