| L(s) = 1 | + (−1.53 + 0.805i)3-s + (0.583 + 3.31i)5-s + (1.47 + 1.23i)7-s + (1.70 − 2.46i)9-s + (−0.352 + 1.99i)11-s + (−5.53 − 2.01i)13-s + (−3.56 − 4.60i)15-s + (1.69 + 2.93i)17-s + (0.0802 − 0.138i)19-s + (−3.25 − 0.708i)21-s + (3.29 − 2.76i)23-s + (−5.92 + 2.15i)25-s + (−0.622 + 5.15i)27-s + (−7.79 + 2.83i)29-s + (−8.07 + 6.77i)31-s + ⋯ |
| L(s) = 1 | + (−0.885 + 0.464i)3-s + (0.261 + 1.48i)5-s + (0.556 + 0.467i)7-s + (0.567 − 0.823i)9-s + (−0.106 + 0.602i)11-s + (−1.53 − 0.558i)13-s + (−0.919 − 1.18i)15-s + (0.411 + 0.712i)17-s + (0.0184 − 0.0318i)19-s + (−0.709 − 0.154i)21-s + (0.686 − 0.576i)23-s + (−1.18 + 0.431i)25-s + (−0.119 + 0.992i)27-s + (−1.44 + 0.526i)29-s + (−1.45 + 1.21i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.763 - 0.645i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.763 - 0.645i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.305282 + 0.833885i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.305282 + 0.833885i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (1.53 - 0.805i)T \) |
| good | 5 | \( 1 + (-0.583 - 3.31i)T + (-4.69 + 1.71i)T^{2} \) |
| 7 | \( 1 + (-1.47 - 1.23i)T + (1.21 + 6.89i)T^{2} \) |
| 11 | \( 1 + (0.352 - 1.99i)T + (-10.3 - 3.76i)T^{2} \) |
| 13 | \( 1 + (5.53 + 2.01i)T + (9.95 + 8.35i)T^{2} \) |
| 17 | \( 1 + (-1.69 - 2.93i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-0.0802 + 0.138i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-3.29 + 2.76i)T + (3.99 - 22.6i)T^{2} \) |
| 29 | \( 1 + (7.79 - 2.83i)T + (22.2 - 18.6i)T^{2} \) |
| 31 | \( 1 + (8.07 - 6.77i)T + (5.38 - 30.5i)T^{2} \) |
| 37 | \( 1 + (-2.17 - 3.77i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-1.21 - 0.441i)T + (31.4 + 26.3i)T^{2} \) |
| 43 | \( 1 + (-1.25 + 7.11i)T + (-40.4 - 14.7i)T^{2} \) |
| 47 | \( 1 + (0.0743 + 0.0623i)T + (8.16 + 46.2i)T^{2} \) |
| 53 | \( 1 - 12.8T + 53T^{2} \) |
| 59 | \( 1 + (-0.422 - 2.39i)T + (-55.4 + 20.1i)T^{2} \) |
| 61 | \( 1 + (-4.41 - 3.70i)T + (10.5 + 60.0i)T^{2} \) |
| 67 | \( 1 + (3.89 + 1.41i)T + (51.3 + 43.0i)T^{2} \) |
| 71 | \( 1 + (-3.09 - 5.36i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (-2.12 + 3.68i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (2.80 - 1.02i)T + (60.5 - 50.7i)T^{2} \) |
| 83 | \( 1 + (-8.95 + 3.26i)T + (63.5 - 53.3i)T^{2} \) |
| 89 | \( 1 + (0.821 - 1.42i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-0.484 + 2.74i)T + (-91.1 - 33.1i)T^{2} \) |
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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
−11.34449823680125856408221286554, −10.48803011958696596900753998619, −10.13572815769603370812022345960, −9.010942021935575238338254115859, −7.42639516675763629618613191742, −6.93647267124791904388290361010, −5.69149447839929175176880756180, −4.99004484720346111715160512842, −3.52616206612647542368602779548, −2.19955949785790563460769300796,
0.62208677206387151259937686463, 1.97848054220907917385042165174, 4.26447478892306852639494574043, 5.14507027898100025592630502687, 5.72882294337900522407274249534, 7.29184332021698780081446166872, 7.78243694775365265100683313457, 9.151159449693636256667940468066, 9.767317003683093328653582064638, 11.12976929038798473415310830237
