| L(s) = 1 | + (−1.55 + 0.694i)2-s + (2.24 − 0.476i)3-s + (0.611 − 0.679i)4-s + (0.425 + 4.04i)5-s + (−3.16 + 2.29i)6-s + (0.572 − 1.76i)8-s + (2.05 − 0.915i)9-s + (−3.47 − 6.01i)10-s + (3.11 − 1.12i)11-s + (1.04 − 1.81i)12-s + (2.65 + 1.92i)13-s + (2.87 + 8.86i)15-s + (0.521 + 4.96i)16-s + (1.20 + 0.537i)17-s + (−2.57 + 2.85i)18-s + (−1.44 − 1.60i)19-s + ⋯ |
| L(s) = 1 | + (−1.10 + 0.490i)2-s + (1.29 − 0.275i)3-s + (0.305 − 0.339i)4-s + (0.190 + 1.80i)5-s + (−1.29 + 0.938i)6-s + (0.202 − 0.623i)8-s + (0.685 − 0.305i)9-s + (−1.09 − 1.90i)10-s + (0.940 − 0.340i)11-s + (0.302 − 0.523i)12-s + (0.735 + 0.534i)13-s + (0.743 + 2.28i)15-s + (0.130 + 1.24i)16-s + (0.292 + 0.130i)17-s + (−0.606 + 0.673i)18-s + (−0.331 − 0.368i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 539 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.100 - 0.994i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 539 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.100 - 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.869147 + 0.961554i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.869147 + 0.961554i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 \) |
| 11 | \( 1 + (-3.11 + 1.12i)T \) |
| good | 2 | \( 1 + (1.55 - 0.694i)T + (1.33 - 1.48i)T^{2} \) |
| 3 | \( 1 + (-2.24 + 0.476i)T + (2.74 - 1.22i)T^{2} \) |
| 5 | \( 1 + (-0.425 - 4.04i)T + (-4.89 + 1.03i)T^{2} \) |
| 13 | \( 1 + (-2.65 - 1.92i)T + (4.01 + 12.3i)T^{2} \) |
| 17 | \( 1 + (-1.20 - 0.537i)T + (11.3 + 12.6i)T^{2} \) |
| 19 | \( 1 + (1.44 + 1.60i)T + (-1.98 + 18.8i)T^{2} \) |
| 23 | \( 1 + (-0.933 + 1.61i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (0.0754 + 0.232i)T + (-23.4 + 17.0i)T^{2} \) |
| 31 | \( 1 + (0.718 - 6.83i)T + (-30.3 - 6.44i)T^{2} \) |
| 37 | \( 1 + (0.250 + 0.0531i)T + (33.8 + 15.0i)T^{2} \) |
| 41 | \( 1 + (-1.77 + 5.45i)T + (-33.1 - 24.0i)T^{2} \) |
| 43 | \( 1 + 8.01T + 43T^{2} \) |
| 47 | \( 1 + (-2.72 - 3.02i)T + (-4.91 + 46.7i)T^{2} \) |
| 53 | \( 1 + (-0.523 + 4.97i)T + (-51.8 - 11.0i)T^{2} \) |
| 59 | \( 1 + (-0.658 + 0.730i)T + (-6.16 - 58.6i)T^{2} \) |
| 61 | \( 1 + (-0.193 - 1.83i)T + (-59.6 + 12.6i)T^{2} \) |
| 67 | \( 1 + (-1.50 - 2.60i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (5.23 - 3.80i)T + (21.9 - 67.5i)T^{2} \) |
| 73 | \( 1 + (-6.45 + 7.17i)T + (-7.63 - 72.6i)T^{2} \) |
| 79 | \( 1 + (5.05 - 2.25i)T + (52.8 - 58.7i)T^{2} \) |
| 83 | \( 1 + (-1.67 + 1.21i)T + (25.6 - 78.9i)T^{2} \) |
| 89 | \( 1 + (-8.31 + 14.4i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (2.09 + 1.51i)T + (29.9 + 92.2i)T^{2} \) |
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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
−10.71064612456919973535417716916, −9.958426836128515752565894855800, −9.004440005495245857865393649769, −8.533726496301519549804441848828, −7.49466224810673779460494688764, −6.86897614892422913945625035570, −6.22355224300197838793600928900, −3.84266617618593282966615717487, −3.13479472545906828633059588083, −1.79471811994133761349202095998,
1.04695443997358377035807372069, 2.05023794095185098993603046970, 3.65000383322510273257276792037, 4.70069795663321695551279076249, 5.85974418939572815942652515241, 7.71907169089302223474880326722, 8.358246960910274692050880596845, 8.900107663301116733016303893302, 9.484847055609453313404185751344, 10.03251881803585092537467155269
