| 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
−10.03251881803585092537467155269, −9.484847055609453313404185751344, −8.900107663301116733016303893302, −8.358246960910274692050880596845, −7.71907169089302223474880326722, −5.85974418939572815942652515241, −4.70069795663321695551279076249, −3.65000383322510273257276792037, −2.05023794095185098993603046970, −1.04695443997358377035807372069,
1.79471811994133761349202095998, 3.13479472545906828633059588083, 3.84266617618593282966615717487, 6.22355224300197838793600928900, 6.86897614892422913945625035570, 7.49466224810673779460494688764, 8.533726496301519549804441848828, 9.004440005495245857865393649769, 9.958426836128515752565894855800, 10.71064612456919973535417716916
