L(s) = 1 | + (0.622 + 1.61i)3-s + (0.5 + 0.866i)5-s + (2.57 − 0.615i)7-s + (−2.22 + 2.01i)9-s + (1.80 + 1.04i)11-s + 0.245i·13-s + (−1.08 + 1.34i)15-s + (0.471 − 0.816i)17-s + (−0.465 + 0.268i)19-s + (2.59 + 3.77i)21-s + (−2.40 + 1.38i)23-s + (−0.499 + 0.866i)25-s + (−4.63 − 2.34i)27-s + 0.267i·29-s + (0.981 + 0.566i)31-s + ⋯ |
L(s) = 1 | + (0.359 + 0.933i)3-s + (0.223 + 0.387i)5-s + (0.972 − 0.232i)7-s + (−0.741 + 0.671i)9-s + (0.544 + 0.314i)11-s + 0.0681i·13-s + (−0.281 + 0.347i)15-s + (0.114 − 0.198i)17-s + (−0.106 + 0.0616i)19-s + (0.566 + 0.823i)21-s + (−0.500 + 0.288i)23-s + (−0.0999 + 0.173i)25-s + (−0.892 − 0.450i)27-s + 0.0496i·29-s + (0.176 + 0.101i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 420 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.312 - 0.949i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 420 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.312 - 0.949i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.39312 + 1.00806i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.39312 + 1.00806i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-0.622 - 1.61i)T \) |
| 5 | \( 1 + (-0.5 - 0.866i)T \) |
| 7 | \( 1 + (-2.57 + 0.615i)T \) |
good | 11 | \( 1 + (-1.80 - 1.04i)T + (5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 - 0.245iT - 13T^{2} \) |
| 17 | \( 1 + (-0.471 + 0.816i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (0.465 - 0.268i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (2.40 - 1.38i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 - 0.267iT - 29T^{2} \) |
| 31 | \( 1 + (-0.981 - 0.566i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-3.08 - 5.33i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 - 2.38T + 41T^{2} \) |
| 43 | \( 1 + 11.4T + 43T^{2} \) |
| 47 | \( 1 + (6.23 + 10.7i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-10.8 - 6.26i)T + (26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-6.25 + 10.8i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-4.96 + 2.86i)T + (30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-2.78 + 4.81i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 10.1iT - 71T^{2} \) |
| 73 | \( 1 + (11.3 + 6.56i)T + (36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-3.17 - 5.49i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + 1.06T + 83T^{2} \) |
| 89 | \( 1 + (-0.463 - 0.803i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + 3.01iT - 97T^{2} \) |
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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
−11.32643443397434733406833034289, −10.33746941773246223361407399271, −9.734625719647060574834935867616, −8.666755611040178030974675453172, −7.892783830284578419336425631178, −6.71945632778300048699555839185, −5.40473149514884314899594790123, −4.49901383670412269086598917687, −3.44581260108048289068143846719, −1.97842063142399830043493053692,
1.24413183542557531365773490939, 2.45053773858738151446264261704, 4.02911342424338911496998107511, 5.39163370169364488282989906557, 6.30979977136803990189497534353, 7.42064142272026158908573332674, 8.346057287332562409362779887684, 8.862959842004412986132161693202, 10.04013668903908933965571646020, 11.35363735977551705348263475778