| L(s) = 1 | + 1.41i·2-s − 2.00·4-s + 2.23i·5-s − 1.37·7-s − 2.82i·8-s − 3.16·10-s − 10.6i·11-s + 9.84·13-s − 1.94i·14-s + 4.00·16-s + 3.83i·17-s + 16.2·19-s − 4.47i·20-s + 14.9·22-s + 31.0i·23-s + ⋯ |
| L(s) = 1 | + 0.707i·2-s − 0.500·4-s + 0.447i·5-s − 0.196·7-s − 0.353i·8-s − 0.316·10-s − 0.963i·11-s + 0.757·13-s − 0.138i·14-s + 0.250·16-s + 0.225i·17-s + 0.856·19-s − 0.223i·20-s + 0.681·22-s + 1.35i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 810 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 810 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.695983573\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.695983573\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 1.41iT \) |
| 3 | \( 1 \) |
| 5 | \( 1 - 2.23iT \) |
| good | 7 | \( 1 + 1.37T + 49T^{2} \) |
| 11 | \( 1 + 10.6iT - 121T^{2} \) |
| 13 | \( 1 - 9.84T + 169T^{2} \) |
| 17 | \( 1 - 3.83iT - 289T^{2} \) |
| 19 | \( 1 - 16.2T + 361T^{2} \) |
| 23 | \( 1 - 31.0iT - 529T^{2} \) |
| 29 | \( 1 + 19.6iT - 841T^{2} \) |
| 31 | \( 1 - 43.8T + 961T^{2} \) |
| 37 | \( 1 - 35.1T + 1.36e3T^{2} \) |
| 41 | \( 1 - 71.1iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 43.8T + 1.84e3T^{2} \) |
| 47 | \( 1 - 72.9iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 62.2iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 29.8iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 108.T + 3.72e3T^{2} \) |
| 67 | \( 1 + 65.0T + 4.48e3T^{2} \) |
| 71 | \( 1 - 90.6iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 71.2T + 5.32e3T^{2} \) |
| 79 | \( 1 - 14.4T + 6.24e3T^{2} \) |
| 83 | \( 1 - 134. iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 102. iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 159.T + 9.40e3T^{2} \) |
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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
−9.999353365731592035114468203520, −9.466348867195349471555869571711, −8.270741295896152019375325449959, −7.85073446532934751305755253294, −6.60138142359804482045796361707, −6.09087914645179746413168756976, −5.11523200692666409645586721759, −3.82748754199139512118378361930, −2.97260873853415694002325496722, −1.07641360370895090022098877124,
0.72727677592140340830011173661, 2.02435220502083696070657925213, 3.23952868352852307015344614488, 4.35186101289700997174545395808, 5.12375123681032759131625141310, 6.29771455681706046909034578517, 7.32244114601128638000420165610, 8.394338805092816030714514256899, 9.060465761080100343148046339260, 9.984227662331636155269457179444
