| L(s) = 1 | + 1.73i·3-s − 2.23·5-s − 0.596i·7-s − 2.99·9-s + 9.27i·11-s + 23.5·13-s − 3.87i·15-s + 3.97·17-s − 7.04i·19-s + 1.03·21-s − 32.0i·23-s + 5.00·25-s − 5.19i·27-s − 35.6·29-s + 59.2i·31-s + ⋯ |
| L(s) = 1 | + 0.577i·3-s − 0.447·5-s − 0.0852i·7-s − 0.333·9-s + 0.843i·11-s + 1.80·13-s − 0.258i·15-s + 0.233·17-s − 0.370i·19-s + 0.0492·21-s − 1.39i·23-s + 0.200·25-s − 0.192i·27-s − 1.23·29-s + 1.91i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.621070139\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.621070139\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - 1.73iT \) |
| 5 | \( 1 + 2.23T \) |
| good | 7 | \( 1 + 0.596iT - 49T^{2} \) |
| 11 | \( 1 - 9.27iT - 121T^{2} \) |
| 13 | \( 1 - 23.5T + 169T^{2} \) |
| 17 | \( 1 - 3.97T + 289T^{2} \) |
| 19 | \( 1 + 7.04iT - 361T^{2} \) |
| 23 | \( 1 + 32.0iT - 529T^{2} \) |
| 29 | \( 1 + 35.6T + 841T^{2} \) |
| 31 | \( 1 - 59.2iT - 961T^{2} \) |
| 37 | \( 1 - 5.38T + 1.36e3T^{2} \) |
| 41 | \( 1 - 40.0T + 1.68e3T^{2} \) |
| 43 | \( 1 - 36.1iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 74.0iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 2.55T + 2.80e3T^{2} \) |
| 59 | \( 1 - 36.4iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 8.73T + 3.72e3T^{2} \) |
| 67 | \( 1 - 69.7iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 59.2iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 83.0T + 5.32e3T^{2} \) |
| 79 | \( 1 + 65.8iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 129. iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 130.T + 7.92e3T^{2} \) |
| 97 | \( 1 - 93.1T + 9.40e3T^{2} \) |
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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
−10.15411319365638836139957431408, −9.076037618389830553820890813603, −8.564250039229579639188608730288, −7.56259887233404045098327827956, −6.61778530518841396881603395108, −5.69668700639894487044779586365, −4.56177812214011305773283867171, −3.89437039912116536280406962352, −2.80339986852901628314326050096, −1.19448600151075283678428800655,
0.59971928685199392444214869114, 1.83089860032269415088676942279, 3.36384702391692608210620722252, 3.96608428300161901765833540380, 5.67436777431011219872516151699, 5.97359337121814050710216048930, 7.21576966372409382169139919955, 7.949759955374747230763320799200, 8.662295067839984068268189460392, 9.451564394967981632954121769577
