L(s) = 1 | + 1.41·2-s + 2.00·4-s + (−2.04 + 4.56i)5-s + 11.8i·7-s + 2.82·8-s + (−2.89 + 6.44i)10-s + 6.43i·11-s − 11.1i·13-s + 16.8i·14-s + 4.00·16-s − 20.5·17-s − 20.4·19-s + (−4.09 + 9.12i)20-s + 9.10i·22-s − 9.29·23-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.500·4-s + (−0.409 + 0.912i)5-s + 1.69i·7-s + 0.353·8-s + (−0.289 + 0.644i)10-s + 0.585i·11-s − 0.857i·13-s + 1.20i·14-s + 0.250·16-s − 1.21·17-s − 1.07·19-s + (−0.204 + 0.456i)20-s + 0.413i·22-s − 0.404·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 810 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.912 - 0.409i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 810 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.912 - 0.409i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(1.646462441\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.646462441\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - 1.41T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (2.04 - 4.56i)T \) |
good | 7 | \( 1 - 11.8iT - 49T^{2} \) |
| 11 | \( 1 - 6.43iT - 121T^{2} \) |
| 13 | \( 1 + 11.1iT - 169T^{2} \) |
| 17 | \( 1 + 20.5T + 289T^{2} \) |
| 19 | \( 1 + 20.4T + 361T^{2} \) |
| 23 | \( 1 + 9.29T + 529T^{2} \) |
| 29 | \( 1 + 13.6iT - 841T^{2} \) |
| 31 | \( 1 - 53.8T + 961T^{2} \) |
| 37 | \( 1 + 18.6iT - 1.36e3T^{2} \) |
| 41 | \( 1 - 0.460iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 35.1iT - 1.84e3T^{2} \) |
| 47 | \( 1 + 64.3T + 2.20e3T^{2} \) |
| 53 | \( 1 + 2.25T + 2.80e3T^{2} \) |
| 59 | \( 1 - 45.8iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 9.89T + 3.72e3T^{2} \) |
| 67 | \( 1 - 41.9iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 121. iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 11.1iT - 5.32e3T^{2} \) |
| 79 | \( 1 - 145.T + 6.24e3T^{2} \) |
| 83 | \( 1 + 21.2T + 6.88e3T^{2} \) |
| 89 | \( 1 - 73.2iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 78.7iT - 9.40e3T^{2} \) |
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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
−10.57417605175968624760495754997, −9.696372918526346451740364470688, −8.542737595935387452236712362893, −7.88451888475745765926729997942, −6.59672019283561891933514261939, −6.17129592624611099284357539758, −5.06581775107701447155046755494, −4.07954279942233851187704829372, −2.75145550006235729896120321245, −2.26115664804336478317938828630,
0.40718124158804702648843292999, 1.76292638401912523802533047369, 3.48487404486226239145268525748, 4.37483068586616125796984487873, 4.74630345690612171964662123823, 6.30502330125325053994373575251, 6.89173176585412993620715866084, 7.966607576325522478546803003253, 8.663341221383494971120084727014, 9.800374237841490184854400486524