| L(s) = 1 | + (−1.70 + 0.292i)3-s + i·5-s − 0.585i·7-s + (2.82 − i)9-s − 2.82·11-s + 2·13-s + (−0.292 − 1.70i)15-s − 3.65i·17-s − 2.82i·19-s + (0.171 + i)21-s + 4.58·23-s − 25-s + (−4.53 + 2.53i)27-s + 8i·29-s + 5.65i·31-s + ⋯ |
| L(s) = 1 | + (−0.985 + 0.169i)3-s + 0.447i·5-s − 0.221i·7-s + (0.942 − 0.333i)9-s − 0.852·11-s + 0.554·13-s + (−0.0756 − 0.440i)15-s − 0.886i·17-s − 0.648i·19-s + (0.0374 + 0.218i)21-s + 0.956·23-s − 0.200·25-s + (−0.872 + 0.487i)27-s + 1.48i·29-s + 1.01i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.985 - 0.169i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.985 - 0.169i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.08729 + 0.0925987i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.08729 + 0.0925987i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (1.70 - 0.292i)T \) |
| 5 | \( 1 - iT \) |
| good | 7 | \( 1 + 0.585iT - 7T^{2} \) |
| 11 | \( 1 + 2.82T + 11T^{2} \) |
| 13 | \( 1 - 2T + 13T^{2} \) |
| 17 | \( 1 + 3.65iT - 17T^{2} \) |
| 19 | \( 1 + 2.82iT - 19T^{2} \) |
| 23 | \( 1 - 4.58T + 23T^{2} \) |
| 29 | \( 1 - 8iT - 29T^{2} \) |
| 31 | \( 1 - 5.65iT - 31T^{2} \) |
| 37 | \( 1 - 11.6T + 37T^{2} \) |
| 41 | \( 1 - 2iT - 41T^{2} \) |
| 43 | \( 1 + 11.8iT - 43T^{2} \) |
| 47 | \( 1 - 2.24T + 47T^{2} \) |
| 53 | \( 1 + 7.65iT - 53T^{2} \) |
| 59 | \( 1 - 9.65T + 59T^{2} \) |
| 61 | \( 1 - 5.65T + 61T^{2} \) |
| 67 | \( 1 - 13.0iT - 67T^{2} \) |
| 71 | \( 1 - 6.82T + 71T^{2} \) |
| 73 | \( 1 - 4.34T + 73T^{2} \) |
| 79 | \( 1 - 12.4iT - 79T^{2} \) |
| 83 | \( 1 - 9.07T + 83T^{2} \) |
| 89 | \( 1 - 15.3iT - 89T^{2} \) |
| 97 | \( 1 + 14.9T + 97T^{2} \) |
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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
−10.24994718925563255928730577089, −9.394232645211362928548210615620, −8.394026820497932386790050152984, −7.07988302790223246056144120900, −6.89046630475133300534130116360, −5.54592286500107858042304277805, −5.01807871026151360302052969832, −3.83298226078509975248661279788, −2.64195822731942701521079276501, −0.865214052208346090838523450650,
0.889344124165243472400658211134, 2.31458868077247472819404367954, 3.94635010344571889670603757365, 4.81733059445413063157327450310, 5.85668004846700883932640289113, 6.23699734120006272994674980479, 7.62464895451334996590191240642, 8.110404819899600024099433735861, 9.310247012749419130747726189770, 10.10055248540965057508858184739
