| L(s) = 1 | − 5.94i·5-s − 2.64i·7-s + 15.0·11-s − 11.2i·13-s + 22.8·17-s + 33.0·19-s + 1.69i·23-s − 10.3·25-s + 28.9i·29-s − 24.7i·31-s − 15.7·35-s + 53.7i·37-s − 30.8·41-s + 44.2·43-s + 37.2i·47-s + ⋯ |
| L(s) = 1 | − 1.18i·5-s − 0.377i·7-s + 1.37·11-s − 0.862i·13-s + 1.34·17-s + 1.74·19-s + 0.0734i·23-s − 0.412·25-s + 0.999i·29-s − 0.797i·31-s − 0.449·35-s + 1.45i·37-s − 0.753·41-s + 1.02·43-s + 0.793i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2016 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.421 + 0.906i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2016 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.421 + 0.906i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(2.643828280\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.643828280\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + 2.64iT \) |
| good | 5 | \( 1 + 5.94iT - 25T^{2} \) |
| 11 | \( 1 - 15.0T + 121T^{2} \) |
| 13 | \( 1 + 11.2iT - 169T^{2} \) |
| 17 | \( 1 - 22.8T + 289T^{2} \) |
| 19 | \( 1 - 33.0T + 361T^{2} \) |
| 23 | \( 1 - 1.69iT - 529T^{2} \) |
| 29 | \( 1 - 28.9iT - 841T^{2} \) |
| 31 | \( 1 + 24.7iT - 961T^{2} \) |
| 37 | \( 1 - 53.7iT - 1.36e3T^{2} \) |
| 41 | \( 1 + 30.8T + 1.68e3T^{2} \) |
| 43 | \( 1 - 44.2T + 1.84e3T^{2} \) |
| 47 | \( 1 - 37.2iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 72.2iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 33.2T + 3.48e3T^{2} \) |
| 61 | \( 1 + 96.8iT - 3.72e3T^{2} \) |
| 67 | \( 1 - 86.0T + 4.48e3T^{2} \) |
| 71 | \( 1 - 40.4iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 28.5T + 5.32e3T^{2} \) |
| 79 | \( 1 + 80.5iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 36.2T + 6.88e3T^{2} \) |
| 89 | \( 1 + 13.9T + 7.92e3T^{2} \) |
| 97 | \( 1 + 66.0T + 9.40e3T^{2} \) |
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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
−8.909340410329548593639233795341, −8.008239390724615056525647405143, −7.45875086472973968590890142698, −6.40433427460767614002929131685, −5.45912592739107982068173708092, −4.91049993549779193345435872592, −3.83462404281486288801520406640, −3.10175481969230219599431964010, −1.29437828986419545767470006248, −0.930291754158093199323701090088,
1.08934282258004654944955165263, 2.27676100906303354470917940721, 3.34881469168604942549727438418, 3.90961643477296324569163229828, 5.24336473629206345817036991246, 6.01329866526141327421564012939, 6.89787067355939566215283539634, 7.28805557691578467273349868170, 8.323441694419740719409330965433, 9.312175959870622876367047018236
