| L(s) = 1 | + 8.89i·5-s + 2.64·7-s + 17.7i·11-s − 2.58·13-s + 25.8i·17-s − 20·19-s − 17.7i·23-s − 54.1·25-s + 11.9i·29-s − 17.1·31-s + 23.5i·35-s − 38·37-s + 15.7i·41-s + 43.4·43-s + 16.9i·47-s + ⋯ |
| L(s) = 1 | + 1.77i·5-s + 0.377·7-s + 1.61i·11-s − 0.198·13-s + 1.52i·17-s − 1.05·19-s − 0.773i·23-s − 2.16·25-s + 0.410i·29-s − 0.553·31-s + 0.672i·35-s − 1.02·37-s + 0.383i·41-s + 1.01·43-s + 0.361i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.577 + 0.816i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.577 + 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.9810555854\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9810555854\) |
| \(L(2)\) |
|
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 | \( 1 - 2.64T \) |
| good | 5 | \( 1 - 8.89iT - 25T^{2} \) |
| 11 | \( 1 - 17.7iT - 121T^{2} \) |
| 13 | \( 1 + 2.58T + 169T^{2} \) |
| 17 | \( 1 - 25.8iT - 289T^{2} \) |
| 19 | \( 1 + 20T + 361T^{2} \) |
| 23 | \( 1 + 17.7iT - 529T^{2} \) |
| 29 | \( 1 - 11.9iT - 841T^{2} \) |
| 31 | \( 1 + 17.1T + 961T^{2} \) |
| 37 | \( 1 + 38T + 1.36e3T^{2} \) |
| 41 | \( 1 - 15.7iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 43.4T + 1.84e3T^{2} \) |
| 47 | \( 1 - 16.9iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 85.5iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 1.64iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 100.T + 3.72e3T^{2} \) |
| 67 | \( 1 + 36.6T + 4.48e3T^{2} \) |
| 71 | \( 1 + 17.7iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 28.9T + 5.32e3T^{2} \) |
| 79 | \( 1 - 118.T + 6.24e3T^{2} \) |
| 83 | \( 1 + 120. iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 139. iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 44.4T + 9.40e3T^{2} \) |
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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
−8.673355054531705477909725082230, −7.80034985472263002905238462136, −7.26956779150637533733651077691, −6.55304720707209999556632008973, −6.11541086999772814026651370118, −4.90941232164210381822519433620, −4.12183189517969045943921005133, −3.36067065923157977044311261685, −2.23964727222917821997494584481, −1.85837865663615916081071298396,
0.22369898245173927290370299681, 0.921733791934033855527297296176, 1.94480625124304014274870119777, 3.13679968233974593969398506987, 4.13735321757102972119277301514, 4.81883338750728410551201701511, 5.52201177982273899771831873180, 6.01835964659609836451912685583, 7.25914574357663174227884243007, 7.941229383533728661491279281081
