| L(s) = 1 | − 1.41i·2-s − 2.00·4-s + 4.24i·5-s + 2.82i·8-s + 6·10-s + 12.7i·11-s + 13-s + 4.00·16-s − 16.9i·17-s − 23·19-s − 8.48i·20-s + 18·22-s + 16.9i·23-s + 7.00·25-s − 1.41i·26-s + ⋯ |
| L(s) = 1 | − 0.707i·2-s − 0.500·4-s + 0.848i·5-s + 0.353i·8-s + 0.600·10-s + 1.15i·11-s + 0.0769·13-s + 0.250·16-s − 0.998i·17-s − 1.21·19-s − 0.424i·20-s + 0.818·22-s + 0.737i·23-s + 0.280·25-s − 0.0543i·26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.816 - 0.577i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.816 - 0.577i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.2726401464\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2726401464\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 1.41iT \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 - 4.24iT - 25T^{2} \) |
| 11 | \( 1 - 12.7iT - 121T^{2} \) |
| 13 | \( 1 - T + 169T^{2} \) |
| 17 | \( 1 + 16.9iT - 289T^{2} \) |
| 19 | \( 1 + 23T + 361T^{2} \) |
| 23 | \( 1 - 16.9iT - 529T^{2} \) |
| 29 | \( 1 + 33.9iT - 841T^{2} \) |
| 31 | \( 1 + 47T + 961T^{2} \) |
| 37 | \( 1 + 55T + 1.36e3T^{2} \) |
| 41 | \( 1 + 46.6iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 23T + 1.84e3T^{2} \) |
| 47 | \( 1 + 4.24iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 50.9iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 84.8iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 104T + 3.72e3T^{2} \) |
| 67 | \( 1 + 97T + 4.48e3T^{2} \) |
| 71 | \( 1 + 97.5iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 65T + 5.32e3T^{2} \) |
| 79 | \( 1 - 113T + 6.24e3T^{2} \) |
| 83 | \( 1 + 29.6iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 135. iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 104T + 9.40e3T^{2} \) |
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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
−10.55453481640419041740529738903, −9.505713386376568673210476808105, −8.910182480217836198499580298705, −7.57262719675420339795077380394, −7.04640739208492592152124389790, −5.89210002556530629649368849919, −4.76948962132770955315423845377, −3.83706515057428053077967521764, −2.72177873511208712586506191645, −1.80454228260531207055942104922,
0.087164595751584953522891153827, 1.54796115841814924399104948350, 3.32598395922472054956012873035, 4.35557174064098322875808167754, 5.29473893408680961965871669588, 6.10542686613735438204102033904, 6.92654382232964623444551127333, 8.176084068298713639108636445033, 8.607746552631514427123925111892, 9.222054175908024045832977661098
