L(s) = 1 | + (−0.920 − 0.733i)2-s + (−0.136 − 0.599i)4-s + (1.46 + 0.703i)5-s + (−2.01 − 1.70i)7-s + (−1.33 + 2.77i)8-s + (−0.828 − 1.71i)10-s + (−2.35 − 1.87i)11-s + (−2.44 − 1.94i)13-s + (0.604 + 3.05i)14-s + (2.15 − 1.03i)16-s + (−0.882 + 3.86i)17-s − 6.34i·19-s + (0.221 − 0.971i)20-s + (0.788 + 3.45i)22-s + (1.23 − 0.280i)23-s + ⋯ |
L(s) = 1 | + (−0.650 − 0.518i)2-s + (−0.0683 − 0.299i)4-s + (0.653 + 0.314i)5-s + (−0.763 − 0.646i)7-s + (−0.472 + 0.980i)8-s + (−0.261 − 0.543i)10-s + (−0.710 − 0.566i)11-s + (−0.676 − 0.539i)13-s + (0.161 + 0.816i)14-s + (0.538 − 0.259i)16-s + (−0.214 + 0.938i)17-s − 1.45i·19-s + (0.0495 − 0.217i)20-s + (0.168 + 0.736i)22-s + (0.256 − 0.0585i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 - 0.0198i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.999 - 0.0198i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.00444918 + 0.447985i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.00444918 + 0.447985i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 + (2.01 + 1.70i)T \) |
good | 2 | \( 1 + (0.920 + 0.733i)T + (0.445 + 1.94i)T^{2} \) |
| 5 | \( 1 + (-1.46 - 0.703i)T + (3.11 + 3.90i)T^{2} \) |
| 11 | \( 1 + (2.35 + 1.87i)T + (2.44 + 10.7i)T^{2} \) |
| 13 | \( 1 + (2.44 + 1.94i)T + (2.89 + 12.6i)T^{2} \) |
| 17 | \( 1 + (0.882 - 3.86i)T + (-15.3 - 7.37i)T^{2} \) |
| 19 | \( 1 + 6.34iT - 19T^{2} \) |
| 23 | \( 1 + (-1.23 + 0.280i)T + (20.7 - 9.97i)T^{2} \) |
| 29 | \( 1 + (7.25 + 1.65i)T + (26.1 + 12.5i)T^{2} \) |
| 31 | \( 1 - 1.49iT - 31T^{2} \) |
| 37 | \( 1 + (1.72 - 7.54i)T + (-33.3 - 16.0i)T^{2} \) |
| 41 | \( 1 + (9.09 + 4.37i)T + (25.5 + 32.0i)T^{2} \) |
| 43 | \( 1 + (0.978 - 0.471i)T + (26.8 - 33.6i)T^{2} \) |
| 47 | \( 1 + (-5.44 + 6.82i)T + (-10.4 - 45.8i)T^{2} \) |
| 53 | \( 1 + (0.104 - 0.0237i)T + (47.7 - 22.9i)T^{2} \) |
| 59 | \( 1 + (1.54 - 0.746i)T + (36.7 - 46.1i)T^{2} \) |
| 61 | \( 1 + (10.5 + 2.41i)T + (54.9 + 26.4i)T^{2} \) |
| 67 | \( 1 - 11.0T + 67T^{2} \) |
| 71 | \( 1 + (3.29 - 0.752i)T + (63.9 - 30.8i)T^{2} \) |
| 73 | \( 1 + (-8.80 + 7.01i)T + (16.2 - 71.1i)T^{2} \) |
| 79 | \( 1 - 14.5T + 79T^{2} \) |
| 83 | \( 1 + (2.15 + 2.70i)T + (-18.4 + 80.9i)T^{2} \) |
| 89 | \( 1 + (-7.65 - 9.59i)T + (-19.8 + 86.7i)T^{2} \) |
| 97 | \( 1 + 6.40iT - 97T^{2} \) |
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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
−10.50884302608201003058006677143, −9.953311531857598544244184980666, −9.133860566410163254312556141562, −8.140509488658543322443497993740, −6.92029699115032819595445813053, −5.95471454525985609822662973260, −4.98158236194811280928560731991, −3.25473790716624353559720578468, −2.14503407282848752967586183448, −0.32297536722017209841099851023,
2.18713574968892956486618996177, 3.56991201274485740567842163224, 5.09973974674906457927412267606, 6.07255393004165609124111616768, 7.13724349519274850869013840386, 7.85704028987633559105069906194, 9.130218012039882114174776265313, 9.441612669297400074589052049997, 10.24036937453144759145156365096, 11.72325723875655263487841104272