L(s) = 1 | − 0.239·2-s + (1.09 − 1.34i)3-s − 1.94·4-s + (0.590 + 1.02i)5-s + (−0.260 + 0.321i)6-s + 0.942·8-s + (−0.619 − 2.93i)9-s + (−0.141 − 0.244i)10-s + (1.85 − 3.20i)11-s + (−2.11 + 2.61i)12-s + (0.5 − 0.866i)13-s + (2.02 + 0.321i)15-s + 3.66·16-s + (−3.47 − 6.01i)17-s + (0.148 + 0.701i)18-s + (0.971 − 1.68i)19-s + ⋯ |
L(s) = 1 | − 0.169·2-s + (0.629 − 0.776i)3-s − 0.971·4-s + (0.264 + 0.457i)5-s + (−0.106 + 0.131i)6-s + 0.333·8-s + (−0.206 − 0.978i)9-s + (−0.0446 − 0.0774i)10-s + (0.558 − 0.967i)11-s + (−0.611 + 0.754i)12-s + (0.138 − 0.240i)13-s + (0.522 + 0.0830i)15-s + 0.915·16-s + (−0.841 − 1.45i)17-s + (0.0349 + 0.165i)18-s + (0.222 − 0.385i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.112 + 0.993i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.112 + 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.944231 - 0.843602i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.944231 - 0.843602i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-1.09 + 1.34i)T \) |
| 7 | \( 1 \) |
good | 2 | \( 1 + 0.239T + 2T^{2} \) |
| 5 | \( 1 + (-0.590 - 1.02i)T + (-2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (-1.85 + 3.20i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (-0.5 + 0.866i)T + (-6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + (3.47 + 6.01i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-0.971 + 1.68i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-2.80 - 4.85i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (0.119 + 0.207i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + 1.66T + 31T^{2} \) |
| 37 | \( 1 + (-4.77 + 8.26i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (5.09 - 8.81i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (1.11 + 1.92i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + 5.82T + 47T^{2} \) |
| 53 | \( 1 + (-5.80 - 10.0i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + 2.60T + 59T^{2} \) |
| 61 | \( 1 - 7.60T + 61T^{2} \) |
| 67 | \( 1 - 3.50T + 67T^{2} \) |
| 71 | \( 1 - 8.60T + 71T^{2} \) |
| 73 | \( 1 + (-7.57 - 13.1i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 - 7.37T + 79T^{2} \) |
| 83 | \( 1 + (3.47 + 6.01i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (-1.37 + 2.37i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-3.58 - 6.20i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−10.98778650013616277055452982355, −9.616800903838214841710037955482, −9.117339039489341182055169137362, −8.291396895544287323128136860353, −7.30124706600830106042133006169, −6.38091920192777813985259009833, −5.19415420416481064096068118927, −3.75760701712055102898928713463, −2.70002845892550777932718434439, −0.873394054344896180135246586070,
1.80334589498930851154051725232, 3.63589354217713771305739119544, 4.46850391429830045957973243690, 5.24434896010162046004734743416, 6.71805888634145325840861049414, 8.162664048376671333225334130436, 8.706570953738336272696781125344, 9.485837872083985142891719841580, 10.12392992174525061398172569478, 11.03983816277787099060980677099