L(s) = 1 | + (0.866 + 0.5i)2-s + (−0.866 − 1.5i)3-s + (0.499 + 0.866i)4-s + (1.73 − 3i)5-s − 1.73i·6-s + (2 + 1.73i)7-s + 0.999i·8-s + (−1.5 + 2.59i)9-s + (3 − 1.73i)10-s + (0.866 − 0.5i)11-s + (0.866 − 1.49i)12-s − 3.46i·13-s + (0.866 + 2.5i)14-s − 6·15-s + (−0.5 + 0.866i)16-s + (−0.866 − 1.5i)17-s + ⋯ |
L(s) = 1 | + (0.612 + 0.353i)2-s + (−0.499 − 0.866i)3-s + (0.249 + 0.433i)4-s + (0.774 − 1.34i)5-s − 0.707i·6-s + (0.755 + 0.654i)7-s + 0.353i·8-s + (−0.5 + 0.866i)9-s + (0.948 − 0.547i)10-s + (0.261 − 0.150i)11-s + (0.250 − 0.433i)12-s − 0.960i·13-s + (0.231 + 0.668i)14-s − 1.54·15-s + (−0.125 + 0.216i)16-s + (−0.210 − 0.363i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.667 + 0.744i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.667 + 0.744i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.82187 - 0.814120i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.82187 - 0.814120i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.866 - 0.5i)T \) |
| 3 | \( 1 + (0.866 + 1.5i)T \) |
| 7 | \( 1 + (-2 - 1.73i)T \) |
| 11 | \( 1 + (-0.866 + 0.5i)T \) |
good | 5 | \( 1 + (-1.73 + 3i)T + (-2.5 - 4.33i)T^{2} \) |
| 13 | \( 1 + 3.46iT - 13T^{2} \) |
| 17 | \( 1 + (0.866 + 1.5i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (1.5 + 0.866i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-2.59 - 1.5i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + 3iT - 29T^{2} \) |
| 31 | \( 1 + (-3 + 1.73i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (3.5 - 6.06i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + 6.92T + 41T^{2} \) |
| 43 | \( 1 - T + 43T^{2} \) |
| 47 | \( 1 + (0.866 - 1.5i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-6.06 - 10.5i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-9 - 5.19i)T + (30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-8 - 13.8i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + 15iT - 71T^{2} \) |
| 73 | \( 1 + (12 - 6.92i)T + (36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (4 - 6.92i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + 17.3T + 83T^{2} \) |
| 89 | \( 1 + (5.19 - 9i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + 15.5iT - 97T^{2} \) |
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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
−11.39135174453451074882207438870, −10.05950086594889462282749100532, −8.682612786787120031946021037168, −8.316742650068183773271706853247, −7.11023929521331669129939663700, −5.91390324463607909486627665871, −5.38032588545960324396405292415, −4.62258253235883166671872217703, −2.54817334375893223770278762989, −1.25485179781589831187800653972,
1.93242226107785101022012739884, 3.37302914381018434139894033212, 4.33412679702544244760459956502, 5.33802346703479719624049578417, 6.48539917634026910444066643953, 6.98815092020727939280786477730, 8.710549140403202264873176760769, 9.869621554511272484706868965951, 10.43888782961734727242794869755, 11.10773251716136037417059808331