L(s) = 1 | + (0.866 − 1.5i)5-s + (2.62 − 0.358i)7-s + (−3.67 + 2.12i)11-s + (2.12 + 1.22i)13-s − 1.73·17-s + 2.44i·19-s + (7.34 + 4.24i)23-s + (1 + 1.73i)25-s + (−3.67 + 2.12i)29-s + (6.36 + 3.67i)31-s + (1.73 − 4.24i)35-s + 37-s + (−0.866 + 1.5i)41-s + (−3.5 − 6.06i)43-s + (6.06 + 10.5i)47-s + ⋯ |
L(s) = 1 | + (0.387 − 0.670i)5-s + (0.990 − 0.135i)7-s + (−1.10 + 0.639i)11-s + (0.588 + 0.339i)13-s − 0.420·17-s + 0.561i·19-s + (1.53 + 0.884i)23-s + (0.200 + 0.346i)25-s + (−0.682 + 0.393i)29-s + (1.14 + 0.659i)31-s + (0.292 − 0.717i)35-s + 0.164·37-s + (−0.135 + 0.234i)41-s + (−0.533 − 0.924i)43-s + (0.884 + 1.53i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.952 - 0.305i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.952 - 0.305i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.093321383\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.093321383\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (-2.62 + 0.358i)T \) |
good | 5 | \( 1 + (-0.866 + 1.5i)T + (-2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (3.67 - 2.12i)T + (5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (-2.12 - 1.22i)T + (6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + 1.73T + 17T^{2} \) |
| 19 | \( 1 - 2.44iT - 19T^{2} \) |
| 23 | \( 1 + (-7.34 - 4.24i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (3.67 - 2.12i)T + (14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (-6.36 - 3.67i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 - T + 37T^{2} \) |
| 41 | \( 1 + (0.866 - 1.5i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (3.5 + 6.06i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (-6.06 - 10.5i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 - 53T^{2} \) |
| 59 | \( 1 + (-4.33 + 7.5i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-2.12 + 1.22i)T + (30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-5 + 8.66i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 8.48iT - 71T^{2} \) |
| 73 | \( 1 - 9.79iT - 73T^{2} \) |
| 79 | \( 1 + (2.5 + 4.33i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (6.06 + 10.5i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + 10.3T + 89T^{2} \) |
| 97 | \( 1 + (2.12 - 1.22i)T + (48.5 - 84.0i)T^{2} \) |
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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
−8.984893543271479215569862300951, −8.346742892031484553681414517795, −7.57826227221924695210614458269, −6.86607501071026925700754265455, −5.67398716419400583616996177967, −5.05960415593083677990536019658, −4.48185857612643045688839476607, −3.25199121856420309671050234511, −2.01260358895977304152859967400, −1.18291154028982388960686095866,
0.833529948232097098051280884403, 2.37063616686154375997038311165, 2.87823328155389908151593711960, 4.20115194507462246049905576623, 5.10686019689295075181035015551, 5.77722530990046798383585980366, 6.68055981004230422503274085548, 7.42777558203863289727155070233, 8.441252155463257811158542277791, 8.638624161595955220778431353484