L(s) = 1 | + (0.866 + 0.5i)2-s + (1.41 + i)3-s + (0.499 + 0.866i)4-s + (0.724 + 1.57i)6-s + (0.389 + 0.224i)7-s + 0.999i·8-s + (1.00 + 2.82i)9-s + (2.44 − 4.24i)11-s + (−0.158 + 1.72i)12-s + (0.389 − 0.224i)13-s + (0.224 + 0.389i)14-s + (−0.5 + 0.866i)16-s + 4.89i·17-s + (−0.548 + 2.94i)18-s − 7.44·19-s + ⋯ |
L(s) = 1 | + (0.612 + 0.353i)2-s + (0.816 + 0.577i)3-s + (0.249 + 0.433i)4-s + (0.295 + 0.642i)6-s + (0.147 + 0.0849i)7-s + 0.353i·8-s + (0.333 + 0.942i)9-s + (0.738 − 1.27i)11-s + (−0.0458 + 0.497i)12-s + (0.107 − 0.0623i)13-s + (0.0600 + 0.104i)14-s + (−0.125 + 0.216i)16-s + 1.18i·17-s + (−0.129 + 0.695i)18-s − 1.70·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.406 - 0.913i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.406 - 0.913i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.14893 + 1.39650i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.14893 + 1.39650i\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (-1.41 - i)T \) |
| 5 | \( 1 \) |
good | 7 | \( 1 + (-0.389 - 0.224i)T + (3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (-2.44 + 4.24i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (-0.389 + 0.224i)T + (6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 - 4.89iT - 17T^{2} \) |
| 19 | \( 1 + 7.44T + 19T^{2} \) |
| 23 | \( 1 + (-2.12 + 1.22i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (1.22 - 2.12i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (2.22 + 3.85i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + 11.3iT - 37T^{2} \) |
| 41 | \( 1 + (4.5 + 7.79i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-2.20 - 1.27i)T + (21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (-9.43 - 5.44i)T + (23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 - 3.55iT - 53T^{2} \) |
| 59 | \( 1 + (2.72 + 4.71i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (4 - 6.92i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-0.301 + 0.174i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 - 13.3T + 71T^{2} \) |
| 73 | \( 1 - iT - 73T^{2} \) |
| 79 | \( 1 + (-8.34 + 14.4i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (4.71 + 2.72i)T + (41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 - 9T + 89T^{2} \) |
| 97 | \( 1 + (7.61 + 4.39i)T + (48.5 + 84.0i)T^{2} \) |
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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
−10.95342318012605137814596465760, −10.67547695296741109331837042287, −9.075445770982452276529562319335, −8.645129430410929377166458770460, −7.71100329850388044243923866260, −6.44605692748288343963793268720, −5.55557151973751445746148560017, −4.18290591960772263574508911061, −3.59583320787432508317988168847, −2.16337020735023492830108763976,
1.55997885681308033660358949425, 2.67407650321425084919376402984, 3.96457159041005449739731872180, 4.87666415536503729289740062123, 6.50077322079107699869411674076, 7.02762341737638020368383076240, 8.174315988342544859197082951166, 9.231460734051338649750693111317, 9.918114618185545332063688787847, 11.11546399189281758558175898594