L(s) = 1 | + 3-s − 5-s + (−0.5 + 0.866i)7-s + 9-s + (0.5 − 0.866i)11-s + (−3.5 − 6.06i)13-s − 15-s + (1.5 + 2.59i)17-s + (2.5 + 4.33i)19-s + (−0.5 + 0.866i)21-s + (1.5 + 2.59i)23-s + 25-s + 27-s + (−0.5 + 0.866i)29-s + (−0.5 + 0.866i)31-s + ⋯ |
L(s) = 1 | + 0.577·3-s − 0.447·5-s + (−0.188 + 0.327i)7-s + 0.333·9-s + (0.150 − 0.261i)11-s + (−0.970 − 1.68i)13-s − 0.258·15-s + (0.363 + 0.630i)17-s + (0.573 + 0.993i)19-s + (−0.109 + 0.188i)21-s + (0.312 + 0.541i)23-s + 0.200·25-s + 0.192·27-s + (−0.0928 + 0.160i)29-s + (−0.0898 + 0.155i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4020 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.978 + 0.205i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4020 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.978 + 0.205i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.023280959\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.023280959\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 67 | \( 1 + (-8 + 1.73i)T \) |
good | 7 | \( 1 + (0.5 - 0.866i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (-0.5 + 0.866i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (3.5 + 6.06i)T + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + (-1.5 - 2.59i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-2.5 - 4.33i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-1.5 - 2.59i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (0.5 - 0.866i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (0.5 - 0.866i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-0.5 - 0.866i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-3.5 + 6.06i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 - 8T + 43T^{2} \) |
| 47 | \( 1 + (1.5 - 2.59i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + 2T + 53T^{2} \) |
| 59 | \( 1 - 4T + 59T^{2} \) |
| 61 | \( 1 + (5.5 + 9.52i)T + (-30.5 + 52.8i)T^{2} \) |
| 71 | \( 1 + (-6.5 + 11.2i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (5.5 + 9.52i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-5.5 + 9.52i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-5.5 - 9.52i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 - 2T + 89T^{2} \) |
| 97 | \( 1 + (3.5 + 6.06i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−8.226092511783942496929075882156, −7.77388080323204718278084139125, −7.27369704541936828360470978865, −6.07080960604687865365446256858, −5.51298566459986640142029710113, −4.63128744477594458141505365172, −3.48608425531599299679690638388, −3.16307681379710851525817851492, −2.05005755495356891937845234828, −0.74278018820985881227813629710,
0.841703966956516799376904572713, 2.19695657364563690658098170859, 2.88527970145881767384816990499, 4.00552501085272860820747205290, 4.49590997965160349904598855369, 5.31736138417466182660038346506, 6.61336029544474662903924863331, 7.09321140788080575918386489883, 7.57926087729692550025178697787, 8.523121869379670869261598197449