L(s) = 1 | + (0.190 − 0.330i)2-s + (0.927 + 1.60i)4-s − 0.381·5-s + (−0.5 − 0.866i)7-s + 1.47·8-s + (−0.0729 + 0.126i)10-s + (−2.42 + 4.20i)11-s + (−2.5 + 2.59i)13-s − 0.381·14-s + (−1.57 + 2.72i)16-s + (3.73 + 6.47i)17-s + (−2.42 − 4.20i)19-s + (−0.354 − 0.613i)20-s + (0.927 + 1.60i)22-s + (2.23 − 3.87i)23-s + ⋯ |
L(s) = 1 | + (0.135 − 0.233i)2-s + (0.463 + 0.802i)4-s − 0.170·5-s + (−0.188 − 0.327i)7-s + 0.520·8-s + (−0.0230 + 0.0399i)10-s + (−0.731 + 1.26i)11-s + (−0.693 + 0.720i)13-s − 0.102·14-s + (−0.393 + 0.681i)16-s + (0.906 + 1.56i)17-s + (−0.556 − 0.964i)19-s + (−0.0791 − 0.137i)20-s + (0.197 + 0.342i)22-s + (0.466 − 0.807i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0128 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0128 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.998608 + 1.01149i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.998608 + 1.01149i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 + (0.5 + 0.866i)T \) |
| 13 | \( 1 + (2.5 - 2.59i)T \) |
good | 2 | \( 1 + (-0.190 + 0.330i)T + (-1 - 1.73i)T^{2} \) |
| 5 | \( 1 + 0.381T + 5T^{2} \) |
| 11 | \( 1 + (2.42 - 4.20i)T + (-5.5 - 9.52i)T^{2} \) |
| 17 | \( 1 + (-3.73 - 6.47i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (2.42 + 4.20i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-2.23 + 3.87i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (2.04 - 3.54i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 - 8.70T + 31T^{2} \) |
| 37 | \( 1 + (2 - 3.46i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-2.61 + 4.53i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (-3.78 - 6.54i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + 2.23T + 47T^{2} \) |
| 53 | \( 1 + 8.23T + 53T^{2} \) |
| 59 | \( 1 + (-1.11 - 1.93i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-3 - 5.19i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (0.354 - 0.613i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (-4.09 - 7.08i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + 2T + 73T^{2} \) |
| 79 | \( 1 - 4T + 79T^{2} \) |
| 83 | \( 1 - 6.70T + 83T^{2} \) |
| 89 | \( 1 + (-8.04 + 13.9i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (6.07 + 10.5i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−10.45110438994788950830886850223, −9.824694925727353730363959051922, −8.609432880264323150680285941877, −7.78293430802580175255287716455, −7.12577117877480719011546405196, −6.30265891331899111803496247732, −4.76644028616231424088351591976, −4.12717149339415676035531642755, −2.88839007343481812204518329279, −1.89232333274381743998561965559,
0.64321717806939043957696105151, 2.39181384031773604804189766146, 3.38013031077345341789765706428, 4.99795370004493506715759759109, 5.60194763657191085290964024000, 6.33743520295249215975961340096, 7.59407467048077706020535351884, 8.012866776448582930040988234484, 9.411136417832784912572796776136, 9.995202407898850301355382269874