L(s) = 1 | − 2.74·2-s + (−0.682 − 1.18i)3-s + 5.51·4-s + (−0.370 − 0.641i)5-s + (1.87 + 3.23i)6-s − 9.63·8-s + (0.568 − 0.984i)9-s + (1.01 + 1.75i)10-s + (0.682 + 1.18i)11-s + (−3.76 − 6.51i)12-s + (0.301 + 3.59i)13-s + (−0.505 + 0.875i)15-s + 15.3·16-s − 4.14·17-s + (−1.55 + 2.69i)18-s + (−3.63 + 6.29i)19-s + ⋯ |
L(s) = 1 | − 1.93·2-s + (−0.393 − 0.682i)3-s + 2.75·4-s + (−0.165 − 0.287i)5-s + (0.763 + 1.32i)6-s − 3.40·8-s + (0.189 − 0.328i)9-s + (0.321 + 0.556i)10-s + (0.205 + 0.356i)11-s + (−1.08 − 1.88i)12-s + (0.0837 + 0.996i)13-s + (−0.130 + 0.226i)15-s + 3.84·16-s − 1.00·17-s + (−0.367 + 0.636i)18-s + (−0.833 + 1.44i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0575 - 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0575 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.156304 + 0.147546i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.156304 + 0.147546i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 \) |
| 13 | \( 1 + (-0.301 - 3.59i)T \) |
good | 2 | \( 1 + 2.74T + 2T^{2} \) |
| 3 | \( 1 + (0.682 + 1.18i)T + (-1.5 + 2.59i)T^{2} \) |
| 5 | \( 1 + (0.370 + 0.641i)T + (-2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (-0.682 - 1.18i)T + (-5.5 + 9.52i)T^{2} \) |
| 17 | \( 1 + 4.14T + 17T^{2} \) |
| 19 | \( 1 + (3.63 - 6.29i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + 2.33T + 23T^{2} \) |
| 29 | \( 1 + (-0.203 + 0.353i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (-1.38 + 2.40i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + 6.10T + 37T^{2} \) |
| 41 | \( 1 + (0.627 - 1.08i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (-0.870 - 1.50i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (-2.92 - 5.07i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (2.28 - 3.95i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + 10.9T + 59T^{2} \) |
| 61 | \( 1 + (3.26 - 5.65i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-6.87 - 11.9i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (-2.40 - 4.17i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (3.03 - 5.25i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-4.56 - 7.90i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 - 11.7T + 83T^{2} \) |
| 89 | \( 1 + 1.76T + 89T^{2} \) |
| 97 | \( 1 + (4.76 + 8.25i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−10.61093791118923783648366102920, −9.791735803128591836154009341356, −9.000171211902498752006739304994, −8.288274946915704553626773549814, −7.40540709921642098573204160479, −6.58078534147424720324816026727, −6.10585348998826593581116284136, −4.09542559838221610750002603225, −2.23057552458400994297437591355, −1.31389159898318514484699604796,
0.23670754907169003333456472033, 2.04551074346973380959626175496, 3.31271096930131691178799743818, 4.99409751990236933387070868652, 6.26837385517952032204545951631, 7.04688237108837418806253634072, 7.955398155893523658604431535533, 8.808918835116419083420014125592, 9.443063469861906985613485885236, 10.62601786688380032238232963445