L(s) = 1 | + (1.10 + 0.880i)2-s + (0.450 + 1.94i)4-s + (−2.03 − 1.17i)5-s + (−2.03 + 1.69i)7-s + (−1.21 + 2.55i)8-s + (−1.21 − 3.08i)10-s + (−2.18 + 1.26i)11-s − 1.48i·13-s + (−3.74 + 0.0828i)14-s + (−3.59 + 1.75i)16-s + (−1.66 + 0.958i)17-s + (0.454 − 0.786i)19-s + (1.36 − 4.48i)20-s + (−3.53 − 0.526i)22-s + (−4.55 − 2.63i)23-s + ⋯ |
L(s) = 1 | + (0.782 + 0.622i)2-s + (0.225 + 0.974i)4-s + (−0.908 − 0.524i)5-s + (−0.768 + 0.639i)7-s + (−0.429 + 0.902i)8-s + (−0.384 − 0.976i)10-s + (−0.659 + 0.380i)11-s − 0.411i·13-s + (−0.999 + 0.0221i)14-s + (−0.898 + 0.439i)16-s + (−0.402 + 0.232i)17-s + (0.104 − 0.180i)19-s + (0.306 − 1.00i)20-s + (−0.753 − 0.112i)22-s + (−0.950 − 0.548i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.963 + 0.267i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.963 + 0.267i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.0923019 - 0.677306i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0923019 - 0.677306i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-1.10 - 0.880i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (2.03 - 1.69i)T \) |
good | 5 | \( 1 + (2.03 + 1.17i)T + (2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (2.18 - 1.26i)T + (5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + 1.48iT - 13T^{2} \) |
| 17 | \( 1 + (1.66 - 0.958i)T + (8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-0.454 + 0.786i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (4.55 + 2.63i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + 4.85T + 29T^{2} \) |
| 31 | \( 1 + (-3.77 - 6.53i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (4.63 - 8.03i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + 12.0iT - 41T^{2} \) |
| 43 | \( 1 - 10.9iT - 43T^{2} \) |
| 47 | \( 1 + (2.04 - 3.54i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-2.83 - 4.91i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-3.98 - 6.90i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-5.72 - 3.30i)T + (30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-1.36 + 0.790i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 - 0.670iT - 71T^{2} \) |
| 73 | \( 1 + (-2.49 + 1.44i)T + (36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-13.5 - 7.81i)T + (39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 - 6.82T + 83T^{2} \) |
| 89 | \( 1 + (3.23 + 1.86i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + 14.0iT - 97T^{2} \) |
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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
−10.98188475496194865520538235567, −9.908034437077806525583282446321, −8.709227149151027608273997731735, −8.205864950865484555247992506364, −7.27886823304908589884588162807, −6.36450710668137336414255291496, −5.41454967988087335009982068468, −4.54261116270745611731514593950, −3.56676923701789302153914910092, −2.49558924668215048320806800381,
0.24883366170221445961734791196, 2.26238877505745540210778232585, 3.54965628555989265461902920283, 3.93365207060359615174004471476, 5.24957323385140626722562484403, 6.30242305402462973090432635675, 7.12852258871575649755805621219, 7.970799348457948144216873719396, 9.371053994035118269895511677090, 10.08632576773046428501726436208