L(s) = 1 | + (1.34 + 2.33i)2-s + (−2.64 + 4.57i)4-s + (−0.794 − 1.37i)5-s + (1.23 + 2.33i)7-s − 8.87·8-s + (2.14 − 3.71i)10-s + (0.150 − 0.260i)11-s + 2.81·13-s + (−3.79 + 6.05i)14-s + (−6.69 − 11.5i)16-s + (2.93 − 5.08i)17-s + (1.14 + 1.98i)19-s + 8.39·20-s + 0.810·22-s + (0.944 + 1.63i)23-s + ⋯ |
L(s) = 1 | + (0.954 + 1.65i)2-s + (−1.32 + 2.28i)4-s + (−0.355 − 0.615i)5-s + (0.468 + 0.883i)7-s − 3.13·8-s + (0.677 − 1.17i)10-s + (0.0452 − 0.0784i)11-s + 0.779·13-s + (−1.01 + 1.61i)14-s + (−1.67 − 2.89i)16-s + (0.712 − 1.23i)17-s + (0.262 + 0.454i)19-s + 1.87·20-s + 0.172·22-s + (0.196 + 0.341i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.811 - 0.583i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.811 - 0.583i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.513720 + 1.59439i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.513720 + 1.59439i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 + (-1.23 - 2.33i)T \) |
good | 2 | \( 1 + (-1.34 - 2.33i)T + (-1 + 1.73i)T^{2} \) |
| 5 | \( 1 + (0.794 + 1.37i)T + (-2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (-0.150 + 0.260i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 - 2.81T + 13T^{2} \) |
| 17 | \( 1 + (-2.93 + 5.08i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-1.14 - 1.98i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-0.944 - 1.63i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + 2.52T + 29T^{2} \) |
| 31 | \( 1 + (-2.40 + 4.16i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-2.23 - 3.87i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + 8.90T + 41T^{2} \) |
| 43 | \( 1 + 9.09T + 43T^{2} \) |
| 47 | \( 1 + (1.60 + 2.78i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-1.00 + 1.74i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (2.44 - 4.23i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (3.78 + 6.56i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (0.356 - 0.616i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 - 12.8T + 71T^{2} \) |
| 73 | \( 1 + (-5.83 + 10.1i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (0.833 + 1.44i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 - 5.43T + 83T^{2} \) |
| 89 | \( 1 + (4.67 + 8.10i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + 12.5T + 97T^{2} \) |
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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
−13.27307347723461686542889933820, −12.18835223749911752231583158147, −11.64102447129663135611859207618, −9.448198817602835847915901480729, −8.433852872728916520660334155356, −7.86419658766660756451435939706, −6.57025391529394888713174948974, −5.47851722756754097897830001174, −4.76684561509377538972656150986, −3.36939575975004998159327043845,
1.44210283564697640358080730706, 3.23232935400843525132708911932, 4.04102450783686135612168721122, 5.28103108177296070175151546898, 6.69762645908457850119297206861, 8.333098547836233402548535942330, 9.768227130883434834230553096197, 10.72392550859315756278631569045, 11.08049947778631818627689567547, 12.10021605066844905513002647246