L(s) = 1 | + (1.61 − 1.28i)2-s + (0.505 − 2.21i)4-s + (3.93 − 1.89i)5-s + (−2.02 − 1.70i)7-s + (−0.245 − 0.509i)8-s + (3.91 − 8.12i)10-s + (−3.17 + 2.53i)11-s + (−1.07 + 0.858i)13-s + (−5.46 − 0.147i)14-s + (3.04 + 1.46i)16-s + (−0.000551 − 0.00241i)17-s + 4.84i·19-s + (−2.20 − 9.67i)20-s + (−1.87 + 8.19i)22-s + (−1.08 − 0.246i)23-s + ⋯ |
L(s) = 1 | + (1.14 − 0.911i)2-s + (0.252 − 1.10i)4-s + (1.75 − 0.846i)5-s + (−0.764 − 0.644i)7-s + (−0.0867 − 0.180i)8-s + (1.23 − 2.56i)10-s + (−0.958 + 0.764i)11-s + (−0.298 + 0.238i)13-s + (−1.46 − 0.0395i)14-s + (0.760 + 0.366i)16-s + (−0.000133 − 0.000585i)17-s + 1.11i·19-s + (−0.493 − 2.16i)20-s + (−0.398 + 1.74i)22-s + (−0.225 − 0.0514i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0164 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0164 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.05908 - 2.02550i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.05908 - 2.02550i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 + (2.02 + 1.70i)T \) |
good | 2 | \( 1 + (-1.61 + 1.28i)T + (0.445 - 1.94i)T^{2} \) |
| 5 | \( 1 + (-3.93 + 1.89i)T + (3.11 - 3.90i)T^{2} \) |
| 11 | \( 1 + (3.17 - 2.53i)T + (2.44 - 10.7i)T^{2} \) |
| 13 | \( 1 + (1.07 - 0.858i)T + (2.89 - 12.6i)T^{2} \) |
| 17 | \( 1 + (0.000551 + 0.00241i)T + (-15.3 + 7.37i)T^{2} \) |
| 19 | \( 1 - 4.84iT - 19T^{2} \) |
| 23 | \( 1 + (1.08 + 0.246i)T + (20.7 + 9.97i)T^{2} \) |
| 29 | \( 1 + (1.58 - 0.361i)T + (26.1 - 12.5i)T^{2} \) |
| 31 | \( 1 - 3.23iT - 31T^{2} \) |
| 37 | \( 1 + (2.10 + 9.23i)T + (-33.3 + 16.0i)T^{2} \) |
| 41 | \( 1 + (7.61 - 3.66i)T + (25.5 - 32.0i)T^{2} \) |
| 43 | \( 1 + (0.421 + 0.203i)T + (26.8 + 33.6i)T^{2} \) |
| 47 | \( 1 + (-1.51 - 1.89i)T + (-10.4 + 45.8i)T^{2} \) |
| 53 | \( 1 + (6.81 + 1.55i)T + (47.7 + 22.9i)T^{2} \) |
| 59 | \( 1 + (-7.40 - 3.56i)T + (36.7 + 46.1i)T^{2} \) |
| 61 | \( 1 + (-14.5 + 3.32i)T + (54.9 - 26.4i)T^{2} \) |
| 67 | \( 1 - 11.7T + 67T^{2} \) |
| 71 | \( 1 + (-4.38 - 1.00i)T + (63.9 + 30.8i)T^{2} \) |
| 73 | \( 1 + (4.34 + 3.46i)T + (16.2 + 71.1i)T^{2} \) |
| 79 | \( 1 + 10.0T + 79T^{2} \) |
| 83 | \( 1 + (10.4 - 13.1i)T + (-18.4 - 80.9i)T^{2} \) |
| 89 | \( 1 + (3.92 - 4.92i)T + (-19.8 - 86.7i)T^{2} \) |
| 97 | \( 1 + 2.91iT - 97T^{2} \) |
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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
−10.81182821164370039470901815615, −10.01754207561275236633983739597, −9.692202828716651038413721946472, −8.301823395146386287801501492511, −6.82543044286158910679937371233, −5.67575770723658312317318180371, −5.10791219035899597233306369732, −3.99978934383725845175700116799, −2.58659097596440301287425577817, −1.63915497767788662789588539553,
2.47761142821073376917615214951, 3.27026924747348996413942690052, 5.16235643209585241070211234680, 5.63307881511744542121458001561, 6.46004634621888468480288889293, 7.07051421057813576556627943928, 8.527274146504520270438733397347, 9.753439580982789704540766857699, 10.22712515997052665659523474849, 11.42086835466664193322954051776