L(s) = 1 | + (1 − 1.41i)3-s + (0.5 + 0.866i)5-s + (0.724 − 1.25i)7-s + (−1.00 − 2.82i)9-s + (1.72 − 2.98i)11-s + (1.94 + 3.37i)13-s + (1.72 + 0.158i)15-s − 4.89·17-s + 4·19-s + (−1.05 − 2.28i)21-s + (0.275 + 0.476i)23-s + (2 − 3.46i)25-s + (−5.00 − 1.41i)27-s + (−4.94 + 8.57i)29-s + (−3.72 − 6.45i)31-s + ⋯ |
L(s) = 1 | + (0.577 − 0.816i)3-s + (0.223 + 0.387i)5-s + (0.273 − 0.474i)7-s + (−0.333 − 0.942i)9-s + (0.520 − 0.900i)11-s + (0.540 + 0.936i)13-s + (0.445 + 0.0410i)15-s − 1.18·17-s + 0.917·19-s + (−0.229 − 0.497i)21-s + (0.0573 + 0.0994i)23-s + (0.400 − 0.692i)25-s + (−0.962 − 0.272i)27-s + (−0.919 + 1.59i)29-s + (−0.668 − 1.15i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.635 + 0.771i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.635 + 0.771i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.46730 - 0.692474i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.46730 - 0.692474i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-1 + 1.41i)T \) |
good | 5 | \( 1 + (-0.5 - 0.866i)T + (-2.5 + 4.33i)T^{2} \) |
| 7 | \( 1 + (-0.724 + 1.25i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (-1.72 + 2.98i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (-1.94 - 3.37i)T + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + 4.89T + 17T^{2} \) |
| 19 | \( 1 - 4T + 19T^{2} \) |
| 23 | \( 1 + (-0.275 - 0.476i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (4.94 - 8.57i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (3.72 + 6.45i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 - 8.89T + 37T^{2} \) |
| 41 | \( 1 + (-1.05 - 1.81i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (6.17 - 10.6i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (4.17 - 7.22i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + 0.898T + 53T^{2} \) |
| 59 | \( 1 + (-0.174 - 0.301i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (0.949 - 1.64i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-1.17 - 2.03i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + 11.7T + 71T^{2} \) |
| 73 | \( 1 - 4.89T + 73T^{2} \) |
| 79 | \( 1 + (4.27 - 7.40i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-2.72 + 4.71i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + 3.10T + 89T^{2} \) |
| 97 | \( 1 + (2.94 - 5.10i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−11.40942559765661940517177777433, −11.14877795275595989431313305343, −9.516153872317038295425137179922, −8.829647466751290307177146738037, −7.76080480194947032409279009545, −6.78469894815488152792174459391, −6.03760891892484447331283984319, −4.22314448988996876319154296437, −2.97200087401570772926206173816, −1.43111160062968314284468391098,
2.07552688324018637090115211045, 3.57442300361465014986418461469, 4.77378150722055495741520014556, 5.66970187538746219855186958130, 7.22753108179000511519004744035, 8.384599591567832178479766498476, 9.127414246479911924559155321964, 9.889367846441103405604076553281, 10.92885317127161543234462025008, 11.82782605066980285517700872194