L(s) = 1 | + (−0.5 − 1.65i)3-s + (0.5 − 0.866i)5-s + (1.68 + 2.92i)7-s + (−2.5 + 1.65i)9-s + (−2.18 − 3.78i)11-s + (3.37 − 5.84i)13-s + (−1.68 − 0.396i)15-s − 1.62·17-s + 2.37·19-s + (4 − 4.25i)21-s + (0.686 − 1.18i)23-s + (−0.499 − 0.866i)25-s + (4 + 3.31i)27-s + (−0.686 − 1.18i)29-s + (2.37 − 4.10i)31-s + ⋯ |
L(s) = 1 | + (−0.288 − 0.957i)3-s + (0.223 − 0.387i)5-s + (0.637 + 1.10i)7-s + (−0.833 + 0.552i)9-s + (−0.659 − 1.14i)11-s + (0.935 − 1.61i)13-s + (−0.435 − 0.102i)15-s − 0.394·17-s + 0.544·19-s + (0.872 − 0.928i)21-s + (0.143 − 0.247i)23-s + (−0.0999 − 0.173i)25-s + (0.769 + 0.638i)27-s + (−0.127 − 0.220i)29-s + (0.426 − 0.737i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.283 + 0.959i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.283 + 0.959i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.799200 - 1.06914i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.799200 - 1.06914i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (0.5 + 1.65i)T \) |
| 5 | \( 1 + (-0.5 + 0.866i)T \) |
good | 7 | \( 1 + (-1.68 - 2.92i)T + (-3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (2.18 + 3.78i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (-3.37 + 5.84i)T + (-6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + 1.62T + 17T^{2} \) |
| 19 | \( 1 - 2.37T + 19T^{2} \) |
| 23 | \( 1 + (-0.686 + 1.18i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (0.686 + 1.18i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (-2.37 + 4.10i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + 4T + 37T^{2} \) |
| 41 | \( 1 + (-1.5 + 2.59i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (2.81 + 4.87i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (3.68 + 6.38i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + 11.4T + 53T^{2} \) |
| 59 | \( 1 + (-2.18 + 3.78i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-4.05 - 7.02i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (3.5 - 6.06i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 - 6T + 71T^{2} \) |
| 73 | \( 1 - 3.11T + 73T^{2} \) |
| 79 | \( 1 + (-1 - 1.73i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-3.68 - 6.38i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 - 16.1T + 89T^{2} \) |
| 97 | \( 1 + (-4.18 - 7.25i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−10.38237882523974941432845660824, −8.939802449387705163626122805743, −8.296947592819329347751373051126, −7.86143608382591532912354156359, −6.42181100335918809527847325685, −5.55820279733234683070623559558, −5.24934320231587197705396059503, −3.25853268780041647815861755521, −2.19985562581886842298166770815, −0.74678151871686385370391301464,
1.67968990075918061043865538968, 3.33608604005334151096161475763, 4.42012588764006285357093255135, 4.90441204968841974734233093047, 6.31064795445806585845113464698, 7.07029882326073477280732772888, 8.092013707512517075329414430753, 9.230408248345664959414744372672, 9.851211446780944221842307233630, 10.76798313301803499142644018688