| L(s) = 1 | + (−1.27 + 1.17i)3-s + (0.0952 − 0.0346i)5-s + (−0.165 − 0.935i)7-s + (0.240 − 2.99i)9-s + (3.80 + 1.38i)11-s + (3.80 + 3.18i)13-s + (−0.0805 + 0.156i)15-s + (−3.11 + 5.39i)17-s + (0.514 + 0.891i)19-s + (1.30 + 0.997i)21-s + (−0.602 + 3.41i)23-s + (−3.82 + 3.20i)25-s + (3.20 + 4.08i)27-s + (−3.04 + 2.55i)29-s + (0.740 − 4.20i)31-s + ⋯ |
| L(s) = 1 | + (−0.734 + 0.678i)3-s + (0.0426 − 0.0155i)5-s + (−0.0623 − 0.353i)7-s + (0.0800 − 0.996i)9-s + (1.14 + 0.417i)11-s + (1.05 + 0.884i)13-s + (−0.0207 + 0.0402i)15-s + (−0.755 + 1.30i)17-s + (0.118 + 0.204i)19-s + (0.285 + 0.217i)21-s + (−0.125 + 0.712i)23-s + (−0.764 + 0.641i)25-s + (0.617 + 0.786i)27-s + (−0.566 + 0.475i)29-s + (0.133 − 0.754i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.318 - 0.948i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.318 - 0.948i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.898806 + 0.646406i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.898806 + 0.646406i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (1.27 - 1.17i)T \) |
| good | 5 | \( 1 + (-0.0952 + 0.0346i)T + (3.83 - 3.21i)T^{2} \) |
| 7 | \( 1 + (0.165 + 0.935i)T + (-6.57 + 2.39i)T^{2} \) |
| 11 | \( 1 + (-3.80 - 1.38i)T + (8.42 + 7.07i)T^{2} \) |
| 13 | \( 1 + (-3.80 - 3.18i)T + (2.25 + 12.8i)T^{2} \) |
| 17 | \( 1 + (3.11 - 5.39i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-0.514 - 0.891i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (0.602 - 3.41i)T + (-21.6 - 7.86i)T^{2} \) |
| 29 | \( 1 + (3.04 - 2.55i)T + (5.03 - 28.5i)T^{2} \) |
| 31 | \( 1 + (-0.740 + 4.20i)T + (-29.1 - 10.6i)T^{2} \) |
| 37 | \( 1 + (-4.19 + 7.26i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-2.15 - 1.80i)T + (7.11 + 40.3i)T^{2} \) |
| 43 | \( 1 + (-6.32 - 2.30i)T + (32.9 + 27.6i)T^{2} \) |
| 47 | \( 1 + (-1.35 - 7.70i)T + (-44.1 + 16.0i)T^{2} \) |
| 53 | \( 1 - 12.9T + 53T^{2} \) |
| 59 | \( 1 + (13.3 - 4.87i)T + (45.1 - 37.9i)T^{2} \) |
| 61 | \( 1 + (0.469 + 2.66i)T + (-57.3 + 20.8i)T^{2} \) |
| 67 | \( 1 + (-2.53 - 2.12i)T + (11.6 + 65.9i)T^{2} \) |
| 71 | \( 1 + (-7.26 + 12.5i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (5.82 + 10.0i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-0.981 + 0.823i)T + (13.7 - 77.7i)T^{2} \) |
| 83 | \( 1 + (4.02 - 3.37i)T + (14.4 - 81.7i)T^{2} \) |
| 89 | \( 1 + (4.80 + 8.32i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (3.70 + 1.34i)T + (74.3 + 62.3i)T^{2} \) |
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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
−11.21521060409307118022039992400, −10.63252967251789099002673422526, −9.397133946179110561011024200586, −9.042276219101878696267483238949, −7.51815752200356563242797822792, −6.38398873155137936649074464101, −5.81375140106108387155239802813, −4.16576427864128705560769318613, −3.89584799879777635669817268491, −1.53305246663911036082134088343,
0.872228767584973094744608470977, 2.54870420950362433695851497245, 4.14438642773871668422906788536, 5.47166802765376189295061344714, 6.25388749317651184493390646313, 7.06184490033569805240135172121, 8.235524669669706540733920943699, 9.043893662922887388426951748904, 10.25031144368483842701473651631, 11.22337745684029170142504977362
