L(s) = 1 | + (2.18 − 2.05i)3-s + (−2.05 − 1.18i)5-s + (4.05 + 7.02i)7-s + (0.558 − 8.98i)9-s + (−17.6 + 10.1i)11-s + (−3.05 + 5.29i)13-s + (−6.94 + 1.63i)15-s − 17.9i·17-s + 9.11·19-s + (23.3 + 7.02i)21-s + (29.0 + 16.7i)23-s + (−9.67 − 16.7i)25-s + (−17.2 − 20.7i)27-s + (14.4 − 8.31i)29-s + (11.1 − 19.3i)31-s + ⋯ |
L(s) = 1 | + (0.728 − 0.684i)3-s + (−0.411 − 0.237i)5-s + (0.579 + 1.00i)7-s + (0.0620 − 0.998i)9-s + (−1.60 + 0.924i)11-s + (−0.235 + 0.407i)13-s + (−0.462 + 0.108i)15-s − 1.05i·17-s + 0.479·19-s + (1.11 + 0.334i)21-s + (1.26 + 0.729i)23-s + (−0.387 − 0.670i)25-s + (−0.638 − 0.769i)27-s + (0.496 − 0.286i)29-s + (0.360 − 0.624i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 36 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.916 + 0.399i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 36 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.916 + 0.399i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(1.15104 - 0.240019i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.15104 - 0.240019i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-2.18 + 2.05i)T \) |
good | 5 | \( 1 + (2.05 + 1.18i)T + (12.5 + 21.6i)T^{2} \) |
| 7 | \( 1 + (-4.05 - 7.02i)T + (-24.5 + 42.4i)T^{2} \) |
| 11 | \( 1 + (17.6 - 10.1i)T + (60.5 - 104. i)T^{2} \) |
| 13 | \( 1 + (3.05 - 5.29i)T + (-84.5 - 146. i)T^{2} \) |
| 17 | \( 1 + 17.9iT - 289T^{2} \) |
| 19 | \( 1 - 9.11T + 361T^{2} \) |
| 23 | \( 1 + (-29.0 - 16.7i)T + (264.5 + 458. i)T^{2} \) |
| 29 | \( 1 + (-14.4 + 8.31i)T + (420.5 - 728. i)T^{2} \) |
| 31 | \( 1 + (-11.1 + 19.3i)T + (-480.5 - 832. i)T^{2} \) |
| 37 | \( 1 + 50.4T + 1.36e3T^{2} \) |
| 41 | \( 1 + (-29.9 - 17.3i)T + (840.5 + 1.45e3i)T^{2} \) |
| 43 | \( 1 + (11.5 + 19.9i)T + (-924.5 + 1.60e3i)T^{2} \) |
| 47 | \( 1 + (33.1 - 19.1i)T + (1.10e3 - 1.91e3i)T^{2} \) |
| 53 | \( 1 + 19.0iT - 2.80e3T^{2} \) |
| 59 | \( 1 + (2.96 + 1.71i)T + (1.74e3 + 3.01e3i)T^{2} \) |
| 61 | \( 1 + (-23.1 - 40.1i)T + (-1.86e3 + 3.22e3i)T^{2} \) |
| 67 | \( 1 + (-3.14 + 5.45i)T + (-2.24e3 - 3.88e3i)T^{2} \) |
| 71 | \( 1 - 35.9iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 47.3T + 5.32e3T^{2} \) |
| 79 | \( 1 + (-42.2 - 73.2i)T + (-3.12e3 + 5.40e3i)T^{2} \) |
| 83 | \( 1 + (33.1 - 19.1i)T + (3.44e3 - 5.96e3i)T^{2} \) |
| 89 | \( 1 - 143. iT - 7.92e3T^{2} \) |
| 97 | \( 1 + (40.3 + 69.9i)T + (-4.70e3 + 8.14e3i)T^{2} \) |
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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
−15.77670011090823402414702689963, −15.09432839383404389371102036669, −13.74847765334451650706021923912, −12.56700517165280561925497593757, −11.63160118847508671367372263975, −9.625803875974266392545386611428, −8.306082582376046738474800724984, −7.27587624631547979538705340350, −5.08419366454016553049499615652, −2.50959671808882599528232416564,
3.28606872193177282185519833530, 5.01102340857181410439282506004, 7.54773451554962177434978772190, 8.480493950718427862156277348009, 10.39074282721924029658561171899, 10.92388431095590021738856817061, 13.04470751672729028906098906347, 14.06129920780672061042583799833, 15.17486733731570531274418622507, 16.11040457678461466610315704530