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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.804 + 0.594i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.804 + 0.594i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(1.795780872\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.795780872\) |
\(L(2)\) |
|
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
−10.66198430104848749874057711301, −9.606536750878993593387996238734, −8.639041273766701099807555873286, −7.51357217833080912680506073067, −6.78895396983433894587846263242, −6.05778431979703888929786292872, −4.74477587192890412294359569831, −4.02460555399002153083339821574, −1.87620493192179352172749446063, −1.09481984422210633427900701050,
1.04830864823181602116335778468, 2.80358446819467384618597747133, 4.04125400605622598471542647018, 5.16660999132995258174730359911, 5.97511816566812498630355170854, 6.62969788338621152845540357883, 8.110431972562570627202861103110, 9.311822350662611281208391148066, 9.391608588706294020791455996269, 10.86749222539001667483346404554