| L(s) = 1 | + (−0.136 − 1.99i)2-s + (−2.85 + 0.910i)3-s + (−3.96 + 0.545i)4-s + (1.49 + 8.48i)5-s + (2.20 + 5.57i)6-s + (−7.40 − 6.21i)7-s + (1.62 + 7.83i)8-s + (7.34 − 5.20i)9-s + (16.7 − 4.14i)10-s + (2.58 − 14.6i)11-s + (10.8 − 5.16i)12-s + (3.35 − 9.21i)13-s + (−11.3 + 15.6i)14-s + (−12.0 − 22.8i)15-s + (15.4 − 4.32i)16-s + (14.8 − 8.59i)17-s + ⋯ |
| L(s) = 1 | + (−0.0683 − 0.997i)2-s + (−0.952 + 0.303i)3-s + (−0.990 + 0.136i)4-s + (0.299 + 1.69i)5-s + (0.367 + 0.929i)6-s + (−1.05 − 0.887i)7-s + (0.203 + 0.979i)8-s + (0.815 − 0.578i)9-s + (1.67 − 0.414i)10-s + (0.234 − 1.33i)11-s + (0.902 − 0.430i)12-s + (0.257 − 0.708i)13-s + (−0.813 + 1.11i)14-s + (−0.800 − 1.52i)15-s + (0.962 − 0.270i)16-s + (0.876 − 0.505i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.149 + 0.988i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.149 + 0.988i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.519682 - 0.603876i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.519682 - 0.603876i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.136 + 1.99i)T \) |
| 3 | \( 1 + (2.85 - 0.910i)T \) |
| good | 5 | \( 1 + (-1.49 - 8.48i)T + (-23.4 + 8.55i)T^{2} \) |
| 7 | \( 1 + (7.40 + 6.21i)T + (8.50 + 48.2i)T^{2} \) |
| 11 | \( 1 + (-2.58 + 14.6i)T + (-113. - 41.3i)T^{2} \) |
| 13 | \( 1 + (-3.35 + 9.21i)T + (-129. - 108. i)T^{2} \) |
| 17 | \( 1 + (-14.8 + 8.59i)T + (144.5 - 250. i)T^{2} \) |
| 19 | \( 1 + (-16.8 - 9.73i)T + (180.5 + 312. i)T^{2} \) |
| 23 | \( 1 + (7.74 + 9.22i)T + (-91.8 + 520. i)T^{2} \) |
| 29 | \( 1 + (-14.6 + 5.32i)T + (644. - 540. i)T^{2} \) |
| 31 | \( 1 + (-14.0 + 11.8i)T + (166. - 946. i)T^{2} \) |
| 37 | \( 1 + (23.4 - 13.5i)T + (684.5 - 1.18e3i)T^{2} \) |
| 41 | \( 1 + (-23.5 + 64.8i)T + (-1.28e3 - 1.08e3i)T^{2} \) |
| 43 | \( 1 + (-20.3 - 3.59i)T + (1.73e3 + 632. i)T^{2} \) |
| 47 | \( 1 + (-3.08 + 3.67i)T + (-383. - 2.17e3i)T^{2} \) |
| 53 | \( 1 + 38.0T + 2.80e3T^{2} \) |
| 59 | \( 1 + (4.87 + 27.6i)T + (-3.27e3 + 1.19e3i)T^{2} \) |
| 61 | \( 1 + (14.4 - 17.2i)T + (-646. - 3.66e3i)T^{2} \) |
| 67 | \( 1 + (-24.9 + 68.4i)T + (-3.43e3 - 2.88e3i)T^{2} \) |
| 71 | \( 1 + (-97.3 + 56.2i)T + (2.52e3 - 4.36e3i)T^{2} \) |
| 73 | \( 1 + (-59.1 + 102. i)T + (-2.66e3 - 4.61e3i)T^{2} \) |
| 79 | \( 1 + (90.3 - 32.8i)T + (4.78e3 - 4.01e3i)T^{2} \) |
| 83 | \( 1 + (79.3 - 28.8i)T + (5.27e3 - 4.42e3i)T^{2} \) |
| 89 | \( 1 + (36.3 + 20.9i)T + (3.96e3 + 6.85e3i)T^{2} \) |
| 97 | \( 1 + (1.76 - 10.0i)T + (-8.84e3 - 3.21e3i)T^{2} \) |
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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
−11.54059349470066535162777584098, −10.70753801280192991210778657898, −10.27343006090682678012512552992, −9.542373915506327418041242172137, −7.68945203714078281259730292924, −6.47892044875072902714363284560, −5.62726266414892619085929567308, −3.68056881771589769427898676163, −3.12904947481381602261026467161, −0.60466315646757691718856632941,
1.27928298810298900029118891747, 4.35388215921753712970267609127, 5.27476862101538361318528669731, 6.04627669329488718759071767242, 7.09586434678897401048116756295, 8.363944009273746734976044846045, 9.510585894318057531686759558344, 9.792618964312907286331929110401, 11.89279465592931423309549194967, 12.58211295056325808278627522795
