L(s) = 1 | + (0.382 − 0.923i)3-s + (2.14 − 0.890i)5-s + (1.10 − 1.10i)7-s + (−0.707 − 0.707i)9-s + (−0.999 − 2.41i)11-s + (−2.03 − 0.841i)13-s − 2.32i·15-s + 5.68i·17-s + (6.02 + 2.49i)19-s + (−0.595 − 1.43i)21-s + (−3.60 − 3.60i)23-s + (0.293 − 0.293i)25-s + (−0.923 + 0.382i)27-s + (3.82 − 9.23i)29-s − 1.98·31-s + ⋯ |
L(s) = 1 | + (0.220 − 0.533i)3-s + (0.961 − 0.398i)5-s + (0.415 − 0.415i)7-s + (−0.235 − 0.235i)9-s + (−0.301 − 0.727i)11-s + (−0.563 − 0.233i)13-s − 0.600i·15-s + 1.37i·17-s + (1.38 + 0.572i)19-s + (−0.129 − 0.313i)21-s + (−0.751 − 0.751i)23-s + (0.0586 − 0.0586i)25-s + (−0.177 + 0.0736i)27-s + (0.710 − 1.71i)29-s − 0.357·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.502 + 0.864i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.502 + 0.864i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.46259 - 0.841380i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.46259 - 0.841380i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-0.382 + 0.923i)T \) |
good | 5 | \( 1 + (-2.14 + 0.890i)T + (3.53 - 3.53i)T^{2} \) |
| 7 | \( 1 + (-1.10 + 1.10i)T - 7iT^{2} \) |
| 11 | \( 1 + (0.999 + 2.41i)T + (-7.77 + 7.77i)T^{2} \) |
| 13 | \( 1 + (2.03 + 0.841i)T + (9.19 + 9.19i)T^{2} \) |
| 17 | \( 1 - 5.68iT - 17T^{2} \) |
| 19 | \( 1 + (-6.02 - 2.49i)T + (13.4 + 13.4i)T^{2} \) |
| 23 | \( 1 + (3.60 + 3.60i)T + 23iT^{2} \) |
| 29 | \( 1 + (-3.82 + 9.23i)T + (-20.5 - 20.5i)T^{2} \) |
| 31 | \( 1 + 1.98T + 31T^{2} \) |
| 37 | \( 1 + (-5.97 + 2.47i)T + (26.1 - 26.1i)T^{2} \) |
| 41 | \( 1 + (4.33 + 4.33i)T + 41iT^{2} \) |
| 43 | \( 1 + (-4.39 - 10.6i)T + (-30.4 + 30.4i)T^{2} \) |
| 47 | \( 1 - 5.32iT - 47T^{2} \) |
| 53 | \( 1 + (-0.802 - 1.93i)T + (-37.4 + 37.4i)T^{2} \) |
| 59 | \( 1 + (5.97 - 2.47i)T + (41.7 - 41.7i)T^{2} \) |
| 61 | \( 1 + (3.53 - 8.54i)T + (-43.1 - 43.1i)T^{2} \) |
| 67 | \( 1 + (-2.25 + 5.44i)T + (-47.3 - 47.3i)T^{2} \) |
| 71 | \( 1 + (2.57 - 2.57i)T - 71iT^{2} \) |
| 73 | \( 1 + (-8.01 - 8.01i)T + 73iT^{2} \) |
| 79 | \( 1 - 14.4iT - 79T^{2} \) |
| 83 | \( 1 + (2.50 + 1.03i)T + (58.6 + 58.6i)T^{2} \) |
| 89 | \( 1 + (2.98 - 2.98i)T - 89iT^{2} \) |
| 97 | \( 1 + 5.81T + 97T^{2} \) |
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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
−11.16971516217714130499188499290, −10.17024277927837696440383013668, −9.433495759242841073865252890419, −8.204268677748913300662825043793, −7.69679673320468513341461260769, −6.19984707987843685726162001860, −5.61446403278287090963928193020, −4.19172225964645055970039175706, −2.61757624992457500937500203501, −1.26998540547536968043481854827,
2.05413398308801740077463021440, 3.12546902703605309462224226063, 4.86641750226909668294362975167, 5.40413329324867721211033495132, 6.83052916314630310884969840304, 7.69732839178554227737755798973, 9.070470346727065907018063480745, 9.660864950984361322722170239312, 10.34624232667640350347244845438, 11.49759804712930245655666163178