L(s) = 1 | + 1.41i·2-s − 2.00·4-s + (−2.44 − 1.41i)5-s + (1.5 − 0.866i)7-s − 2.82i·8-s + (2.00 − 3.46i)10-s + (−2.44 − 4.24i)11-s + (0.5 − 0.866i)13-s + (1.22 + 2.12i)14-s + 4.00·16-s + 2.82i·17-s − 5.19i·19-s + (4.89 + 2.82i)20-s + (6 − 3.46i)22-s + (2.44 − 4.24i)23-s + ⋯ |
L(s) = 1 | + 0.999i·2-s − 1.00·4-s + (−1.09 − 0.632i)5-s + (0.566 − 0.327i)7-s − 1.00i·8-s + (0.632 − 1.09i)10-s + (−0.738 − 1.27i)11-s + (0.138 − 0.240i)13-s + (0.327 + 0.566i)14-s + 1.00·16-s + 0.685i·17-s − 1.19i·19-s + (1.09 + 0.632i)20-s + (1.27 − 0.738i)22-s + (0.510 − 0.884i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.642 + 0.766i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.642 + 0.766i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.658770 - 0.307189i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.658770 - 0.307189i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - 1.41iT \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + (2.44 + 1.41i)T + (2.5 + 4.33i)T^{2} \) |
| 7 | \( 1 + (-1.5 + 0.866i)T + (3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (2.44 + 4.24i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (-0.5 + 0.866i)T + (-6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 - 2.82iT - 17T^{2} \) |
| 19 | \( 1 + 5.19iT - 19T^{2} \) |
| 23 | \( 1 + (-2.44 + 4.24i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (4.89 - 2.82i)T + (14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (3 + 1.73i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + T + 37T^{2} \) |
| 41 | \( 1 + (-4.89 - 2.82i)T + (20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (3 - 1.73i)T + (21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (2.44 + 4.24i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + 5.65iT - 53T^{2} \) |
| 59 | \( 1 + (-2.44 + 4.24i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (5.5 + 9.52i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-10.5 - 6.06i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 + T + 73T^{2} \) |
| 79 | \( 1 + (-1.5 + 0.866i)T + (39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-4.89 - 8.48i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 - 2.82iT - 89T^{2} \) |
| 97 | \( 1 + (-6.5 - 11.2i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−11.36081896964728209465023609624, −10.70556165399551463341873636399, −9.192323164593118278122176897566, −8.284311391516580194323789608815, −7.929258475368335618894650602707, −6.75033457302711834331528391974, −5.45311809145221515340641399476, −4.59254157472082617100481322493, −3.48440990481653988386812738108, −0.53330560875433874219015734496,
1.97650555028363986605070807297, 3.34942834674788102263952355676, 4.40533686011668830291725867325, 5.46929717547565204087265273560, 7.35364872910435467386172281747, 7.88503049448383049836368763587, 9.124927284261025437705122696470, 10.09054906296567798192004599657, 10.99376660422508138126631484593, 11.68844777513245264019934923161