L(s) = 1 | + (1.36 + 0.366i)2-s + (1.73 + i)4-s + (1.73 + i)5-s + (−2 − 3.46i)7-s + (1.99 + 2i)8-s + (1.99 + 2i)10-s + (−2.59 + 1.5i)11-s + (1.73 + i)13-s + (−1.46 − 5.46i)14-s + (1.99 + 3.46i)16-s − 5·17-s − i·19-s + (1.99 + 3.46i)20-s + (−4.09 + 1.09i)22-s + (1 − 1.73i)23-s + ⋯ |
L(s) = 1 | + (0.965 + 0.258i)2-s + (0.866 + 0.5i)4-s + (0.774 + 0.447i)5-s + (−0.755 − 1.30i)7-s + (0.707 + 0.707i)8-s + (0.632 + 0.632i)10-s + (−0.783 + 0.452i)11-s + (0.480 + 0.277i)13-s + (−0.391 − 1.46i)14-s + (0.499 + 0.866i)16-s − 1.21·17-s − 0.229i·19-s + (0.447 + 0.774i)20-s + (−0.873 + 0.234i)22-s + (0.208 − 0.361i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.906 - 0.422i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.906 - 0.422i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.07033 + 0.458981i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.07033 + 0.458981i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-1.36 - 0.366i)T \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + (-1.73 - i)T + (2.5 + 4.33i)T^{2} \) |
| 7 | \( 1 + (2 + 3.46i)T + (-3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (2.59 - 1.5i)T + (5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (-1.73 - i)T + (6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + 5T + 17T^{2} \) |
| 19 | \( 1 + iT - 19T^{2} \) |
| 23 | \( 1 + (-1 + 1.73i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (-2 + 3.46i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + 2iT - 37T^{2} \) |
| 41 | \( 1 + (2.5 - 4.33i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (9.52 - 5.5i)T + (21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (3 + 5.19i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 - 53T^{2} \) |
| 59 | \( 1 + (-0.866 - 0.5i)T + (29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-10.3 + 6i)T + (30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-2.59 - 1.5i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - 6T + 71T^{2} \) |
| 73 | \( 1 - 9T + 73T^{2} \) |
| 79 | \( 1 + (-7 - 12.1i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (3.46 - 2i)T + (41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 - 14T + 89T^{2} \) |
| 97 | \( 1 + (0.5 + 0.866i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−12.88463597714765421131100185509, −11.39181425957411070834864329612, −10.55889326384078634580707331686, −9.770116090951147465538491338719, −8.107254999974141425733612226723, −6.82722978419181667164956850862, −6.44053010502898545213009758947, −4.92878379485521351559472110760, −3.76294563482214022757096069319, −2.36332118847499301302913558219,
2.11554593785227139652540245546, 3.29884724948060516704191743938, 5.08061973141520009686416015880, 5.78243200103484308714693569016, 6.68144037078655771021164295697, 8.431662777003552260151574237361, 9.425844864693216776246554208264, 10.44085417466216955794832872097, 11.50284980469511585215836280874, 12.48335729396583527390590189808