L(s) = 1 | − i·2-s + (−0.866 + 0.5i)3-s − 4-s + (0.594 + 2.15i)5-s + (0.5 + 0.866i)6-s + (−0.759 + 0.438i)7-s + i·8-s + (0.499 − 0.866i)9-s + (2.15 − 0.594i)10-s + (1.33 − 2.31i)11-s + (0.866 − 0.5i)12-s + (0.519 + 0.299i)13-s + (0.438 + 0.759i)14-s + (−1.59 − 1.56i)15-s + 16-s + (−1.86 + 1.07i)17-s + ⋯ |
L(s) = 1 | − 0.707i·2-s + (−0.499 + 0.288i)3-s − 0.5·4-s + (0.265 + 0.964i)5-s + (0.204 + 0.353i)6-s + (−0.287 + 0.165i)7-s + 0.353i·8-s + (0.166 − 0.288i)9-s + (0.681 − 0.187i)10-s + (0.402 − 0.697i)11-s + (0.249 − 0.144i)12-s + (0.144 + 0.0831i)13-s + (0.117 + 0.202i)14-s + (−0.411 − 0.405i)15-s + 0.250·16-s + (−0.451 + 0.260i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.149 - 0.988i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.149 - 0.988i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.683863 + 0.588466i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.683863 + 0.588466i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + iT \) |
| 3 | \( 1 + (0.866 - 0.5i)T \) |
| 5 | \( 1 + (-0.594 - 2.15i)T \) |
| 31 | \( 1 + (-5.42 + 1.25i)T \) |
good | 7 | \( 1 + (0.759 - 0.438i)T + (3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (-1.33 + 2.31i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (-0.519 - 0.299i)T + (6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + (1.86 - 1.07i)T + (8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-0.673 - 1.16i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 - 5.93iT - 23T^{2} \) |
| 29 | \( 1 + 1.51T + 29T^{2} \) |
| 37 | \( 1 + (9.54 - 5.50i)T + (18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (3.64 - 6.31i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (6.33 - 3.65i)T + (21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 - 6.66iT - 47T^{2} \) |
| 53 | \( 1 + (-0.455 - 0.263i)T + (26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-3.97 - 6.87i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 - 2.40T + 61T^{2} \) |
| 67 | \( 1 + (3.85 + 2.22i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (-7.57 + 13.1i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (1.34 + 0.774i)T + (36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (3.08 + 5.34i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-14.7 - 8.53i)T + (41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + 5.54T + 89T^{2} \) |
| 97 | \( 1 + 2.08iT - 97T^{2} \) |
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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
−10.26504008154481067923686585145, −9.727023817003692454066981130192, −8.850974400186334249490752954435, −7.78340798729456608584748893167, −6.57488528520090018292615761717, −6.02018125404031370665630270808, −4.93149478578961289145648529820, −3.68400538324414638214195870992, −3.01707811616802594402847086514, −1.55491706266004461880914724442,
0.46392655091913745019161847757, 1.98483337946052097076702264742, 3.85954595439568854646001557734, 4.84719143418307604231550171407, 5.46453543955122001867897060829, 6.62919089844447462155802355412, 7.03358584243322844000420045969, 8.300383586566917824270316282899, 8.847828730483151715874547136906, 9.808729074775381940682953479712