| L(s) = 1 | + (0.434 + 0.566i)2-s + (2.71 + 1.28i)3-s + (0.903 − 3.37i)4-s + (1.70 − 1.49i)5-s + (0.451 + 2.09i)6-s + (−0.856 − 0.751i)7-s + (4.93 − 2.04i)8-s + (5.70 + 6.96i)9-s + (1.58 + 0.316i)10-s + (−0.488 − 7.45i)11-s + (6.77 − 7.98i)12-s + (−5.82 − 1.56i)13-s + (0.0531 − 0.811i)14-s + (6.55 − 1.86i)15-s + (−8.78 − 5.07i)16-s + (14.4 + 8.92i)17-s + ⋯ |
| L(s) = 1 | + (0.217 + 0.283i)2-s + (0.903 + 0.427i)3-s + (0.225 − 0.842i)4-s + (0.341 − 0.299i)5-s + (0.0751 + 0.348i)6-s + (−0.122 − 0.107i)7-s + (0.617 − 0.255i)8-s + (0.633 + 0.773i)9-s + (0.158 + 0.0316i)10-s + (−0.0444 − 0.677i)11-s + (0.564 − 0.665i)12-s + (−0.447 − 0.120i)13-s + (0.00379 − 0.0579i)14-s + (0.436 − 0.124i)15-s + (−0.549 − 0.317i)16-s + (0.850 + 0.525i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 153 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.00747i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 153 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.999 + 0.00747i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(2.31091 - 0.00863366i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.31091 - 0.00863366i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-2.71 - 1.28i)T \) |
| 17 | \( 1 + (-14.4 - 8.92i)T \) |
| good | 2 | \( 1 + (-0.434 - 0.566i)T + (-1.03 + 3.86i)T^{2} \) |
| 5 | \( 1 + (-1.70 + 1.49i)T + (3.26 - 24.7i)T^{2} \) |
| 7 | \( 1 + (0.856 + 0.751i)T + (6.39 + 48.5i)T^{2} \) |
| 11 | \( 1 + (0.488 + 7.45i)T + (-119. + 15.7i)T^{2} \) |
| 13 | \( 1 + (5.82 + 1.56i)T + (146. + 84.5i)T^{2} \) |
| 19 | \( 1 + (7.99 - 19.2i)T + (-255. - 255. i)T^{2} \) |
| 23 | \( 1 + (7.81 - 15.8i)T + (-322. - 419. i)T^{2} \) |
| 29 | \( 1 + (-9.18 + 27.0i)T + (-667. - 511. i)T^{2} \) |
| 31 | \( 1 + (-0.862 + 13.1i)T + (-952. - 125. i)T^{2} \) |
| 37 | \( 1 + (30.4 - 45.5i)T + (-523. - 1.26e3i)T^{2} \) |
| 41 | \( 1 + (22.9 - 7.78i)T + (1.33e3 - 1.02e3i)T^{2} \) |
| 43 | \( 1 + (24.4 + 3.21i)T + (1.78e3 + 478. i)T^{2} \) |
| 47 | \( 1 + (10.6 + 39.7i)T + (-1.91e3 + 1.10e3i)T^{2} \) |
| 53 | \( 1 + (-6.58 + 15.8i)T + (-1.98e3 - 1.98e3i)T^{2} \) |
| 59 | \( 1 + (-4.04 - 3.10i)T + (900. + 3.36e3i)T^{2} \) |
| 61 | \( 1 + (36.7 - 41.8i)T + (-485. - 3.68e3i)T^{2} \) |
| 67 | \( 1 + (37.7 - 21.7i)T + (2.24e3 - 3.88e3i)T^{2} \) |
| 71 | \( 1 + (-12.5 + 18.7i)T + (-1.92e3 - 4.65e3i)T^{2} \) |
| 73 | \( 1 + (-15.7 - 78.9i)T + (-4.92e3 + 2.03e3i)T^{2} \) |
| 79 | \( 1 + (7.63 + 116. i)T + (-6.18e3 + 814. i)T^{2} \) |
| 83 | \( 1 + (-95.4 + 73.2i)T + (1.78e3 - 6.65e3i)T^{2} \) |
| 89 | \( 1 + (39.2 + 39.2i)T + 7.92e3iT^{2} \) |
| 97 | \( 1 + (-33.7 + 99.4i)T + (-7.46e3 - 5.72e3i)T^{2} \) |
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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
−13.20510118337090728200955845283, −11.70340884467470340657482158880, −10.22601162930978323206907747246, −9.939818200568865325795455654383, −8.611799718729489971778326869190, −7.55006985527144945791786634705, −6.08289459781741193764634281923, −5.04593371874367721919350798236, −3.56110425914053203023448433900, −1.73575959411510229784574734579,
2.16290664733802653744065084842, 3.15696403792188951566770901626, 4.61398900424394988510642366809, 6.67058535075576410678035067781, 7.44384936973138271291296361147, 8.525438832570191239607801923825, 9.588251519282994015816163161660, 10.74523727736625784493314012701, 12.26112112626540348003827108147, 12.52296088988584237863495200000
