L(s) = 1 | + 2-s + (−1.72 + 0.158i)3-s + 4-s + (0.724 − 1.25i)5-s + (−1.72 + 0.158i)6-s + 8-s + (2.94 − 0.548i)9-s + (0.724 − 1.25i)10-s + (−1 − 1.73i)11-s + (−1.72 + 0.158i)12-s + (−2.44 − 4.24i)13-s + (−1.05 + 2.28i)15-s + 16-s + (−1 + 1.73i)17-s + (2.94 − 0.548i)18-s + (−1.27 − 2.20i)19-s + ⋯ |
L(s) = 1 | + 0.707·2-s + (−0.995 + 0.0917i)3-s + 0.5·4-s + (0.324 − 0.561i)5-s + (−0.704 + 0.0648i)6-s + 0.353·8-s + (0.983 − 0.182i)9-s + (0.229 − 0.396i)10-s + (−0.301 − 0.522i)11-s + (−0.497 + 0.0458i)12-s + (−0.679 − 1.17i)13-s + (−0.271 + 0.588i)15-s + 0.250·16-s + (−0.242 + 0.420i)17-s + (0.695 − 0.129i)18-s + (−0.292 − 0.506i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.135 + 0.990i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.135 + 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.19236 - 1.04033i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.19236 - 1.04033i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 3 | \( 1 + (1.72 - 0.158i)T \) |
| 7 | \( 1 \) |
good | 5 | \( 1 + (-0.724 + 1.25i)T + (-2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (1 + 1.73i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (2.44 + 4.24i)T + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + (1 - 1.73i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (1.27 + 2.20i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-0.5 + 0.866i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-3.44 + 5.97i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 - 6T + 31T^{2} \) |
| 37 | \( 1 + (5.89 + 10.2i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (4.89 + 8.48i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (3.44 - 5.97i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 - 9.79T + 47T^{2} \) |
| 53 | \( 1 + (-5.44 + 9.43i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + 2T + 59T^{2} \) |
| 61 | \( 1 - 6.55T + 61T^{2} \) |
| 67 | \( 1 + 12.8T + 67T^{2} \) |
| 71 | \( 1 - 0.101T + 71T^{2} \) |
| 73 | \( 1 + (-3.44 + 5.97i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + 1.89T + 79T^{2} \) |
| 83 | \( 1 + (1 - 1.73i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (-8.44 - 14.6i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (1.44 - 2.51i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−10.39300879905634525222816490908, −9.213379530163089314980586640564, −8.140204847228069737593245202754, −7.14538418315273409030869738086, −6.20935706270545946990250286488, −5.40635787501367757965196985665, −4.88630363271291972867257963614, −3.76309616241093245233201141054, −2.35795815038552955639303282708, −0.68360126664365061279217092826,
1.68101488821066779600766274575, 2.87919173027555136321973424140, 4.40647303453820468015884355694, 4.91579774995962534684418416263, 6.02658707998982627249246281286, 6.82464746729667166205671043561, 7.22075390300609497800803571250, 8.626209874821450577097698950190, 10.02653560047303350788951212117, 10.26586248811661746390767734590