L(s) = 1 | + (−0.939 + 0.342i)3-s + (0.139 − 0.793i)5-s + (−0.580 + 3.29i)7-s + (0.766 − 0.642i)9-s + (−2.31 − 4.01i)11-s + (3.55 + 2.98i)13-s + (0.139 + 0.793i)15-s + (−3.21 + 2.70i)17-s + (−1.11 + 0.406i)19-s + (−0.580 − 3.29i)21-s + (−1.67 + 2.89i)23-s + (4.08 + 1.48i)25-s + (−0.500 + 0.866i)27-s + (−0.410 − 0.710i)29-s − 10.7·31-s + ⋯ |
L(s) = 1 | + (−0.542 + 0.197i)3-s + (0.0625 − 0.354i)5-s + (−0.219 + 1.24i)7-s + (0.255 − 0.214i)9-s + (−0.698 − 1.20i)11-s + (0.985 + 0.827i)13-s + (0.0361 + 0.204i)15-s + (−0.780 + 0.654i)17-s + (−0.255 + 0.0931i)19-s + (−0.126 − 0.717i)21-s + (−0.348 + 0.603i)23-s + (0.817 + 0.297i)25-s + (−0.0962 + 0.166i)27-s + (−0.0761 − 0.131i)29-s − 1.92·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 888 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.631 - 0.775i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 888 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.631 - 0.775i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.303145 + 0.637380i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.303145 + 0.637380i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (0.939 - 0.342i)T \) |
| 37 | \( 1 + (2.49 - 5.54i)T \) |
good | 5 | \( 1 + (-0.139 + 0.793i)T + (-4.69 - 1.71i)T^{2} \) |
| 7 | \( 1 + (0.580 - 3.29i)T + (-6.57 - 2.39i)T^{2} \) |
| 11 | \( 1 + (2.31 + 4.01i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (-3.55 - 2.98i)T + (2.25 + 12.8i)T^{2} \) |
| 17 | \( 1 + (3.21 - 2.70i)T + (2.95 - 16.7i)T^{2} \) |
| 19 | \( 1 + (1.11 - 0.406i)T + (14.5 - 12.2i)T^{2} \) |
| 23 | \( 1 + (1.67 - 2.89i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (0.410 + 0.710i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + 10.7T + 31T^{2} \) |
| 41 | \( 1 + (3.23 + 2.71i)T + (7.11 + 40.3i)T^{2} \) |
| 43 | \( 1 - 4.12T + 43T^{2} \) |
| 47 | \( 1 + (6.37 - 11.0i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-0.807 - 4.57i)T + (-49.8 + 18.1i)T^{2} \) |
| 59 | \( 1 + (-0.449 - 2.54i)T + (-55.4 + 20.1i)T^{2} \) |
| 61 | \( 1 + (4.78 + 4.01i)T + (10.5 + 60.0i)T^{2} \) |
| 67 | \( 1 + (2.54 - 14.4i)T + (-62.9 - 22.9i)T^{2} \) |
| 71 | \( 1 + (-11.0 + 4.01i)T + (54.3 - 45.6i)T^{2} \) |
| 73 | \( 1 + 9.68T + 73T^{2} \) |
| 79 | \( 1 + (2.45 - 13.9i)T + (-74.2 - 27.0i)T^{2} \) |
| 83 | \( 1 + (-2.98 + 2.50i)T + (14.4 - 81.7i)T^{2} \) |
| 89 | \( 1 + (0.904 + 5.13i)T + (-83.6 + 30.4i)T^{2} \) |
| 97 | \( 1 + (-4.48 + 7.77i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−10.70684321827884742049872676291, −9.358197084820567942326042538802, −8.851732090276741381893529921784, −8.165917821351403326884702883636, −6.76115736801076191153671127207, −5.88100370418439239321481971453, −5.45288040710486383884774812437, −4.20105568413993608032319986698, −3.07707396941853316418751787199, −1.65223024048429996448772823056,
0.35424335450922232747783177456, 2.01558026175829111353004000240, 3.45412262871672899211688228300, 4.49429389964487140969459542948, 5.39953036601898315156200917439, 6.61137864669296206515754774219, 7.11833955255895518444046464150, 7.904477515761004951173702362344, 9.055521230662672904138020701520, 10.18343299781996494651935968088