L(s) = 1 | + (0.870 + 1.11i)2-s + (−0.483 + 1.94i)4-s + (−1.71 − 0.988i)5-s + (2.27 − 1.35i)7-s + (−2.58 + 1.15i)8-s + (−0.389 − 2.76i)10-s + (−4.16 + 2.40i)11-s + 6.20i·13-s + (3.48 + 1.35i)14-s + (−3.53 − 1.87i)16-s + (−0.655 + 0.378i)17-s + (−3.65 + 6.33i)19-s + (2.74 − 2.84i)20-s + (−6.30 − 2.54i)22-s + (4.09 + 2.36i)23-s + ⋯ |
L(s) = 1 | + (0.615 + 0.787i)2-s + (−0.241 + 0.970i)4-s + (−0.765 − 0.441i)5-s + (0.858 − 0.512i)7-s + (−0.913 + 0.407i)8-s + (−0.123 − 0.875i)10-s + (−1.25 + 0.725i)11-s + 1.72i·13-s + (0.932 + 0.361i)14-s + (−0.883 − 0.469i)16-s + (−0.159 + 0.0918i)17-s + (−0.838 + 1.45i)19-s + (0.613 − 0.635i)20-s + (−1.34 − 0.543i)22-s + (0.854 + 0.493i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.937 - 0.348i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.937 - 0.348i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.226084 + 1.25520i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.226084 + 1.25520i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.870 - 1.11i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (-2.27 + 1.35i)T \) |
good | 5 | \( 1 + (1.71 + 0.988i)T + (2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (4.16 - 2.40i)T + (5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 - 6.20iT - 13T^{2} \) |
| 17 | \( 1 + (0.655 - 0.378i)T + (8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (3.65 - 6.33i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-4.09 - 2.36i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + 3.09T + 29T^{2} \) |
| 31 | \( 1 + (-3.85 - 6.67i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-3.98 + 6.90i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 - 6.55iT - 41T^{2} \) |
| 43 | \( 1 - 1.68iT - 43T^{2} \) |
| 47 | \( 1 + (-1.30 + 2.25i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (7.08 + 12.2i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (0.705 + 1.22i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (5.09 + 2.93i)T + (30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-4.74 + 2.73i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 - 13.5iT - 71T^{2} \) |
| 73 | \( 1 + (-4.77 + 2.75i)T + (36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-3.07 - 1.77i)T + (39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 - 1.69T + 83T^{2} \) |
| 89 | \( 1 + (-3.63 - 2.09i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + 2.34iT - 97T^{2} \) |
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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
−10.95565402775104442082868545500, −9.749896369332129905313830005283, −8.611844094365053201326264230535, −7.983298512661762012935303796854, −7.34459851576459385813423196071, −6.42233751579499962099804375782, −5.09433681528324801138188819240, −4.51561720818196407978283439154, −3.74533093570806019292292109035, −2.01716332333906886603812027827,
0.51492739997048549537843378179, 2.53579374249763224477692874664, 3.09479823041289240581649969383, 4.49508695389808707244809528956, 5.25661136868360347413224495627, 6.10966045778806558337178900165, 7.52782092953700438537570497000, 8.239460917887425880503192766640, 9.150375851549884440148695676530, 10.48550403934473927385541450802