L(s) = 1 | + 1.41·2-s + (−0.707 + 1.58i)3-s − 2.23i·5-s + (−1.00 + 2.23i)6-s + 7-s − 2.82·8-s + (−2.00 − 2.23i)9-s − 3.16i·10-s + 2.23i·11-s − 3.16i·13-s + 1.41·14-s + (3.53 + 1.58i)15-s − 4.00·16-s + 6.70i·17-s + (−2.82 − 3.16i)18-s + (3 + 3.16i)19-s + ⋯ |
L(s) = 1 | + 1.00·2-s + (−0.408 + 0.912i)3-s − 0.999i·5-s + (−0.408 + 0.912i)6-s + 0.377·7-s − 0.999·8-s + (−0.666 − 0.745i)9-s − 1.00i·10-s + 0.674i·11-s − 0.877i·13-s + 0.377·14-s + (0.912 + 0.408i)15-s − 1.00·16-s + 1.62i·17-s + (−0.666 − 0.745i)18-s + (0.688 + 0.725i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 57 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.943 - 0.332i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 57 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.943 - 0.332i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.07822 + 0.184271i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.07822 + 0.184271i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (0.707 - 1.58i)T \) |
| 19 | \( 1 + (-3 - 3.16i)T \) |
good | 2 | \( 1 - 1.41T + 2T^{2} \) |
| 5 | \( 1 + 2.23iT - 5T^{2} \) |
| 7 | \( 1 - T + 7T^{2} \) |
| 11 | \( 1 - 2.23iT - 11T^{2} \) |
| 13 | \( 1 + 3.16iT - 13T^{2} \) |
| 17 | \( 1 - 6.70iT - 17T^{2} \) |
| 23 | \( 1 + 4.47iT - 23T^{2} \) |
| 29 | \( 1 + 5.65T + 29T^{2} \) |
| 31 | \( 1 - 3.16iT - 31T^{2} \) |
| 37 | \( 1 + 9.48iT - 37T^{2} \) |
| 41 | \( 1 - 9.89T + 41T^{2} \) |
| 43 | \( 1 + 5T + 43T^{2} \) |
| 47 | \( 1 + 2.23iT - 47T^{2} \) |
| 53 | \( 1 + 4.24T + 53T^{2} \) |
| 59 | \( 1 + 1.41T + 59T^{2} \) |
| 61 | \( 1 + T + 61T^{2} \) |
| 67 | \( 1 + 6.32iT - 67T^{2} \) |
| 71 | \( 1 - 4.24T + 71T^{2} \) |
| 73 | \( 1 + 3T + 73T^{2} \) |
| 79 | \( 1 - 12.6iT - 79T^{2} \) |
| 83 | \( 1 - 8.94iT - 83T^{2} \) |
| 89 | \( 1 - 12.7T + 89T^{2} \) |
| 97 | \( 1 + 3.16iT - 97T^{2} \) |
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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
−15.07775049480717081918669936708, −14.41579157887449807636719255788, −12.77043670922727425119801816799, −12.36580195854198475157715701813, −10.82293300995852641350550235410, −9.498651377320540499453261903596, −8.331695407974657162829287151151, −5.87162159242412698248818548088, −4.93785631535670169891209178417, −3.83538579096647599912719981865,
2.96675602736984127460157644008, 5.03864915657837585618048477899, 6.35413710902425330001465723755, 7.47792633547664623918598299666, 9.269834507866149855137805291709, 11.34595879883026444447957287624, 11.65809790001365106892915998469, 13.27774734279471841401361491382, 13.88418300241703166462681872186, 14.69119031246940625234437704326