L(s) = 1 | − 1.42i·2-s − 0.0397·4-s − 0.0606i·5-s + 1.70i·7-s − 2.79i·8-s − 0.0866·10-s + i·11-s + (3.52 − 0.738i)13-s + 2.43·14-s − 4.07·16-s + 3.75·17-s + 2.02i·19-s + 0.00241i·20-s + 1.42·22-s + 0.704·23-s + ⋯ |
L(s) = 1 | − 1.00i·2-s − 0.0198·4-s − 0.0271i·5-s + 0.644i·7-s − 0.989i·8-s − 0.0273·10-s + 0.301i·11-s + (0.978 − 0.204i)13-s + 0.651·14-s − 1.01·16-s + 0.910·17-s + 0.464i·19-s + 0.000538i·20-s + 0.304·22-s + 0.146·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1287 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.204 + 0.978i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1287 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.204 + 0.978i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.007614876\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.007614876\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 11 | \( 1 - iT \) |
| 13 | \( 1 + (-3.52 + 0.738i)T \) |
good | 2 | \( 1 + 1.42iT - 2T^{2} \) |
| 5 | \( 1 + 0.0606iT - 5T^{2} \) |
| 7 | \( 1 - 1.70iT - 7T^{2} \) |
| 17 | \( 1 - 3.75T + 17T^{2} \) |
| 19 | \( 1 - 2.02iT - 19T^{2} \) |
| 23 | \( 1 - 0.704T + 23T^{2} \) |
| 29 | \( 1 - 0.0346T + 29T^{2} \) |
| 31 | \( 1 - 1.85iT - 31T^{2} \) |
| 37 | \( 1 + 8.82iT - 37T^{2} \) |
| 41 | \( 1 + 3.32iT - 41T^{2} \) |
| 43 | \( 1 + 5.29T + 43T^{2} \) |
| 47 | \( 1 - 6.04iT - 47T^{2} \) |
| 53 | \( 1 - 10.6T + 53T^{2} \) |
| 59 | \( 1 - 3.34iT - 59T^{2} \) |
| 61 | \( 1 - 3.26T + 61T^{2} \) |
| 67 | \( 1 + 10.9iT - 67T^{2} \) |
| 71 | \( 1 + 1.96iT - 71T^{2} \) |
| 73 | \( 1 + 15.3iT - 73T^{2} \) |
| 79 | \( 1 - 10.5T + 79T^{2} \) |
| 83 | \( 1 + 4.19iT - 83T^{2} \) |
| 89 | \( 1 - 14.3iT - 89T^{2} \) |
| 97 | \( 1 - 5.86iT - 97T^{2} \) |
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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
−9.559243618568092007126864642245, −8.940987898960669436926444664321, −7.972230530173257235111686264132, −7.00883317381444504542698022805, −6.10280254389741306116293173296, −5.23699252849555963927390913025, −3.95064077140124545403951598532, −3.18582673760313975159287550798, −2.16787345251671201112128405969, −1.05884031433445127830112777446,
1.19451461927875386340240680812, 2.77257122820474371714405894331, 3.87764780920459698396157904110, 5.01902096894301210844495896425, 5.79925028410368710491061359831, 6.72350348034355632116002523594, 7.15981707996558595074164786457, 8.252375306654125501552103220941, 8.597493705350615521825598296725, 9.799280814696057900366585907138