L(s) = 1 | + (0.5 − 0.866i)2-s + (0.707 + 1.22i)3-s + (−0.499 − 0.866i)4-s + (−0.707 + 1.22i)5-s + 1.41·6-s − 0.999·8-s + (0.500 − 0.866i)9-s + (0.707 + 1.22i)10-s + (0.5 + 0.866i)11-s + (0.707 − 1.22i)12-s + 2.82·13-s − 2·15-s + (−0.5 + 0.866i)16-s + (2.82 + 4.89i)17-s + (−0.499 − 0.866i)18-s + (−1.41 + 2.44i)19-s + ⋯ |
L(s) = 1 | + (0.353 − 0.612i)2-s + (0.408 + 0.707i)3-s + (−0.249 − 0.433i)4-s + (−0.316 + 0.547i)5-s + 0.577·6-s − 0.353·8-s + (0.166 − 0.288i)9-s + (0.223 + 0.387i)10-s + (0.150 + 0.261i)11-s + (0.204 − 0.353i)12-s + 0.784·13-s − 0.516·15-s + (−0.125 + 0.216i)16-s + (0.685 + 1.18i)17-s + (−0.117 − 0.204i)18-s + (−0.324 + 0.561i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1078 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.900 - 0.435i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1078 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.900 - 0.435i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.143871110\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.143871110\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.5 + 0.866i)T \) |
| 7 | \( 1 \) |
| 11 | \( 1 + (-0.5 - 0.866i)T \) |
good | 3 | \( 1 + (-0.707 - 1.22i)T + (-1.5 + 2.59i)T^{2} \) |
| 5 | \( 1 + (0.707 - 1.22i)T + (-2.5 - 4.33i)T^{2} \) |
| 13 | \( 1 - 2.82T + 13T^{2} \) |
| 17 | \( 1 + (-2.82 - 4.89i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (1.41 - 2.44i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-1 + 1.73i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + 6T + 29T^{2} \) |
| 31 | \( 1 + (-0.707 - 1.22i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-1 + 1.73i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 - 5.65T + 41T^{2} \) |
| 43 | \( 1 - 4T + 43T^{2} \) |
| 47 | \( 1 + (2.12 - 3.67i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-6 - 10.3i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (2.12 + 3.67i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-2.82 + 4.89i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-1 - 1.73i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + 10T + 71T^{2} \) |
| 73 | \( 1 + (-7.07 - 12.2i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-4 + 6.92i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + 2.82T + 83T^{2} \) |
| 89 | \( 1 + (-3.53 + 6.12i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + 15.5T + 97T^{2} \) |
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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
−10.06676973981224527880344011634, −9.274681054265508098785914302674, −8.526733060473410109455209796330, −7.50666885545617956566347665268, −6.41612098810812933989532338711, −5.59123617697526498404469089651, −4.22470240037869523938341830626, −3.78314046931653629738575030484, −2.90454780051418567248627330187, −1.43254125869397259724395104169,
0.939337805131261358351192474322, 2.47782074971000954998203089855, 3.66286346539957141566501866204, 4.70370248902019579022030748297, 5.54364405496541873922930200555, 6.60033486625472391882394591366, 7.37785589597875927836448293496, 8.025883698479201246532506719682, 8.753731866630101218626904909304, 9.492900956805119002456625676851