L(s) = 1 | + (−0.707 − 1.22i)2-s + (1.5 + 0.866i)3-s + (−0.999 + 1.73i)4-s − 2.44i·6-s + (6.38 − 2.86i)7-s + 2.82·8-s + (1.5 + 2.59i)9-s + (−9.98 + 17.2i)11-s + (−2.99 + 1.73i)12-s − 3.49i·13-s + (−8.02 − 5.80i)14-s + (−2.00 − 3.46i)16-s + (−15.7 − 9.12i)17-s + (2.12 − 3.67i)18-s + (−21.3 + 12.3i)19-s + ⋯ |
L(s) = 1 | + (−0.353 − 0.612i)2-s + (0.5 + 0.288i)3-s + (−0.249 + 0.433i)4-s − 0.408i·6-s + (0.912 − 0.408i)7-s + 0.353·8-s + (0.166 + 0.288i)9-s + (−0.907 + 1.57i)11-s + (−0.249 + 0.144i)12-s − 0.269i·13-s + (−0.572 − 0.414i)14-s + (−0.125 − 0.216i)16-s + (−0.929 − 0.536i)17-s + (0.117 − 0.204i)18-s + (−1.12 + 0.647i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.520 - 0.853i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.520 - 0.853i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.7138345842\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7138345842\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.707 + 1.22i)T \) |
| 3 | \( 1 + (-1.5 - 0.866i)T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (-6.38 + 2.86i)T \) |
good | 11 | \( 1 + (9.98 - 17.2i)T + (-60.5 - 104. i)T^{2} \) |
| 13 | \( 1 + 3.49iT - 169T^{2} \) |
| 17 | \( 1 + (15.7 + 9.12i)T + (144.5 + 250. i)T^{2} \) |
| 19 | \( 1 + (21.3 - 12.3i)T + (180.5 - 312. i)T^{2} \) |
| 23 | \( 1 + (-12.5 - 21.8i)T + (-264.5 + 458. i)T^{2} \) |
| 29 | \( 1 + 53.1T + 841T^{2} \) |
| 31 | \( 1 + (-26.0 - 15.0i)T + (480.5 + 832. i)T^{2} \) |
| 37 | \( 1 + (23.3 + 40.5i)T + (-684.5 + 1.18e3i)T^{2} \) |
| 41 | \( 1 + 31.5iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 64.4T + 1.84e3T^{2} \) |
| 47 | \( 1 + (-24.3 + 14.0i)T + (1.10e3 - 1.91e3i)T^{2} \) |
| 53 | \( 1 + (32.4 - 56.1i)T + (-1.40e3 - 2.43e3i)T^{2} \) |
| 59 | \( 1 + (-86.7 - 50.0i)T + (1.74e3 + 3.01e3i)T^{2} \) |
| 61 | \( 1 + (-6.94 + 4.01i)T + (1.86e3 - 3.22e3i)T^{2} \) |
| 67 | \( 1 + (-8.13 + 14.0i)T + (-2.24e3 - 3.88e3i)T^{2} \) |
| 71 | \( 1 + 107.T + 5.04e3T^{2} \) |
| 73 | \( 1 + (44.7 + 25.8i)T + (2.66e3 + 4.61e3i)T^{2} \) |
| 79 | \( 1 + (-10.9 - 19.0i)T + (-3.12e3 + 5.40e3i)T^{2} \) |
| 83 | \( 1 + 0.417iT - 6.88e3T^{2} \) |
| 89 | \( 1 + (96.3 - 55.6i)T + (3.96e3 - 6.85e3i)T^{2} \) |
| 97 | \( 1 + 74.2iT - 9.40e3T^{2} \) |
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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
−10.14220226969386093250939698066, −9.207695462697528528762420483698, −8.483264371984271612879834835323, −7.55005738808120852405539398559, −7.13299098086113807400289239994, −5.37228019663819243086269216979, −4.58987826585938900911824796632, −3.78701490285282346567488929110, −2.39424717330093148122486938913, −1.70980523777215170586983426079,
0.21498287632065604004210717235, 1.79106334731846615941163192384, 2.86451425987261879624135929775, 4.28681227066640265336044490038, 5.22996407577117797790734299036, 6.18008377239939312370006667471, 6.93724698794978331451449951217, 8.166206956547328630548696404784, 8.395670333009227196368730976731, 9.008816671144870323689516008990