L(s) = 1 | + (−1.61 − 1.18i)2-s + (0.455 − 0.788i)3-s + (1.21 + 3.81i)4-s + (−3.17 − 5.49i)5-s + (−1.66 + 0.735i)6-s + (2.54 − 7.58i)8-s + (4.08 + 7.07i)9-s + (−1.36 + 12.6i)10-s + (11.4 + 6.60i)11-s + (3.55 + 0.780i)12-s + 19.4·13-s − 5.77·15-s + (−13.0 + 9.23i)16-s + (−13.7 − 7.96i)17-s + (1.76 − 16.2i)18-s + (8.22 + 14.2i)19-s + ⋯ |
L(s) = 1 | + (−0.807 − 0.590i)2-s + (0.151 − 0.262i)3-s + (0.302 + 0.953i)4-s + (−0.634 − 1.09i)5-s + (−0.277 + 0.122i)6-s + (0.318 − 0.948i)8-s + (0.453 + 0.786i)9-s + (−0.136 + 1.26i)10-s + (1.04 + 0.600i)11-s + (0.296 + 0.0650i)12-s + 1.49·13-s − 0.385·15-s + (−0.816 + 0.577i)16-s + (−0.811 − 0.468i)17-s + (0.0977 − 0.902i)18-s + (0.433 + 0.750i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 392 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.136 + 0.990i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 392 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.136 + 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.906858 - 0.790853i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.906858 - 0.790853i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (1.61 + 1.18i)T \) |
| 7 | \( 1 \) |
good | 3 | \( 1 + (-0.455 + 0.788i)T + (-4.5 - 7.79i)T^{2} \) |
| 5 | \( 1 + (3.17 + 5.49i)T + (-12.5 + 21.6i)T^{2} \) |
| 11 | \( 1 + (-11.4 - 6.60i)T + (60.5 + 104. i)T^{2} \) |
| 13 | \( 1 - 19.4T + 169T^{2} \) |
| 17 | \( 1 + (13.7 + 7.96i)T + (144.5 + 250. i)T^{2} \) |
| 19 | \( 1 + (-8.22 - 14.2i)T + (-180.5 + 312. i)T^{2} \) |
| 23 | \( 1 + (11.9 + 20.7i)T + (-264.5 + 458. i)T^{2} \) |
| 29 | \( 1 + 16.6iT - 841T^{2} \) |
| 31 | \( 1 + (-11.1 - 6.42i)T + (480.5 + 832. i)T^{2} \) |
| 37 | \( 1 + (-41.1 + 23.7i)T + (684.5 - 1.18e3i)T^{2} \) |
| 41 | \( 1 + 6.49iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 33.2iT - 1.84e3T^{2} \) |
| 47 | \( 1 + (-18.9 + 10.9i)T + (1.10e3 - 1.91e3i)T^{2} \) |
| 53 | \( 1 + (32.2 + 18.5i)T + (1.40e3 + 2.43e3i)T^{2} \) |
| 59 | \( 1 + (-27.3 + 47.3i)T + (-1.74e3 - 3.01e3i)T^{2} \) |
| 61 | \( 1 + (-5.12 - 8.87i)T + (-1.86e3 + 3.22e3i)T^{2} \) |
| 67 | \( 1 + (14.8 + 8.56i)T + (2.24e3 + 3.88e3i)T^{2} \) |
| 71 | \( 1 - 32.0T + 5.04e3T^{2} \) |
| 73 | \( 1 + (92.8 + 53.5i)T + (2.66e3 + 4.61e3i)T^{2} \) |
| 79 | \( 1 + (-29.1 - 50.4i)T + (-3.12e3 + 5.40e3i)T^{2} \) |
| 83 | \( 1 - 36.3T + 6.88e3T^{2} \) |
| 89 | \( 1 + (0.929 - 0.536i)T + (3.96e3 - 6.85e3i)T^{2} \) |
| 97 | \( 1 - 169. iT - 9.40e3T^{2} \) |
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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
−10.93867422452305307972513010079, −9.891942395290899821047558136377, −8.898950901490782470873182548858, −8.339173438581897215221228606611, −7.49220931641106808431787350603, −6.37282795981071077482567802815, −4.56161486252284203096401337787, −3.83931112591923107550689551751, −2.00705464639102490011323047553, −0.880447081404240070904149591196,
1.17047059021069097020873482148, 3.19984253623598338571961038893, 4.20963944319674493667498166256, 6.11164189579774779288245402596, 6.56931782619038632348489698921, 7.54014460950376802057040077547, 8.645073721819464196794030275244, 9.283933989975493082683914961457, 10.32959736109194144706548419754, 11.29340978596425563920381843265