L(s) = 1 | + (−1.22 + 0.707i)2-s + (0.866 − 1.5i)3-s + (0.999 − 1.73i)4-s + 2.44i·6-s + (6.77 + 1.74i)7-s + 2.82i·8-s + (−1.5 − 2.59i)9-s + (−3 + 5.19i)11-s + (−1.73 − 2.99i)12-s − 21.3·13-s + (−9.53 + 2.65i)14-s + (−2.00 − 3.46i)16-s + (4.47 − 7.75i)17-s + (3.67 + 2.12i)18-s + (6.25 − 3.61i)19-s + ⋯ |
L(s) = 1 | + (−0.612 + 0.353i)2-s + (0.288 − 0.5i)3-s + (0.249 − 0.433i)4-s + 0.408i·6-s + (0.968 + 0.248i)7-s + 0.353i·8-s + (−0.166 − 0.288i)9-s + (−0.272 + 0.472i)11-s + (−0.144 − 0.249i)12-s − 1.64·13-s + (−0.681 + 0.189i)14-s + (−0.125 − 0.216i)16-s + (0.263 − 0.456i)17-s + (0.204 + 0.117i)18-s + (0.329 − 0.190i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.831 + 0.555i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.831 + 0.555i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(1.588804230\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.588804230\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (1.22 - 0.707i)T \) |
| 3 | \( 1 + (-0.866 + 1.5i)T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (-6.77 - 1.74i)T \) |
good | 11 | \( 1 + (3 - 5.19i)T + (-60.5 - 104. i)T^{2} \) |
| 13 | \( 1 + 21.3T + 169T^{2} \) |
| 17 | \( 1 + (-4.47 + 7.75i)T + (-144.5 - 250. i)T^{2} \) |
| 19 | \( 1 + (-6.25 + 3.61i)T + (180.5 - 312. i)T^{2} \) |
| 23 | \( 1 + (-32.4 + 18.7i)T + (264.5 - 458. i)T^{2} \) |
| 29 | \( 1 - 33.9T + 841T^{2} \) |
| 31 | \( 1 + (-38.2 - 22.0i)T + (480.5 + 832. i)T^{2} \) |
| 37 | \( 1 + (24.2 - 13.9i)T + (684.5 - 1.18e3i)T^{2} \) |
| 41 | \( 1 + 54.8iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 1.48iT - 1.84e3T^{2} \) |
| 47 | \( 1 + (21.5 + 37.2i)T + (-1.10e3 + 1.91e3i)T^{2} \) |
| 53 | \( 1 + (74.0 + 42.7i)T + (1.40e3 + 2.43e3i)T^{2} \) |
| 59 | \( 1 + (-35.6 - 20.6i)T + (1.74e3 + 3.01e3i)T^{2} \) |
| 61 | \( 1 + (1.02 - 0.594i)T + (1.86e3 - 3.22e3i)T^{2} \) |
| 67 | \( 1 + (3.80 + 2.19i)T + (2.24e3 + 3.88e3i)T^{2} \) |
| 71 | \( 1 - 137.T + 5.04e3T^{2} \) |
| 73 | \( 1 + (-39.4 + 68.3i)T + (-2.66e3 - 4.61e3i)T^{2} \) |
| 79 | \( 1 + (-49.1 - 85.1i)T + (-3.12e3 + 5.40e3i)T^{2} \) |
| 83 | \( 1 - 110.T + 6.88e3T^{2} \) |
| 89 | \( 1 + (-18 + 10.3i)T + (3.96e3 - 6.85e3i)T^{2} \) |
| 97 | \( 1 - 10.9T + 9.40e3T^{2} \) |
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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
−9.558210882295394819062528533467, −8.631498158723159049682440164235, −8.026252323217177589718588459769, −7.17540768076072136876608530722, −6.68341362253227700226773026254, −5.11557096138511869118585182003, −4.85401286299366214340289296645, −2.89600400218659627488735502906, −2.05255201060356710295083610828, −0.71331569752112497765568035548,
1.02784114524428917915344318758, 2.38634561439140460998650701089, 3.30410531069305064906965322222, 4.60485434481237927420600197370, 5.19873582557427132843906372560, 6.60822613870403790946312828938, 7.78225414062527027102533356222, 8.009982992118415638886761708635, 9.119278719831802937221590370015, 9.795199090720382398816405936809