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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.943 + 0.332i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.943 + 0.332i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(2.182707732\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.182707732\) |
\(L(2)\) |
|
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
−9.732291741581433462295983323399, −8.485224253753813185817781039729, −8.106440198294511066126916879392, −6.77559767657560830633940140364, −6.26306112564707637536575936595, −5.68723740315211049786303477665, −4.41911722867365113228223963720, −3.40396672457536603883092739752, −2.42862841591821593245333338417, −0.71830839443367901878976019797,
1.01139455512015196930870956304, 2.59614708834694592103916590168, 3.72282298382036659245105463868, 4.22150721966356071134630387444, 5.52073985096567679832440907831, 6.17079868245786276784284368189, 6.94392970098664243449257133739, 8.203893111072822860552220147149, 9.137926158243656245773436769154, 10.06806260244502660295218305546