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 + (−2.55 − 6.51i)7-s + 2.82i·8-s + (−1.5 − 2.59i)9-s + (−5.79 + 10.0i)11-s + (1.73 + 2.99i)12-s + 7.86·13-s + (7.73 + 6.17i)14-s + (−2.00 − 3.46i)16-s + (13.8 − 23.9i)17-s + (3.67 + 2.12i)18-s + (−27.2 + 15.7i)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 + (−0.365 − 0.930i)7-s + 0.353i·8-s + (−0.166 − 0.288i)9-s + (−0.526 + 0.912i)11-s + (0.144 + 0.249i)12-s + 0.604·13-s + (0.552 + 0.440i)14-s + (−0.125 − 0.216i)16-s + (0.811 − 1.40i)17-s + (0.204 + 0.117i)18-s + (−1.43 + 0.826i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.821 - 0.570i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.821 - 0.570i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.5383822714\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5383822714\) |
\(L(2)\) |
|
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 | \( 1 + (2.55 + 6.51i)T \) |
good | 11 | \( 1 + (5.79 - 10.0i)T + (-60.5 - 104. i)T^{2} \) |
| 13 | \( 1 - 7.86T + 169T^{2} \) |
| 17 | \( 1 + (-13.8 + 23.9i)T + (-144.5 - 250. i)T^{2} \) |
| 19 | \( 1 + (27.2 - 15.7i)T + (180.5 - 312. i)T^{2} \) |
| 23 | \( 1 + (-15.7 + 9.07i)T + (264.5 - 458. i)T^{2} \) |
| 29 | \( 1 - 2.30T + 841T^{2} \) |
| 31 | \( 1 + (4.55 + 2.63i)T + (480.5 + 832. i)T^{2} \) |
| 37 | \( 1 + (1.72 - 0.993i)T + (684.5 - 1.18e3i)T^{2} \) |
| 41 | \( 1 + 22.1iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 49.8iT - 1.84e3T^{2} \) |
| 47 | \( 1 + (-38.3 - 66.3i)T + (-1.10e3 + 1.91e3i)T^{2} \) |
| 53 | \( 1 + (49.4 + 28.5i)T + (1.40e3 + 2.43e3i)T^{2} \) |
| 59 | \( 1 + (-60.9 - 35.1i)T + (1.74e3 + 3.01e3i)T^{2} \) |
| 61 | \( 1 + (58.5 - 33.8i)T + (1.86e3 - 3.22e3i)T^{2} \) |
| 67 | \( 1 + (-85.0 - 49.0i)T + (2.24e3 + 3.88e3i)T^{2} \) |
| 71 | \( 1 + 34.2T + 5.04e3T^{2} \) |
| 73 | \( 1 + (9.74 - 16.8i)T + (-2.66e3 - 4.61e3i)T^{2} \) |
| 79 | \( 1 + (-45.2 - 78.3i)T + (-3.12e3 + 5.40e3i)T^{2} \) |
| 83 | \( 1 + 133.T + 6.88e3T^{2} \) |
| 89 | \( 1 + (-9.58 + 5.53i)T + (3.96e3 - 6.85e3i)T^{2} \) |
| 97 | \( 1 + 72.3T + 9.40e3T^{2} \) |
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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
−9.990154604987793175918048684247, −9.425242828664298385431449223263, −8.387794999909608812972569025437, −7.51673929217553430767330431425, −6.82345164435447854905244890800, −5.91199899341472534495346593444, −4.86747499111893211289137093367, −4.01774729260445540381396049142, −2.71992779326231998194119570702, −1.10207093158621250197072954102,
0.24222668257609907771258657264, 1.66339015223245026271960105979, 2.75221202303341096445561786419, 3.74584907757843245804125130019, 5.30909118360717144333989317080, 6.08737577764803401549566624638, 6.79611914777569838091921809540, 8.047682809826791589283031053920, 8.527748008133269980545924847929, 9.219223942616093999380076597649