L(s) = 1 | + (−0.866 + 0.5i)2-s + (0.499 − 0.866i)4-s + (0.0340 + 0.0590i)5-s + (−2.19 + 1.48i)7-s + 0.999i·8-s + (−0.0590 − 0.0340i)10-s + (−0.866 − 0.5i)11-s − 2.44i·13-s + (1.15 − 2.38i)14-s + (−0.5 − 0.866i)16-s + (−0.817 + 1.41i)17-s + (6.28 − 3.62i)19-s + 0.0681·20-s + 0.999·22-s + (−0.405 + 0.233i)23-s + ⋯ |
L(s) = 1 | + (−0.612 + 0.353i)2-s + (0.249 − 0.433i)4-s + (0.0152 + 0.0263i)5-s + (−0.827 + 0.560i)7-s + 0.353i·8-s + (−0.0186 − 0.0107i)10-s + (−0.261 − 0.150i)11-s − 0.679i·13-s + (0.308 − 0.636i)14-s + (−0.125 − 0.216i)16-s + (−0.198 + 0.343i)17-s + (1.44 − 0.832i)19-s + 0.0152·20-s + 0.213·22-s + (−0.0844 + 0.0487i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.807 - 0.589i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.807 - 0.589i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.049283179\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.049283179\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.866 - 0.5i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (2.19 - 1.48i)T \) |
| 11 | \( 1 + (0.866 + 0.5i)T \) |
good | 5 | \( 1 + (-0.0340 - 0.0590i)T + (-2.5 + 4.33i)T^{2} \) |
| 13 | \( 1 + 2.44iT - 13T^{2} \) |
| 17 | \( 1 + (0.817 - 1.41i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-6.28 + 3.62i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (0.405 - 0.233i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 - 7.23iT - 29T^{2} \) |
| 31 | \( 1 + (0.201 + 0.116i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-2.74 - 4.75i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + 5.38T + 41T^{2} \) |
| 43 | \( 1 - 9.64T + 43T^{2} \) |
| 47 | \( 1 + (-4.31 - 7.46i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-7.34 - 4.23i)T + (26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (0.439 - 0.760i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (8.63 - 4.98i)T + (30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-1.52 + 2.64i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 15.8iT - 71T^{2} \) |
| 73 | \( 1 + (-6.76 - 3.90i)T + (36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-5.76 - 9.98i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 - 17.1T + 83T^{2} \) |
| 89 | \( 1 + (-5.33 - 9.24i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + 9.99iT - 97T^{2} \) |
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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
−9.430888034658628789965556044720, −8.990718614279377723079638030287, −8.064888609109936508059847412130, −7.27363042339905237491006976137, −6.43695799447843433256731935242, −5.67224830968862221825978454976, −4.83766164608747653543670574780, −3.31889641775730839643376172842, −2.54249615772210774631468221955, −0.867751934221223898710504501988,
0.75538822869407592697024860327, 2.17521287269402542179947594059, 3.32386733151507176540267794291, 4.10966042269207682219255683872, 5.37281168744074261285009442664, 6.38576189370233411156094302232, 7.27617458955811080473404533169, 7.76416539064474246709114293438, 8.957519951047449062750325330263, 9.535388700749631801783190881723