L(s) = 1 | + (−0.5 − 0.866i)2-s + (−0.499 + 0.866i)4-s + (2.63 − 0.189i)7-s + 0.999·8-s + (−3.44 − 1.99i)11-s + 0.0681·13-s + (−1.48 − 2.19i)14-s + (−0.5 − 0.866i)16-s + (−6.34 − 3.66i)17-s + (1.76 − 1.01i)19-s + 3.98i·22-s + (1.86 + 3.23i)23-s + (−0.0340 − 0.0590i)26-s + (−1.15 + 2.38i)28-s + 0.898i·29-s + ⋯ |
L(s) = 1 | + (−0.353 − 0.612i)2-s + (−0.249 + 0.433i)4-s + (0.997 − 0.0716i)7-s + 0.353·8-s + (−1.03 − 0.600i)11-s + 0.0189·13-s + (−0.396 − 0.585i)14-s + (−0.125 − 0.216i)16-s + (−1.53 − 0.888i)17-s + (0.404 − 0.233i)19-s + 0.848i·22-s + (0.389 + 0.673i)23-s + (−0.00668 − 0.0115i)26-s + (−0.218 + 0.449i)28-s + 0.166i·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.766 - 0.641i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.766 - 0.641i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.1387751541\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.1387751541\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.5 + 0.866i)T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (-2.63 + 0.189i)T \) |
good | 11 | \( 1 + (3.44 + 1.99i)T + (5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 - 0.0681T + 13T^{2} \) |
| 17 | \( 1 + (6.34 + 3.66i)T + (8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-1.76 + 1.01i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-1.86 - 3.23i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 - 0.898iT - 29T^{2} \) |
| 31 | \( 1 + (4.18 + 2.41i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (3.52 - 2.03i)T + (18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + 1.68T + 41T^{2} \) |
| 43 | \( 1 - 0.964iT - 43T^{2} \) |
| 47 | \( 1 + (1.43 - 0.830i)T + (23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (6.61 - 11.4i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (5.32 - 9.21i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-6.51 + 3.76i)T + (30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (9.23 + 5.33i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + 9.93iT - 71T^{2} \) |
| 73 | \( 1 + (5.82 - 10.0i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-8.77 - 15.1i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + 14.3iT - 83T^{2} \) |
| 89 | \( 1 + (0.913 + 1.58i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + 17.1T + 97T^{2} \) |
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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
−8.332356067450404587230774362672, −7.59473421475479303520270086300, −7.01193892079372281291650690493, −5.77382422204865904865808342127, −4.99226144364956942486852076942, −4.37314444019068317810661614550, −3.18632256833160326565675341360, −2.42815388847553360499543314273, −1.39589328307325554929370418436, −0.04697120524042884131429270921,
1.61802173360731140562121077526, 2.40428609281301037885508827510, 3.84207035168136673172734155855, 4.81539518245941789874125620495, 5.20367176380586134269775683530, 6.23351308514720574883471462293, 6.99966811456239666341151337140, 7.69703838884630418186534675080, 8.381594521162556864466191939736, 8.856494269705602525619952039104