L(s) = 1 | + (−0.309 + 0.951i)3-s + (−1.96 + 1.06i)5-s − 1.74·7-s + (−0.809 − 0.587i)9-s + (−1.87 + 1.36i)11-s + (−3.99 − 2.90i)13-s + (−0.407 − 2.19i)15-s + (1.15 + 3.55i)17-s + (−0.523 − 1.61i)19-s + (0.537 − 1.65i)21-s + (−7.02 + 5.10i)23-s + (2.72 − 4.19i)25-s + (0.809 − 0.587i)27-s + (0.964 − 2.96i)29-s + (2.95 + 9.09i)31-s + ⋯ |
L(s) = 1 | + (−0.178 + 0.549i)3-s + (−0.878 + 0.477i)5-s − 0.657·7-s + (−0.269 − 0.195i)9-s + (−0.565 + 0.411i)11-s + (−1.10 − 0.805i)13-s + (−0.105 − 0.567i)15-s + (0.280 + 0.862i)17-s + (−0.120 − 0.369i)19-s + (0.117 − 0.361i)21-s + (−1.46 + 1.06i)23-s + (0.544 − 0.838i)25-s + (0.155 − 0.113i)27-s + (0.179 − 0.551i)29-s + (0.530 + 1.63i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.990 - 0.135i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.990 - 0.135i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.0240342 + 0.353723i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0240342 + 0.353723i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (0.309 - 0.951i)T \) |
| 5 | \( 1 + (1.96 - 1.06i)T \) |
good | 7 | \( 1 + 1.74T + 7T^{2} \) |
| 11 | \( 1 + (1.87 - 1.36i)T + (3.39 - 10.4i)T^{2} \) |
| 13 | \( 1 + (3.99 + 2.90i)T + (4.01 + 12.3i)T^{2} \) |
| 17 | \( 1 + (-1.15 - 3.55i)T + (-13.7 + 9.99i)T^{2} \) |
| 19 | \( 1 + (0.523 + 1.61i)T + (-15.3 + 11.1i)T^{2} \) |
| 23 | \( 1 + (7.02 - 5.10i)T + (7.10 - 21.8i)T^{2} \) |
| 29 | \( 1 + (-0.964 + 2.96i)T + (-23.4 - 17.0i)T^{2} \) |
| 31 | \( 1 + (-2.95 - 9.09i)T + (-25.0 + 18.2i)T^{2} \) |
| 37 | \( 1 + (4.34 + 3.15i)T + (11.4 + 35.1i)T^{2} \) |
| 41 | \( 1 + (-7.05 - 5.12i)T + (12.6 + 38.9i)T^{2} \) |
| 43 | \( 1 - 2.86T + 43T^{2} \) |
| 47 | \( 1 + (2.61 - 8.04i)T + (-38.0 - 27.6i)T^{2} \) |
| 53 | \( 1 + (0.415 - 1.27i)T + (-42.8 - 31.1i)T^{2} \) |
| 59 | \( 1 + (3.54 + 2.57i)T + (18.2 + 56.1i)T^{2} \) |
| 61 | \( 1 + (-12.4 + 9.01i)T + (18.8 - 58.0i)T^{2} \) |
| 67 | \( 1 + (-2.85 - 8.77i)T + (-54.2 + 39.3i)T^{2} \) |
| 71 | \( 1 + (-0.00728 + 0.0224i)T + (-57.4 - 41.7i)T^{2} \) |
| 73 | \( 1 + (-0.827 + 0.601i)T + (22.5 - 69.4i)T^{2} \) |
| 79 | \( 1 + (0.246 - 0.759i)T + (-63.9 - 46.4i)T^{2} \) |
| 83 | \( 1 + (-0.732 - 2.25i)T + (-67.1 + 48.7i)T^{2} \) |
| 89 | \( 1 + (-3.93 + 2.85i)T + (27.5 - 84.6i)T^{2} \) |
| 97 | \( 1 + (1.06 - 3.29i)T + (-78.4 - 57.0i)T^{2} \) |
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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
−12.25161118791317148911882143323, −11.15742599612600576979893187460, −10.24685205528830097791428801148, −9.701330579381335190511372448974, −8.226112987733870430657098434504, −7.48112775244180707069518033604, −6.29829094039378119788463429900, −5.06965294413434562002964270325, −3.87789613103136790405302001379, −2.79268241098197412373312381121,
0.25077322556532707292336558227, 2.49924201694392635208321723206, 4.02781204414478963335257559368, 5.21430579520330894216592877292, 6.48239131029764526719600456074, 7.47153097747855061326452286686, 8.253254708707529574507832465129, 9.377507137038105562070200644057, 10.37192419377727630579668248665, 11.65266321902540839412356431521