| L(s) = 1 | + 12.7i·2-s − 98·4-s + 63.6i·5-s + 524·7-s − 432. i·8-s − 810·10-s − 865. i·11-s + 344·13-s + 6.66e3i·14-s − 763.·16-s − 7.14e3i·17-s − 2.32e3·19-s − 6.23e3i·20-s + 1.10e4·22-s + 5.75e3i·23-s + ⋯ |
| L(s) = 1 | + 1.59i·2-s − 1.53·4-s + 0.509i·5-s + 1.52·7-s − 0.845i·8-s − 0.810·10-s − 0.650i·11-s + 0.156·13-s + 2.43i·14-s − 0.186·16-s − 1.45i·17-s − 0.338·19-s − 0.779i·20-s + 1.03·22-s + 0.472i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.577 - 0.816i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.577 - 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(0.590868 + 1.14146i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.590868 + 1.14146i\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| good | 2 | \( 1 - 12.7iT - 64T^{2} \) |
| 5 | \( 1 - 63.6iT - 1.56e4T^{2} \) |
| 7 | \( 1 - 524T + 1.17e5T^{2} \) |
| 11 | \( 1 + 865. iT - 1.77e6T^{2} \) |
| 13 | \( 1 - 344T + 4.82e6T^{2} \) |
| 17 | \( 1 + 7.14e3iT - 2.41e7T^{2} \) |
| 19 | \( 1 + 2.32e3T + 4.70e7T^{2} \) |
| 23 | \( 1 - 5.75e3iT - 1.48e8T^{2} \) |
| 29 | \( 1 - 2.31e4iT - 5.94e8T^{2} \) |
| 31 | \( 1 + 1.05e4T + 8.87e8T^{2} \) |
| 37 | \( 1 + 2.40e4T + 2.56e9T^{2} \) |
| 41 | \( 1 + 1.08e5iT - 4.75e9T^{2} \) |
| 43 | \( 1 + 9.09e4T + 6.32e9T^{2} \) |
| 47 | \( 1 - 1.28e5iT - 1.07e10T^{2} \) |
| 53 | \( 1 + 1.96e5iT - 2.21e10T^{2} \) |
| 59 | \( 1 - 3.98e4iT - 4.21e10T^{2} \) |
| 61 | \( 1 - 2.51e5T + 5.15e10T^{2} \) |
| 67 | \( 1 + 2.16e5T + 9.04e10T^{2} \) |
| 71 | \( 1 - 5.39e4iT - 1.28e11T^{2} \) |
| 73 | \( 1 + 3.08e5T + 1.51e11T^{2} \) |
| 79 | \( 1 + 5.40e5T + 2.43e11T^{2} \) |
| 83 | \( 1 - 9.32e5iT - 3.26e11T^{2} \) |
| 89 | \( 1 - 2.23e5iT - 4.96e11T^{2} \) |
| 97 | \( 1 + 3.71e4T + 8.32e11T^{2} \) |
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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
−20.76642025431293266364603068703, −18.49480449467101151869866197576, −17.55864277512361017371785462896, −16.15156916506410243693197017187, −14.75963706360005170911829272614, −13.91551896228703271751997671573, −11.22862469181549875908626782687, −8.611555529409450854191616753602, −7.17174394499640583861555333531, −5.18314750542268666231308173041,
1.65958232111328699218612838507, 4.50346005045654745950699163127, 8.508179297419580287920139270594, 10.42861804748758202934261723378, 11.72915108110501962597986674901, 13.01382348813606708720995813806, 14.79295483199091260110438881137, 17.24619132401129755765817406537, 18.47492421821085225434895206409, 19.97536038971398110781273009528
