| L(s) = 1 | + (−2.65 + 1.40i)3-s − 2.23i·5-s − 12.5·7-s + (5.07 − 7.43i)9-s + 0.188i·11-s + 10.5·13-s + (3.13 + 5.93i)15-s − 27.8i·17-s + 19.7·19-s + (33.2 − 17.5i)21-s − 15.8i·23-s − 5.00·25-s + (−3.06 + 26.8i)27-s + 27.2i·29-s − 45.3·31-s + ⋯ |
| L(s) = 1 | + (−0.884 + 0.466i)3-s − 0.447i·5-s − 1.79·7-s + (0.564 − 0.825i)9-s + 0.0171i·11-s + 0.810·13-s + (0.208 + 0.395i)15-s − 1.63i·17-s + 1.04·19-s + (1.58 − 0.836i)21-s − 0.690i·23-s − 0.200·25-s + (−0.113 + 0.993i)27-s + 0.940i·29-s − 1.46·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.466 - 0.884i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.466 - 0.884i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.3750654028\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3750654028\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (2.65 - 1.40i)T \) |
| 5 | \( 1 + 2.23iT \) |
| good | 7 | \( 1 + 12.5T + 49T^{2} \) |
| 11 | \( 1 - 0.188iT - 121T^{2} \) |
| 13 | \( 1 - 10.5T + 169T^{2} \) |
| 17 | \( 1 + 27.8iT - 289T^{2} \) |
| 19 | \( 1 - 19.7T + 361T^{2} \) |
| 23 | \( 1 + 15.8iT - 529T^{2} \) |
| 29 | \( 1 - 27.2iT - 841T^{2} \) |
| 31 | \( 1 + 45.3T + 961T^{2} \) |
| 37 | \( 1 + 43.8T + 1.36e3T^{2} \) |
| 41 | \( 1 - 76.1iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 23.6T + 1.84e3T^{2} \) |
| 47 | \( 1 + 25.8iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 45.0iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 7.81iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 6.83T + 3.72e3T^{2} \) |
| 67 | \( 1 + 24.4T + 4.48e3T^{2} \) |
| 71 | \( 1 - 110. iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 67.7T + 5.32e3T^{2} \) |
| 79 | \( 1 + 20.9T + 6.24e3T^{2} \) |
| 83 | \( 1 - 92.8iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 8.05iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 22.8T + 9.40e3T^{2} \) |
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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
−9.955159982460902860388120298330, −9.451808017823387420038586982359, −8.791978596010121645054854291770, −7.22564899514556590459496821077, −6.67128891527683900045071048637, −5.72557202533033580552119578930, −5.02083252331855823585434153549, −3.78934275197769113122629889385, −3.02449798871406228744663470451, −0.954141422234860755565980910763,
0.17127459308686970500302393070, 1.72117138926676271253164190846, 3.24451493040396486239859889217, 3.99564245264179513614288571281, 5.77993383421693879917201644472, 5.92273780419295335571144617203, 6.93577267660180588028429997100, 7.54645813113544647182450455600, 8.825567512263758045201860161133, 9.727874055429830013386198921000
