| L(s) = 1 | + (2.69 − 1.32i)3-s − 0.640i·5-s + 2.72·7-s + (5.47 − 7.14i)9-s − 11.2i·11-s − 5.25·13-s + (−0.849 − 1.72i)15-s − 14.8i·17-s + 15.0·19-s + (7.31 − 3.61i)21-s + 36.4i·23-s + 24.5·25-s + (5.24 − 26.4i)27-s − 51.7i·29-s − 36.5·31-s + ⋯ |
| L(s) = 1 | + (0.896 − 0.442i)3-s − 0.128i·5-s + 0.388·7-s + (0.608 − 0.793i)9-s − 1.02i·11-s − 0.403·13-s + (−0.0566 − 0.114i)15-s − 0.874i·17-s + 0.793·19-s + (0.348 − 0.171i)21-s + 1.58i·23-s + 0.983·25-s + (0.194 − 0.980i)27-s − 1.78i·29-s − 1.17·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.442 + 0.896i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.442 + 0.896i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(2.00038 - 1.24359i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.00038 - 1.24359i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (-2.69 + 1.32i)T \) |
| good | 5 | \( 1 + 0.640iT - 25T^{2} \) |
| 7 | \( 1 - 2.72T + 49T^{2} \) |
| 11 | \( 1 + 11.2iT - 121T^{2} \) |
| 13 | \( 1 + 5.25T + 169T^{2} \) |
| 17 | \( 1 + 14.8iT - 289T^{2} \) |
| 19 | \( 1 - 15.0T + 361T^{2} \) |
| 23 | \( 1 - 36.4iT - 529T^{2} \) |
| 29 | \( 1 + 51.7iT - 841T^{2} \) |
| 31 | \( 1 + 36.5T + 961T^{2} \) |
| 37 | \( 1 - 63.6T + 1.36e3T^{2} \) |
| 41 | \( 1 - 12.1iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 11.8T + 1.84e3T^{2} \) |
| 47 | \( 1 - 61.1iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 59.1iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 37.2iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 58.1T + 3.72e3T^{2} \) |
| 67 | \( 1 - 23.0T + 4.48e3T^{2} \) |
| 71 | \( 1 - 7.29iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 73.4T + 5.32e3T^{2} \) |
| 79 | \( 1 - 58.5T + 6.24e3T^{2} \) |
| 83 | \( 1 - 32.3iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 112. iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 80.0T + 9.40e3T^{2} \) |
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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
−11.18126769190662395472055903590, −9.675474251168946409471505379977, −9.210713390941147470972317155430, −7.997744306463461348559513379156, −7.52800056249670874774621814691, −6.26021929297287542581539893143, −5.05272162940116153190431927820, −3.65344511831264177838484568559, −2.58937487059671341088290038810, −1.03453143870669941120534814728,
1.78971122078102450730894375999, 3.03074045857510159345439850913, 4.31706384912619951346323518839, 5.14798194466945600148460143175, 6.79234979990685284082896741918, 7.63769048206776040205521456717, 8.589095580498849476982579243694, 9.411349989541419422721403150707, 10.33788258856438132672243452206, 10.99689260678798524602447144080
