| L(s) = 1 | + 0.585·3-s + 9.03i·5-s − 8.65·9-s − 12.4·11-s − 9.03i·13-s + 5.29i·15-s − 12.3·17-s + 28.8·19-s − 24.6i·23-s − 56.5·25-s − 10.3·27-s − 22.4i·29-s + 16.7i·31-s − 7.31·33-s − 16.2i·37-s + ⋯ |
| L(s) = 1 | + 0.195·3-s + 1.80i·5-s − 0.961·9-s − 1.13·11-s − 0.694i·13-s + 0.352i·15-s − 0.726·17-s + 1.51·19-s − 1.07i·23-s − 2.26·25-s − 0.383·27-s − 0.774i·29-s + 0.541i·31-s − 0.221·33-s − 0.439i·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1568 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.353 + 0.935i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1568 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.353 + 0.935i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.7520273807\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7520273807\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 - 0.585T + 9T^{2} \) |
| 5 | \( 1 - 9.03iT - 25T^{2} \) |
| 11 | \( 1 + 12.4T + 121T^{2} \) |
| 13 | \( 1 + 9.03iT - 169T^{2} \) |
| 17 | \( 1 + 12.3T + 289T^{2} \) |
| 19 | \( 1 - 28.8T + 361T^{2} \) |
| 23 | \( 1 + 24.6iT - 529T^{2} \) |
| 29 | \( 1 + 22.4iT - 841T^{2} \) |
| 31 | \( 1 - 16.7iT - 961T^{2} \) |
| 37 | \( 1 + 16.2iT - 1.36e3T^{2} \) |
| 41 | \( 1 + 6.97T + 1.68e3T^{2} \) |
| 43 | \( 1 - 22.8T + 1.84e3T^{2} \) |
| 47 | \( 1 - 6.19iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 8.01iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 30.4T + 3.48e3T^{2} \) |
| 61 | \( 1 + 15.2iT - 3.72e3T^{2} \) |
| 67 | \( 1 - 78.6T + 4.48e3T^{2} \) |
| 71 | \( 1 + 17.5iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 46.6T + 5.32e3T^{2} \) |
| 79 | \( 1 + 81.0iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 40.3T + 6.88e3T^{2} \) |
| 89 | \( 1 + 111.T + 7.92e3T^{2} \) |
| 97 | \( 1 - 164.T + 9.40e3T^{2} \) |
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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
−9.119009305161967970057964300223, −8.054314451404670742149599377212, −7.57852688123134749335136104064, −6.69011069200226699563550480188, −5.91196079355685377851569931092, −5.10878750584344960458469799639, −3.65492876357410112784320651035, −2.82570362486683027134447115199, −2.40992075052210234939570105991, −0.21795214343187376064699599784,
1.03383878028779031900908622495, 2.21753728605638925480511114858, 3.43288795277321093810966424696, 4.56308682617528990735439866493, 5.30188623110801713557867722721, 5.76788267882417320775758172606, 7.19883636540775947223505209493, 8.025802733764869588724995326852, 8.619866401519962917408746081273, 9.290628988913372923004121171988
