| L(s) = 1 | − 5.18·5-s − 3.67·7-s + 12.8·11-s − 8.72i·13-s − 18.6i·17-s + 30.7i·19-s − 20.7i·23-s + 1.88·25-s − 27.8·29-s + 16.6·31-s + 19.0·35-s + 60.5i·37-s + 28.2i·41-s + 51.8i·43-s + 69.2i·47-s + ⋯ |
| L(s) = 1 | − 1.03·5-s − 0.524·7-s + 1.17·11-s − 0.670i·13-s − 1.09i·17-s + 1.61i·19-s − 0.903i·23-s + 0.0752·25-s − 0.961·29-s + 0.536·31-s + 0.544·35-s + 1.63i·37-s + 0.690i·41-s + 1.20i·43-s + 1.47i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0610 - 0.998i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.0610 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.8790834939\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8790834939\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + 5.18T + 25T^{2} \) |
| 7 | \( 1 + 3.67T + 49T^{2} \) |
| 11 | \( 1 - 12.8T + 121T^{2} \) |
| 13 | \( 1 + 8.72iT - 169T^{2} \) |
| 17 | \( 1 + 18.6iT - 289T^{2} \) |
| 19 | \( 1 - 30.7iT - 361T^{2} \) |
| 23 | \( 1 + 20.7iT - 529T^{2} \) |
| 29 | \( 1 + 27.8T + 841T^{2} \) |
| 31 | \( 1 - 16.6T + 961T^{2} \) |
| 37 | \( 1 - 60.5iT - 1.36e3T^{2} \) |
| 41 | \( 1 - 28.2iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 51.8iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 69.2iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 28.5T + 2.80e3T^{2} \) |
| 59 | \( 1 - 61.1T + 3.48e3T^{2} \) |
| 61 | \( 1 - 105. iT - 3.72e3T^{2} \) |
| 67 | \( 1 - 16.9iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 26.1iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 102.T + 5.32e3T^{2} \) |
| 79 | \( 1 - 92.2T + 6.24e3T^{2} \) |
| 83 | \( 1 + 36.4T + 6.88e3T^{2} \) |
| 89 | \( 1 + 73.5iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 186.T + 9.40e3T^{2} \) |
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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
−10.03636079365424689538631472746, −9.461925919538469361008312482550, −8.325106170185498777043250057476, −7.76627939484203069022028040264, −6.74210981637201167403456181069, −5.97807227352605601528292760406, −4.66636934234438052566677196502, −3.79189537710122309498654629439, −2.93237805403089410064888819028, −1.13316767003147058429016751628,
0.33644015989172098564050735859, 1.97769646067554746484348826782, 3.64271520530884667706869018013, 3.97516071198196436787025068960, 5.28711405276495472063421485406, 6.52381480689869743481132800751, 7.06965334032095674087267294244, 8.053263635141160170117192107349, 9.034874662352132552407668464286, 9.497522093805043436812598976134
