| L(s) = 1 | − 1.16i·2-s + 2.64·4-s + 5.40i·5-s + 2.64·7-s − 7.73i·8-s + 6.29·10-s + 18.1i·11-s + 10.2·13-s − 3.07i·14-s + 1.58·16-s + 4.24i·17-s + 3.35·19-s + 14.3i·20-s + 21.1·22-s − 28.9i·23-s + ⋯ |
| L(s) = 1 | − 0.581i·2-s + 0.661·4-s + 1.08i·5-s + 0.377·7-s − 0.966i·8-s + 0.629·10-s + 1.64i·11-s + 0.786·13-s − 0.219i·14-s + 0.0989·16-s + 0.249i·17-s + 0.176·19-s + 0.715i·20-s + 0.959·22-s − 1.25i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.88453\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.88453\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 - 2.64T \) |
| good | 2 | \( 1 + 1.16iT - 4T^{2} \) |
| 5 | \( 1 - 5.40iT - 25T^{2} \) |
| 11 | \( 1 - 18.1iT - 121T^{2} \) |
| 13 | \( 1 - 10.2T + 169T^{2} \) |
| 17 | \( 1 - 4.24iT - 289T^{2} \) |
| 19 | \( 1 - 3.35T + 361T^{2} \) |
| 23 | \( 1 + 28.9iT - 529T^{2} \) |
| 29 | \( 1 + 36.0iT - 841T^{2} \) |
| 31 | \( 1 - 34.5T + 961T^{2} \) |
| 37 | \( 1 + 60.0T + 1.36e3T^{2} \) |
| 41 | \( 1 - 71.9iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 46.6T + 1.84e3T^{2} \) |
| 47 | \( 1 - 18.2iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 43.3iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 53.1iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 65.9T + 3.72e3T^{2} \) |
| 67 | \( 1 - 36.3T + 4.48e3T^{2} \) |
| 71 | \( 1 + 0.219iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 11.8T + 5.32e3T^{2} \) |
| 79 | \( 1 + 144.T + 6.24e3T^{2} \) |
| 83 | \( 1 + 99.0iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 92.7iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 96.5T + 9.40e3T^{2} \) |
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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
−12.10753514049795733371178876534, −11.33512943306578078093292845958, −10.41665928850652686436126641845, −9.857845069247318393370913030119, −8.161514052114040203208925141423, −7.00345307192654192995220586709, −6.35013629387965750045088782790, −4.46562978923733278612105752396, −3.02454824001006349512758653221, −1.83344230492164416300197464281,
1.32055352489388941270409839888, 3.35682206103020906336606590577, 5.16018856626205032548801493639, 5.89053352468769319973986974840, 7.19115274670839642728840404840, 8.441428314539462904843751559097, 8.808837469738253742789583324322, 10.55867077038773093179422245888, 11.41745568565540258869291228899, 12.20556134884321639714268654325
