L(s) = 1 | + 4.24·5-s − 12.6·7-s + 17.8·11-s + 10i·13-s + 24.0i·17-s − 25.2i·19-s − 17.8i·23-s − 7.00·25-s − 15.5·29-s + 12.6·31-s − 53.6·35-s + 64i·37-s + 12.7i·41-s + 50.5i·43-s + 17.8i·47-s + ⋯ |
L(s) = 1 | + 0.848·5-s − 1.80·7-s + 1.62·11-s + 0.769i·13-s + 1.41i·17-s − 1.33i·19-s − 0.777i·23-s − 0.280·25-s − 0.536·29-s + 0.408·31-s − 1.53·35-s + 1.72i·37-s + 0.310i·41-s + 1.17i·43-s + 0.380i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.169 - 0.985i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.169 - 0.985i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(1.564210071\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.564210071\) |
\(L(2)\) |
|
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 | \( 1 - 4.24T + 25T^{2} \) |
| 7 | \( 1 + 12.6T + 49T^{2} \) |
| 11 | \( 1 - 17.8T + 121T^{2} \) |
| 13 | \( 1 - 10iT - 169T^{2} \) |
| 17 | \( 1 - 24.0iT - 289T^{2} \) |
| 19 | \( 1 + 25.2iT - 361T^{2} \) |
| 23 | \( 1 + 17.8iT - 529T^{2} \) |
| 29 | \( 1 + 15.5T + 841T^{2} \) |
| 31 | \( 1 - 12.6T + 961T^{2} \) |
| 37 | \( 1 - 64iT - 1.36e3T^{2} \) |
| 41 | \( 1 - 12.7iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 50.5iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 17.8iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 18.3T + 2.80e3T^{2} \) |
| 59 | \( 1 + 3.48e3T^{2} \) |
| 61 | \( 1 - 3.72e3T^{2} \) |
| 67 | \( 1 - 75.8iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 125. iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 96T + 5.32e3T^{2} \) |
| 79 | \( 1 + 63.2T + 6.24e3T^{2} \) |
| 83 | \( 1 - 125.T + 6.88e3T^{2} \) |
| 89 | \( 1 - 55.1iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 64T + 9.40e3T^{2} \) |
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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
−9.547295865562299371459466629576, −9.313808145752265573075141345404, −8.386402759490663400220152123125, −6.76268872463092142203536010711, −6.56320176994865821160491370057, −5.91875460685441175597083953963, −4.44635992719445859373515296997, −3.60293468988032955247884298400, −2.51912314891987799888956849521, −1.23190976933036467652879010057,
0.49631227341847242140424733338, 1.96892279335393385139732870892, 3.26757556611605649619726333978, 3.86058739048793352179836062495, 5.46657946832472904688423521972, 6.05544624814131004600765577036, 6.77694779435063096073700722007, 7.60288618057482812162888220833, 9.027045903909763215774862144169, 9.485930469088586886478369441776