| L(s) = 1 | + 1.41i·2-s − 2.00·4-s + 8.89i·5-s − 2.82i·8-s − 12.5·10-s − 17.7i·11-s − 2.58·13-s + 4.00·16-s − 25.8i·17-s − 20·19-s − 17.7i·20-s + 25.1·22-s − 17.7i·23-s − 54.1·25-s − 3.65i·26-s + ⋯ |
| L(s) = 1 | + 0.707i·2-s − 0.500·4-s + 1.77i·5-s − 0.353i·8-s − 1.25·10-s − 1.61i·11-s − 0.198·13-s + 0.250·16-s − 1.52i·17-s − 1.05·19-s − 0.889i·20-s + 1.14·22-s − 0.773i·23-s − 2.16·25-s − 0.140i·26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.816 + 0.577i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.816 + 0.577i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.8624167340\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8624167340\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 1.41iT \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 - 8.89iT - 25T^{2} \) |
| 11 | \( 1 + 17.7iT - 121T^{2} \) |
| 13 | \( 1 + 2.58T + 169T^{2} \) |
| 17 | \( 1 + 25.8iT - 289T^{2} \) |
| 19 | \( 1 + 20T + 361T^{2} \) |
| 23 | \( 1 + 17.7iT - 529T^{2} \) |
| 29 | \( 1 + 11.9iT - 841T^{2} \) |
| 31 | \( 1 - 17.1T + 961T^{2} \) |
| 37 | \( 1 - 38T + 1.36e3T^{2} \) |
| 41 | \( 1 + 15.7iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 43.4T + 1.84e3T^{2} \) |
| 47 | \( 1 + 16.9iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 85.5iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 1.64iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 100.T + 3.72e3T^{2} \) |
| 67 | \( 1 - 36.6T + 4.48e3T^{2} \) |
| 71 | \( 1 + 17.7iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 28.9T + 5.32e3T^{2} \) |
| 79 | \( 1 - 118.T + 6.24e3T^{2} \) |
| 83 | \( 1 + 120. iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 139. iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 44.4T + 9.40e3T^{2} \) |
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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
−9.896466213157686488732588377237, −8.899172980072316951531512822956, −7.989160345681236192658396277988, −7.18894495424599368834722920403, −6.39581867388779198405957859918, −5.88854674261695690186399994311, −4.52631805254222033115541580933, −3.29645970813294176616998095564, −2.58594777894966329508181404296, −0.29130725124109142278209907130,
1.32421606524410398397416818128, 2.08873890762636267050547469465, 3.89632410405985058936055770376, 4.56808557060107565393695849427, 5.27579683939561764961659537325, 6.47302127890892796511219176598, 7.88504752118313328368449733275, 8.409080484206988633236237406237, 9.377791944693159306251872105370, 9.842635270816689860794496660884
