| L(s) = 1 | − 1.41i·2-s − 2.00·4-s + 4.01i·5-s + 2.64·7-s + 2.82i·8-s + 5.67·10-s − 3.36i·11-s − 20.7·13-s − 3.74i·14-s + 4.00·16-s − 9.33i·17-s + 9.84·19-s − 8.02i·20-s − 4.75·22-s + 35.8i·23-s + ⋯ |
| L(s) = 1 | − 0.707i·2-s − 0.500·4-s + 0.802i·5-s + 0.377·7-s + 0.353i·8-s + 0.567·10-s − 0.305i·11-s − 1.59·13-s − 0.267i·14-s + 0.250·16-s − 0.548i·17-s + 0.518·19-s − 0.401i·20-s − 0.216·22-s + 1.55i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1134 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1134 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.4104147647\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4104147647\) |
| \(L(2)\) |
|
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 | \( 1 - 2.64T \) |
| good | 5 | \( 1 - 4.01iT - 25T^{2} \) |
| 11 | \( 1 + 3.36iT - 121T^{2} \) |
| 13 | \( 1 + 20.7T + 169T^{2} \) |
| 17 | \( 1 + 9.33iT - 289T^{2} \) |
| 19 | \( 1 - 9.84T + 361T^{2} \) |
| 23 | \( 1 - 35.8iT - 529T^{2} \) |
| 29 | \( 1 + 28.7iT - 841T^{2} \) |
| 31 | \( 1 + 3.37T + 961T^{2} \) |
| 37 | \( 1 + 65.0T + 1.36e3T^{2} \) |
| 41 | \( 1 + 24.2iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 24.6T + 1.84e3T^{2} \) |
| 47 | \( 1 + 46.9iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 63.1iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 50.4iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 37.6T + 3.72e3T^{2} \) |
| 67 | \( 1 + 53.6T + 4.48e3T^{2} \) |
| 71 | \( 1 + 84.1iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 119.T + 5.32e3T^{2} \) |
| 79 | \( 1 + 146.T + 6.24e3T^{2} \) |
| 83 | \( 1 - 97.9iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 78.6iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 91.4T + 9.40e3T^{2} \) |
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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
−9.511664484942586804815300233237, −8.486115783234551124532732344916, −7.46101818288597290025556931680, −6.95533051214868296992086645296, −5.54209506112722601964796545783, −4.89860948489894191326090532779, −3.63420762987466477465184135933, −2.80766003218689705701660999662, −1.79262622368074241373606525439, −0.12312886795513731682198576572,
1.39055310999081565225151309102, 2.83436213427158386849070820543, 4.38624004360122892799020101504, 4.87226943640608093670635774723, 5.69299502801188230021007127980, 6.89070335552484669806331150745, 7.46227320843652911089255967097, 8.486555793154766885979272219808, 8.933627065816165535423761953283, 9.947331674794641890703915174071
