| L(s) = 1 | − 2.82i·2-s − 8.00·4-s − 12.6i·5-s + 22.6i·8-s − 35.8·10-s + 166. i·11-s + 140.·13-s + 64.0·16-s − 38.7i·17-s − 359.·19-s + 101. i·20-s + 470.·22-s − 22.2i·23-s + 464.·25-s − 398. i·26-s + ⋯ |
| L(s) = 1 | − 0.707i·2-s − 0.500·4-s − 0.506i·5-s + 0.353i·8-s − 0.358·10-s + 1.37i·11-s + 0.834·13-s + 0.250·16-s − 0.134i·17-s − 0.995·19-s + 0.253i·20-s + 0.972·22-s − 0.0419i·23-s + 0.743·25-s − 0.589i·26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.816 - 0.577i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.816 - 0.577i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(0.2997376689\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2997376689\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 2.82iT \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 + 12.6iT - 625T^{2} \) |
| 11 | \( 1 - 166. iT - 1.46e4T^{2} \) |
| 13 | \( 1 - 140.T + 2.85e4T^{2} \) |
| 17 | \( 1 + 38.7iT - 8.35e4T^{2} \) |
| 19 | \( 1 + 359.T + 1.30e5T^{2} \) |
| 23 | \( 1 + 22.2iT - 2.79e5T^{2} \) |
| 29 | \( 1 + 83.2iT - 7.07e5T^{2} \) |
| 31 | \( 1 + 20.6T + 9.23e5T^{2} \) |
| 37 | \( 1 + 1.36e3T + 1.87e6T^{2} \) |
| 41 | \( 1 + 2.21e3iT - 2.82e6T^{2} \) |
| 43 | \( 1 + 1.45e3T + 3.41e6T^{2} \) |
| 47 | \( 1 + 1.41e3iT - 4.87e6T^{2} \) |
| 53 | \( 1 + 2.59e3iT - 7.89e6T^{2} \) |
| 59 | \( 1 - 2.77e3iT - 1.21e7T^{2} \) |
| 61 | \( 1 + 1.73e3T + 1.38e7T^{2} \) |
| 67 | \( 1 + 952.T + 2.01e7T^{2} \) |
| 71 | \( 1 + 3.84e3iT - 2.54e7T^{2} \) |
| 73 | \( 1 + 3.60e3T + 2.83e7T^{2} \) |
| 79 | \( 1 + 6.83e3T + 3.89e7T^{2} \) |
| 83 | \( 1 - 9.44e3iT - 4.74e7T^{2} \) |
| 89 | \( 1 + 1.36e4iT - 6.27e7T^{2} \) |
| 97 | \( 1 - 1.03e4T + 8.85e7T^{2} \) |
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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
−8.999741842631636119065720223985, −8.571744396165299031425938080480, −7.41321173405789875211202699085, −6.48219854517573184001794535897, −5.26398636473346234669601092238, −4.49228817420609826068380005626, −3.60332900252689326777679340790, −2.26656738260703343754757269760, −1.39618718283296033140183797703, −0.06904011393766617503463954894,
1.26936997648480537122715784960, 2.91787153205054782031411374807, 3.76987727882478382036165005034, 4.91715091546209735597524891360, 6.06452959142565530116705319275, 6.42974346768012079636300183015, 7.50295316634611817656920463440, 8.504940864925170940870779711152, 8.834303775448100190560270364629, 10.10101460541444255098824387700
