| L(s) = 1 | − 2.56i·2-s − 2.59·4-s + 0.662i·5-s − 12.8·7-s − 3.60i·8-s + 1.70·10-s − 20.5i·11-s + 4.92·13-s + 32.9i·14-s − 19.6·16-s − 3.69i·17-s − 1.02·19-s − 1.71i·20-s − 52.7·22-s + 26.1i·23-s + ⋯ |
| L(s) = 1 | − 1.28i·2-s − 0.648·4-s + 0.132i·5-s − 1.83·7-s − 0.451i·8-s + 0.170·10-s − 1.86i·11-s + 0.378·13-s + 2.35i·14-s − 1.22·16-s − 0.217i·17-s − 0.0539·19-s − 0.0859i·20-s − 2.39·22-s + 1.13i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1143 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.577 - 0.816i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1143 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.577 - 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.08995723319\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.08995723319\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 127 | \( 1 - 11.2T \) |
| good | 2 | \( 1 + 2.56iT - 4T^{2} \) |
| 5 | \( 1 - 0.662iT - 25T^{2} \) |
| 7 | \( 1 + 12.8T + 49T^{2} \) |
| 11 | \( 1 + 20.5iT - 121T^{2} \) |
| 13 | \( 1 - 4.92T + 169T^{2} \) |
| 17 | \( 1 + 3.69iT - 289T^{2} \) |
| 19 | \( 1 + 1.02T + 361T^{2} \) |
| 23 | \( 1 - 26.1iT - 529T^{2} \) |
| 29 | \( 1 + 27.6iT - 841T^{2} \) |
| 31 | \( 1 + 29.6T + 961T^{2} \) |
| 37 | \( 1 + 2.50T + 1.36e3T^{2} \) |
| 41 | \( 1 - 69.8iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 66.7T + 1.84e3T^{2} \) |
| 47 | \( 1 + 81.6iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 81.3iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 66.1iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 5.87T + 3.72e3T^{2} \) |
| 67 | \( 1 - 60.9T + 4.48e3T^{2} \) |
| 71 | \( 1 - 115. iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 104.T + 5.32e3T^{2} \) |
| 79 | \( 1 - 15.1T + 6.24e3T^{2} \) |
| 83 | \( 1 - 0.533iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 76.1iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 89.0T + 9.40e3T^{2} \) |
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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
−9.216127035751659810780769359619, −8.471672175864838461329076999270, −7.08043668951345535526325321407, −6.34695169055240640314512291251, −5.58656629384187394814649026859, −3.91673668425074989050622121727, −3.28173947016045737727071282683, −2.75173735921039957018686996189, −1.10218533293676934122466209763, −0.02923020464062628188304600518,
2.09098153778356892153410903706, 3.34933565860313476736507233024, 4.53784898959342419201306212493, 5.40242392517966454751643637891, 6.55527190591980874194344799323, 6.74507802591025629704142349842, 7.53206402319084476250491297183, 8.669223431082526222283316598926, 9.281007610666203777204344828514, 10.08131703897920622016581936273
