| L(s) = 1 | − 1.93i·2-s + 0.267·4-s − 3.19·5-s + 13.7i·7-s − 8.24i·8-s + 6.17i·10-s − 13.6i·13-s + 26.5·14-s − 14.8·16-s + 19.6i·17-s − 3.58i·19-s − 0.856·20-s − 10.0·23-s − 14.7·25-s − 26.3·26-s + ⋯ |
| L(s) = 1 | − 0.965i·2-s + 0.0669·4-s − 0.639·5-s + 1.96i·7-s − 1.03i·8-s + 0.617i·10-s − 1.04i·13-s + 1.89·14-s − 0.928·16-s + 1.15i·17-s − 0.188i·19-s − 0.0428·20-s − 0.437·23-s − 0.591·25-s − 1.01·26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.522 - 0.852i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.522 - 0.852i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.1710830001\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1710830001\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 + 1.93iT - 4T^{2} \) |
| 5 | \( 1 + 3.19T + 25T^{2} \) |
| 7 | \( 1 - 13.7iT - 49T^{2} \) |
| 13 | \( 1 + 13.6iT - 169T^{2} \) |
| 17 | \( 1 - 19.6iT - 289T^{2} \) |
| 19 | \( 1 + 3.58iT - 361T^{2} \) |
| 23 | \( 1 + 10.0T + 529T^{2} \) |
| 29 | \( 1 + 47.6iT - 841T^{2} \) |
| 31 | \( 1 + 2.67T + 961T^{2} \) |
| 37 | \( 1 - 1.87T + 1.36e3T^{2} \) |
| 41 | \( 1 - 62.2iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 2.92iT - 1.84e3T^{2} \) |
| 47 | \( 1 + 62.8T + 2.20e3T^{2} \) |
| 53 | \( 1 + 14.2T + 2.80e3T^{2} \) |
| 59 | \( 1 + 103.T + 3.48e3T^{2} \) |
| 61 | \( 1 - 51.6iT - 3.72e3T^{2} \) |
| 67 | \( 1 + 16.0T + 4.48e3T^{2} \) |
| 71 | \( 1 + 54.6T + 5.04e3T^{2} \) |
| 73 | \( 1 + 85.4iT - 5.32e3T^{2} \) |
| 79 | \( 1 + 8.83iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 56.0iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 44.1T + 7.92e3T^{2} \) |
| 97 | \( 1 - 78.2T + 9.40e3T^{2} \) |
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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
−9.995515862863051749974410766736, −9.334841563112299982545979136074, −8.277867583121321646290098081705, −7.81843616574880928080474572937, −6.28792683705608164688395567919, −5.84535759338333642616535808360, −4.55125055109742667482681102180, −3.40029446483691192393550627949, −2.64246848362855782807365103152, −1.69398571414790167437025757844,
0.04859158720820017295682216074, 1.62353034739046005726148586318, 3.31601914337248157848305019901, 4.27013769572892179156987509144, 5.02837919956169004585417874934, 6.34009513734194511648063722386, 7.15593449337968077073908255116, 7.37206740388653702520148700110, 8.234327927966653014484370539046, 9.268257176219432479295149002283
