| L(s) = 1 | + 1.41i·2-s + 1.99·4-s − 5.65i·5-s − 7-s + 8.48i·8-s + 8.00·10-s − 16·13-s − 1.41i·14-s − 4.00·16-s − 11.3i·17-s − 33·19-s − 11.3i·20-s − 24.0i·23-s − 7.00·25-s − 22.6i·26-s + ⋯ |
| L(s) = 1 | + 0.707i·2-s + 0.499·4-s − 1.13i·5-s − 0.142·7-s + 1.06i·8-s + 0.800·10-s − 1.23·13-s − 0.101i·14-s − 0.250·16-s − 0.665i·17-s − 1.73·19-s − 0.565i·20-s − 1.04i·23-s − 0.280·25-s − 0.870i·26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.577 + 0.816i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.577 + 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.5573253808\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5573253808\) |
| \(L(2)\) |
|
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 | \( 1 - 1.41iT - 4T^{2} \) |
| 5 | \( 1 + 5.65iT - 25T^{2} \) |
| 7 | \( 1 + T + 49T^{2} \) |
| 13 | \( 1 + 16T + 169T^{2} \) |
| 17 | \( 1 + 11.3iT - 289T^{2} \) |
| 19 | \( 1 + 33T + 361T^{2} \) |
| 23 | \( 1 + 24.0iT - 529T^{2} \) |
| 29 | \( 1 - 12.7iT - 841T^{2} \) |
| 31 | \( 1 - 31T + 961T^{2} \) |
| 37 | \( 1 + 57T + 1.36e3T^{2} \) |
| 41 | \( 1 + 15.5iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 1.84e3T^{2} \) |
| 47 | \( 1 - 60.8iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 89.0iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 69.2iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 105T + 3.72e3T^{2} \) |
| 67 | \( 1 + 103T + 4.48e3T^{2} \) |
| 71 | \( 1 + 118. iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 47T + 5.32e3T^{2} \) |
| 79 | \( 1 + 23T + 6.24e3T^{2} \) |
| 83 | \( 1 + 106. iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 50.9iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 25T + 9.40e3T^{2} \) |
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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
−9.085985262835343717118712015008, −8.521736553846322655351591783424, −7.71163392046260910560159710946, −6.82678411706001966354791793641, −6.12963317728884603205604906866, −4.97072970831393252833162455447, −4.59064631179632840893732752467, −2.86935628219156447016442233487, −1.81960435233667253479874611678, −0.14728230772656822312402314515,
1.77430814633204411547754937482, 2.62946867691116239107675466686, 3.44245055068357922361408638735, 4.50078700777250284682363521745, 5.94020861529580906403635350847, 6.70164544711601184623792006964, 7.27264667033755079949860006464, 8.269620517384439595298854174812, 9.484760928183775008286914215954, 10.28555928844182302158162980085
