L(s) = 1 | + 2.48i·2-s − 2.17·4-s − 4.68i·5-s + 3.30·7-s + 4.53i·8-s + 11.6·10-s − 1.00·13-s + 8.20i·14-s − 19.9·16-s − 6.45i·17-s − 25.1·19-s + 10.2i·20-s + 24.4i·23-s + 3.01·25-s − 2.50i·26-s + ⋯ |
L(s) = 1 | + 1.24i·2-s − 0.543·4-s − 0.937i·5-s + 0.471·7-s + 0.566i·8-s + 1.16·10-s − 0.0775·13-s + 0.586i·14-s − 1.24·16-s − 0.379i·17-s − 1.32·19-s + 0.510i·20-s + 1.06i·23-s + 0.120·25-s − 0.0963i·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
\(L(\frac{3}{2})\) |
\(\approx\) |
\(1.907076243\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.907076243\) |
\(L(2)\) |
|
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 | \( 1 - 2.48iT - 4T^{2} \) |
| 5 | \( 1 + 4.68iT - 25T^{2} \) |
| 7 | \( 1 - 3.30T + 49T^{2} \) |
| 13 | \( 1 + 1.00T + 169T^{2} \) |
| 17 | \( 1 + 6.45iT - 289T^{2} \) |
| 19 | \( 1 + 25.1T + 361T^{2} \) |
| 23 | \( 1 - 24.4iT - 529T^{2} \) |
| 29 | \( 1 - 50.8iT - 841T^{2} \) |
| 31 | \( 1 - 48.4T + 961T^{2} \) |
| 37 | \( 1 - 65.8T + 1.36e3T^{2} \) |
| 41 | \( 1 + 6.41iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 48.5T + 1.84e3T^{2} \) |
| 47 | \( 1 - 56.8iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 67.3iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 0.307iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 86.8T + 3.72e3T^{2} \) |
| 67 | \( 1 - 29.6T + 4.48e3T^{2} \) |
| 71 | \( 1 - 40.9iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 61.0T + 5.32e3T^{2} \) |
| 79 | \( 1 + 85.4T + 6.24e3T^{2} \) |
| 83 | \( 1 - 64.7iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 64.1iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 86.8T + 9.40e3T^{2} \) |
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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
−9.602671851733748584202988357242, −8.823404000264242040631583076783, −8.231070646689558493452980726792, −7.51304520075474098791678022699, −6.60983704179629627118324897553, −5.76582700507448089898935772172, −4.93469389187632282907998585914, −4.31330953303413963696878208966, −2.60054413359373242370893058149, −1.18217901337544388344756200804,
0.64028139504803290942636210930, 2.17481980871747849021926304509, 2.67490069841919472020590064014, 3.91761390785414923583727688380, 4.60164065866721138024346182551, 6.21361942683462395313680094436, 6.69084915218318066567006756645, 7.890245298462604183644716229644, 8.654968114027224237812155860105, 9.924445177132792655019097191286