| L(s) = 1 | + 1.41i·2-s + 3-s − 2.00·4-s − 5-s + 1.41i·6-s − 8.48i·7-s − 2.82i·8-s − 8·9-s − 1.41i·10-s + (7 + 8.48i)11-s − 2.00·12-s + 8.48i·13-s + 12·14-s − 15-s + 4.00·16-s + 25.4i·17-s + ⋯ |
| L(s) = 1 | + 0.707i·2-s + 0.333·3-s − 0.500·4-s − 0.200·5-s + 0.235i·6-s − 1.21i·7-s − 0.353i·8-s − 0.888·9-s − 0.141i·10-s + (0.636 + 0.771i)11-s − 0.166·12-s + 0.652i·13-s + 0.857·14-s − 0.0666·15-s + 0.250·16-s + 1.49i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 22 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.771 - 0.636i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 22 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.771 - 0.636i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.841851 + 0.302431i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.841851 + 0.302431i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 1.41iT \) |
| 11 | \( 1 + (-7 - 8.48i)T \) |
| good | 3 | \( 1 - T + 9T^{2} \) |
| 5 | \( 1 + T + 25T^{2} \) |
| 7 | \( 1 + 8.48iT - 49T^{2} \) |
| 13 | \( 1 - 8.48iT - 169T^{2} \) |
| 17 | \( 1 - 25.4iT - 289T^{2} \) |
| 19 | \( 1 + 25.4iT - 361T^{2} \) |
| 23 | \( 1 - 17T + 529T^{2} \) |
| 29 | \( 1 + 33.9iT - 841T^{2} \) |
| 31 | \( 1 - 17T + 961T^{2} \) |
| 37 | \( 1 - 47T + 1.36e3T^{2} \) |
| 41 | \( 1 - 8.48iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 16.9iT - 1.84e3T^{2} \) |
| 47 | \( 1 + 58T + 2.20e3T^{2} \) |
| 53 | \( 1 - 2T + 2.80e3T^{2} \) |
| 59 | \( 1 + 55T + 3.48e3T^{2} \) |
| 61 | \( 1 + 84.8iT - 3.72e3T^{2} \) |
| 67 | \( 1 - 89T + 4.48e3T^{2} \) |
| 71 | \( 1 + 7T + 5.04e3T^{2} \) |
| 73 | \( 1 - 127. iT - 5.32e3T^{2} \) |
| 79 | \( 1 + 33.9iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 33.9iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 97T + 7.92e3T^{2} \) |
| 97 | \( 1 + 121T + 9.40e3T^{2} \) |
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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
−17.40262567579664285546855982053, −16.98888891540953619011722420661, −15.30839630622531230607954830551, −14.29477702480245575692025488584, −13.19338430644637928044773230956, −11.31560845057126233909142094495, −9.559499288456436870377875571410, −8.025953340221647977566280143571, −6.58172764741867550155760262949, −4.22152064120141929741538303524,
3.03710504950651394798101376906, 5.62852715127506949887252766425, 8.295168302387767806850181246400, 9.401252051667500770697801645388, 11.30230148601888066512357865305, 12.21185689093297284948418677747, 13.82938194981975101368914151988, 14.94091070994451609486410455958, 16.46854806215560065084358066920, 18.03853100477620250796432743122
