L(s) = 1 | + 3.92i·2-s − 11.4·4-s + 7.06i·5-s − 2.64·7-s − 29.1i·8-s − 27.7·10-s + 4.96i·11-s + 5.02·13-s − 10.3i·14-s + 68.9·16-s − 5.51i·17-s − 18.2·19-s − 80.7i·20-s − 19.5·22-s − 10.6i·23-s + ⋯ |
L(s) = 1 | + 1.96i·2-s − 2.85·4-s + 1.41i·5-s − 0.377·7-s − 3.64i·8-s − 2.77·10-s + 0.451i·11-s + 0.386·13-s − 0.742i·14-s + 4.30·16-s − 0.324i·17-s − 0.961·19-s − 4.03i·20-s − 0.887·22-s − 0.464i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.538647 - 0.538647i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.538647 - 0.538647i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 + 2.64T \) |
good | 2 | \( 1 - 3.92iT - 4T^{2} \) |
| 5 | \( 1 - 7.06iT - 25T^{2} \) |
| 11 | \( 1 - 4.96iT - 121T^{2} \) |
| 13 | \( 1 - 5.02T + 169T^{2} \) |
| 17 | \( 1 + 5.51iT - 289T^{2} \) |
| 19 | \( 1 + 18.2T + 361T^{2} \) |
| 23 | \( 1 + 10.6iT - 529T^{2} \) |
| 29 | \( 1 - 39.9iT - 841T^{2} \) |
| 31 | \( 1 + 2.71T + 961T^{2} \) |
| 37 | \( 1 + 13.0T + 1.36e3T^{2} \) |
| 41 | \( 1 - 33.1iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 61.4T + 1.84e3T^{2} \) |
| 47 | \( 1 - 73.8iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 5.94iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 44.6iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 13.3T + 3.72e3T^{2} \) |
| 67 | \( 1 + 105.T + 4.48e3T^{2} \) |
| 71 | \( 1 - 10.1iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 133.T + 5.32e3T^{2} \) |
| 79 | \( 1 - 50.4T + 6.24e3T^{2} \) |
| 83 | \( 1 - 41.5iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 53.3iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 97.8T + 9.40e3T^{2} \) |
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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
−13.48333115110264557221176066668, −12.54704818687192883433573607016, −10.77074343569310957413250413548, −9.829985536764908063799729727966, −8.759434392604492251171836964579, −7.64781056151194879082916491044, −6.73886002824209458229323632330, −6.25346784630534494444541687076, −4.81493181686830312357725408420, −3.42124542233371734198814536990,
0.46016566572886150955268687412, 1.88366755440864623884734840587, 3.57122058266748149423559425346, 4.56242662922455167796602471835, 5.70082996107911901854815434096, 8.281744236232546009845424005662, 8.840944808088591777742826353415, 9.767855926859392698493969096716, 10.69664064481124004475977370257, 11.76167739781350625107717884699