L(s) = 1 | − 2.37i·5-s − 8.11·7-s + 20.3i·11-s + 6.11·13-s + 17.9i·17-s + 9.11·19-s + 33.5i·23-s + 19.3·25-s − 16.6i·29-s − 22.3·31-s + 19.2i·35-s − 50.4·37-s + 34.6i·41-s + 23·43-s + 38.3i·47-s + ⋯ |
L(s) = 1 | − 0.475i·5-s − 1.15·7-s + 1.84i·11-s + 0.470·13-s + 1.05i·17-s + 0.479·19-s + 1.45i·23-s + 0.774·25-s − 0.573i·29-s − 0.720·31-s + 0.551i·35-s − 1.36·37-s + 0.843i·41-s + 0.534·43-s + 0.815i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.762362 + 0.762362i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.762362 + 0.762362i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + 2.37iT - 25T^{2} \) |
| 7 | \( 1 + 8.11T + 49T^{2} \) |
| 11 | \( 1 - 20.3iT - 121T^{2} \) |
| 13 | \( 1 - 6.11T + 169T^{2} \) |
| 17 | \( 1 - 17.9iT - 289T^{2} \) |
| 19 | \( 1 - 9.11T + 361T^{2} \) |
| 23 | \( 1 - 33.5iT - 529T^{2} \) |
| 29 | \( 1 + 16.6iT - 841T^{2} \) |
| 31 | \( 1 + 22.3T + 961T^{2} \) |
| 37 | \( 1 + 50.4T + 1.36e3T^{2} \) |
| 41 | \( 1 - 34.6iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 23T + 1.84e3T^{2} \) |
| 47 | \( 1 - 38.3iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 19.0iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 3.42iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 46.3T + 3.72e3T^{2} \) |
| 67 | \( 1 + 6.29T + 4.48e3T^{2} \) |
| 71 | \( 1 + 35.9iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 47.3T + 5.32e3T^{2} \) |
| 79 | \( 1 + 84.5T + 6.24e3T^{2} \) |
| 83 | \( 1 - 38.3iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 143. iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 80.7T + 9.40e3T^{2} \) |
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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
−11.80277507325511279981514119749, −10.51811604014312381602901535057, −9.691884296257739292788621801642, −9.051827263378672205427022657995, −7.72157609487153893416715170032, −6.83181674942823919159700493660, −5.74834424476126437298961056084, −4.51132617722209451797820682056, −3.35088214689862707442628255705, −1.65486421922183200267033660836,
0.51426997315600282235032715588, 2.86821267507314394170490024332, 3.58077172662876276003753205685, 5.34002526284874600342590502619, 6.35400463893652349091081111013, 7.09312567732156340873864445744, 8.528121693911632092793531164549, 9.180255919945636622206154124359, 10.41796482864287774638433016033, 11.00219826253919654228419403823