| L(s) = 1 | + (0.837 + 0.998i)2-s + (−0.987 + 2.83i)3-s + (0.399 − 2.26i)4-s + (0.149 + 0.410i)5-s + (−3.65 + 1.38i)6-s + (−1.05 − 5.99i)7-s + (7.11 − 4.10i)8-s + (−7.05 − 5.59i)9-s + (−0.284 + 0.493i)10-s + (−5.19 + 14.2i)11-s + (6.02 + 3.37i)12-s + (−7.31 − 6.13i)13-s + (5.09 − 6.07i)14-s + (−1.31 + 0.0180i)15-s + (1.40 + 0.510i)16-s + (4.20 + 2.42i)17-s + ⋯ |
| L(s) = 1 | + (0.418 + 0.499i)2-s + (−0.329 + 0.944i)3-s + (0.0999 − 0.566i)4-s + (0.0298 + 0.0821i)5-s + (−0.609 + 0.231i)6-s + (−0.150 − 0.855i)7-s + (0.888 − 0.513i)8-s + (−0.783 − 0.621i)9-s + (−0.0284 + 0.0493i)10-s + (−0.472 + 1.29i)11-s + (0.502 + 0.280i)12-s + (−0.562 − 0.472i)13-s + (0.364 − 0.433i)14-s + (−0.0873 + 0.00120i)15-s + (0.0876 + 0.0319i)16-s + (0.247 + 0.142i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.698 - 0.715i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.698 - 0.715i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.972858 + 0.409967i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.972858 + 0.409967i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (0.987 - 2.83i)T \) |
| good | 2 | \( 1 + (-0.837 - 0.998i)T + (-0.694 + 3.93i)T^{2} \) |
| 5 | \( 1 + (-0.149 - 0.410i)T + (-19.1 + 16.0i)T^{2} \) |
| 7 | \( 1 + (1.05 + 5.99i)T + (-46.0 + 16.7i)T^{2} \) |
| 11 | \( 1 + (5.19 - 14.2i)T + (-92.6 - 77.7i)T^{2} \) |
| 13 | \( 1 + (7.31 + 6.13i)T + (29.3 + 166. i)T^{2} \) |
| 17 | \( 1 + (-4.20 - 2.42i)T + (144.5 + 250. i)T^{2} \) |
| 19 | \( 1 + (-17.7 - 30.7i)T + (-180.5 + 312. i)T^{2} \) |
| 23 | \( 1 + (26.7 + 4.71i)T + (497. + 180. i)T^{2} \) |
| 29 | \( 1 + (13.7 + 16.3i)T + (-146. + 828. i)T^{2} \) |
| 31 | \( 1 + (2.87 - 16.3i)T + (-903. - 328. i)T^{2} \) |
| 37 | \( 1 + (-5.31 + 9.20i)T + (-684.5 - 1.18e3i)T^{2} \) |
| 41 | \( 1 + (14.1 - 16.8i)T + (-291. - 1.65e3i)T^{2} \) |
| 43 | \( 1 + (-5.22 - 1.90i)T + (1.41e3 + 1.18e3i)T^{2} \) |
| 47 | \( 1 + (-85.0 + 14.9i)T + (2.07e3 - 755. i)T^{2} \) |
| 53 | \( 1 - 31.9iT - 2.80e3T^{2} \) |
| 59 | \( 1 + (-18.5 - 51.0i)T + (-2.66e3 + 2.23e3i)T^{2} \) |
| 61 | \( 1 + (5.68 + 32.2i)T + (-3.49e3 + 1.27e3i)T^{2} \) |
| 67 | \( 1 + (-2.15 - 1.81i)T + (779. + 4.42e3i)T^{2} \) |
| 71 | \( 1 + (100. + 57.8i)T + (2.52e3 + 4.36e3i)T^{2} \) |
| 73 | \( 1 + (-1.01 - 1.75i)T + (-2.66e3 + 4.61e3i)T^{2} \) |
| 79 | \( 1 + (56.6 - 47.5i)T + (1.08e3 - 6.14e3i)T^{2} \) |
| 83 | \( 1 + (79.5 + 94.7i)T + (-1.19e3 + 6.78e3i)T^{2} \) |
| 89 | \( 1 + (31.0 - 17.9i)T + (3.96e3 - 6.85e3i)T^{2} \) |
| 97 | \( 1 + (75.5 + 27.4i)T + (7.20e3 + 6.04e3i)T^{2} \) |
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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
−16.90235623634511587385756300109, −15.96305266224562808500451183513, −14.90459535099642855356943155069, −14.04286789828471625827847249465, −12.28497692736939750470915542112, −10.34999267441978786689929548816, −9.994219146448734973075918411994, −7.44031329592358783742848329546, −5.71837132740723821756512330515, −4.30155528854535694425044312350,
2.73518659756481902349788120243, 5.45181572270281261140019347869, 7.32784167575366518191649173107, 8.768170069418175993214218223224, 11.17208721294659502667703645728, 11.96511319008944266432927014544, 13.07613118265773612085950142675, 13.97948414871050550759678887985, 15.97388236827564522835803835915, 17.05636550335266301872217509010
