L(s) = 1 | + (−1.46 − 1.35i)2-s + (2.10 + 5.07i)3-s + (0.309 + 3.98i)4-s + (1.74 − 4.21i)5-s + (3.80 − 10.3i)6-s + (−0.392 + 0.392i)7-s + (4.96 − 6.27i)8-s + (−14.9 + 14.9i)9-s + (−8.29 + 3.81i)10-s + (2.90 − 7.02i)11-s + (−19.5 + 9.95i)12-s + (−4.50 − 10.8i)13-s + (1.10 − 0.0429i)14-s + 25.0·15-s + (−15.8 + 2.46i)16-s − 10.5i·17-s + ⋯ |
L(s) = 1 | + (−0.733 − 0.679i)2-s + (0.700 + 1.69i)3-s + (0.0772 + 0.997i)4-s + (0.349 − 0.843i)5-s + (0.634 − 1.71i)6-s + (−0.0560 + 0.0560i)7-s + (0.620 − 0.784i)8-s + (−1.66 + 1.66i)9-s + (−0.829 + 0.381i)10-s + (0.264 − 0.638i)11-s + (−1.63 + 0.829i)12-s + (−0.346 − 0.836i)13-s + (0.0792 − 0.00306i)14-s + 1.67·15-s + (−0.988 + 0.154i)16-s − 0.620i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 32 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.925 - 0.379i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 32 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.925 - 0.379i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.865643 + 0.170835i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.865643 + 0.170835i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (1.46 + 1.35i)T \) |
good | 3 | \( 1 + (-2.10 - 5.07i)T + (-6.36 + 6.36i)T^{2} \) |
| 5 | \( 1 + (-1.74 + 4.21i)T + (-17.6 - 17.6i)T^{2} \) |
| 7 | \( 1 + (0.392 - 0.392i)T - 49iT^{2} \) |
| 11 | \( 1 + (-2.90 + 7.02i)T + (-85.5 - 85.5i)T^{2} \) |
| 13 | \( 1 + (4.50 + 10.8i)T + (-119. + 119. i)T^{2} \) |
| 17 | \( 1 + 10.5iT - 289T^{2} \) |
| 19 | \( 1 + (1.88 - 0.781i)T + (255. - 255. i)T^{2} \) |
| 23 | \( 1 + (0.445 + 0.445i)T + 529iT^{2} \) |
| 29 | \( 1 + (-0.741 + 0.307i)T + (594. - 594. i)T^{2} \) |
| 31 | \( 1 - 47.6iT - 961T^{2} \) |
| 37 | \( 1 + (-14.5 + 35.0i)T + (-968. - 968. i)T^{2} \) |
| 41 | \( 1 + (11.3 - 11.3i)T - 1.68e3iT^{2} \) |
| 43 | \( 1 + (14.6 - 35.3i)T + (-1.30e3 - 1.30e3i)T^{2} \) |
| 47 | \( 1 + 80.5T + 2.20e3T^{2} \) |
| 53 | \( 1 + (66.6 + 27.5i)T + (1.98e3 + 1.98e3i)T^{2} \) |
| 59 | \( 1 + (-65.0 - 26.9i)T + (2.46e3 + 2.46e3i)T^{2} \) |
| 61 | \( 1 + (-87.4 + 36.2i)T + (2.63e3 - 2.63e3i)T^{2} \) |
| 67 | \( 1 + (7.12 + 17.1i)T + (-3.17e3 + 3.17e3i)T^{2} \) |
| 71 | \( 1 + (14.8 - 14.8i)T - 5.04e3iT^{2} \) |
| 73 | \( 1 + (-18.6 + 18.6i)T - 5.32e3iT^{2} \) |
| 79 | \( 1 + 36.2T + 6.24e3T^{2} \) |
| 83 | \( 1 + (-27.0 + 11.2i)T + (4.87e3 - 4.87e3i)T^{2} \) |
| 89 | \( 1 + (56.4 + 56.4i)T + 7.92e3iT^{2} \) |
| 97 | \( 1 - 158.T + 9.40e3T^{2} \) |
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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
−16.50713496598866133573351137848, −15.91700694169994988462309512106, −14.38068684906211839532606332485, −12.94467596810521957861839441768, −11.23834401345801972996154279877, −10.05906556408778074513518198921, −9.178258874500397801103760337072, −8.265384662239433217877938028687, −4.92146256886603644578690108329, −3.20936813046540260023704168544,
2.05590251460062945785704805259, 6.37817601642710346285054049975, 7.10514832812135039619032661397, 8.341259671726478146067360136763, 9.778115405911324973689308132119, 11.61002317399512354926906365199, 13.20073484167474110420668015775, 14.31790805975848871246872124182, 14.94720478903031442934515936608, 16.98481732903970448616218446634