L(s) = 1 | + (−0.108 − 1.99i)2-s + (4.35 + 1.80i)3-s + (−3.97 + 0.432i)4-s + (−2.81 + 1.16i)5-s + (3.12 − 8.88i)6-s + (−6.23 − 6.23i)7-s + (1.29 + 7.89i)8-s + (9.32 + 9.32i)9-s + (2.63 + 5.49i)10-s + (−8.06 + 3.33i)11-s + (−18.0 − 5.28i)12-s + (13.3 + 5.51i)13-s + (−11.7 + 13.1i)14-s − 14.3·15-s + (15.6 − 3.43i)16-s + 4.56i·17-s + ⋯ |
L(s) = 1 | + (−0.0540 − 0.998i)2-s + (1.45 + 0.600i)3-s + (−0.994 + 0.108i)4-s + (−0.563 + 0.233i)5-s + (0.521 − 1.48i)6-s + (−0.890 − 0.890i)7-s + (0.161 + 0.986i)8-s + (1.03 + 1.03i)9-s + (0.263 + 0.549i)10-s + (−0.732 + 0.303i)11-s + (−1.50 − 0.440i)12-s + (1.02 + 0.424i)13-s + (−0.841 + 0.937i)14-s − 0.957·15-s + (0.976 − 0.214i)16-s + 0.268i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 32 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.757 + 0.652i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 32 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.757 + 0.652i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(1.08581 - 0.403218i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.08581 - 0.403218i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.108 + 1.99i)T \) |
good | 3 | \( 1 + (-4.35 - 1.80i)T + (6.36 + 6.36i)T^{2} \) |
| 5 | \( 1 + (2.81 - 1.16i)T + (17.6 - 17.6i)T^{2} \) |
| 7 | \( 1 + (6.23 + 6.23i)T + 49iT^{2} \) |
| 11 | \( 1 + (8.06 - 3.33i)T + (85.5 - 85.5i)T^{2} \) |
| 13 | \( 1 + (-13.3 - 5.51i)T + (119. + 119. i)T^{2} \) |
| 17 | \( 1 - 4.56iT - 289T^{2} \) |
| 19 | \( 1 + (-13.4 + 32.4i)T + (-255. - 255. i)T^{2} \) |
| 23 | \( 1 + (6.75 - 6.75i)T - 529iT^{2} \) |
| 29 | \( 1 + (0.266 - 0.643i)T + (-594. - 594. i)T^{2} \) |
| 31 | \( 1 - 0.326iT - 961T^{2} \) |
| 37 | \( 1 + (-31.5 + 13.0i)T + (968. - 968. i)T^{2} \) |
| 41 | \( 1 + (-15.7 - 15.7i)T + 1.68e3iT^{2} \) |
| 43 | \( 1 + (-4.83 + 2.00i)T + (1.30e3 - 1.30e3i)T^{2} \) |
| 47 | \( 1 + 49.7T + 2.20e3T^{2} \) |
| 53 | \( 1 + (-4.45 - 10.7i)T + (-1.98e3 + 1.98e3i)T^{2} \) |
| 59 | \( 1 + (-13.1 - 31.6i)T + (-2.46e3 + 2.46e3i)T^{2} \) |
| 61 | \( 1 + (35.4 - 85.4i)T + (-2.63e3 - 2.63e3i)T^{2} \) |
| 67 | \( 1 + (41.3 + 17.1i)T + (3.17e3 + 3.17e3i)T^{2} \) |
| 71 | \( 1 + (-37.6 - 37.6i)T + 5.04e3iT^{2} \) |
| 73 | \( 1 + (52.2 + 52.2i)T + 5.32e3iT^{2} \) |
| 79 | \( 1 - 26.9T + 6.24e3T^{2} \) |
| 83 | \( 1 + (-10.6 + 25.6i)T + (-4.87e3 - 4.87e3i)T^{2} \) |
| 89 | \( 1 + (103. - 103. i)T - 7.92e3iT^{2} \) |
| 97 | \( 1 - 77.9T + 9.40e3T^{2} \) |
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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
−16.20080446426020293281452430641, −15.14629624036278650982976926619, −13.71067204990037591086588202251, −13.18874574810191204981791724649, −11.21271303711703240018100762927, −10.00633936203852823256844760757, −9.015333638372154875427759326388, −7.61488754787857234766419170193, −4.15721541689612805112289620141, −3.05428787510512237165024686232,
3.42285750510656632986763037939, 6.02408515740820598320853770376, 7.81608413819510064163439460728, 8.467396903621889249104208060165, 9.723170581960950214267581351321, 12.45677699765347080535746958203, 13.35313427310362507429797200012, 14.41239559253462970938049691538, 15.62745563306526914096289554297, 16.17666860968696463641567316477