L(s) = 1 | + (2.10 + 5.07i)3-s + (−1.74 + 4.21i)5-s + (0.392 − 0.392i)7-s + (−14.9 + 14.9i)9-s + (2.90 − 7.02i)11-s + (4.50 + 10.8i)13-s − 25.0·15-s − 10.5i·17-s + (−1.88 + 0.781i)19-s + (2.81 + 1.16i)21-s + (0.445 + 0.445i)23-s + (2.94 + 2.94i)25-s + (−61.7 − 25.5i)27-s + (−0.741 + 0.307i)29-s − 47.6i·31-s + ⋯ |
L(s) = 1 | + (0.700 + 1.69i)3-s + (−0.349 + 0.843i)5-s + (0.0560 − 0.0560i)7-s + (−1.66 + 1.66i)9-s + (0.264 − 0.638i)11-s + (0.346 + 0.836i)13-s − 1.67·15-s − 0.620i·17-s + (−0.0993 + 0.0411i)19-s + (0.134 + 0.0555i)21-s + (0.0193 + 0.0193i)23-s + (0.117 + 0.117i)25-s + (−2.28 − 0.947i)27-s + (−0.0255 + 0.0105i)29-s − 1.53i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 256 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.871 - 0.489i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 256 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.871 - 0.489i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.431016 + 1.64768i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.431016 + 1.64768i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
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
−11.68318293409418411845013276699, −11.08714966430025694693701920558, −10.27533054122305461229444860652, −9.334485560624493918670339608759, −8.621424002193044917548841569110, −7.43274404614813083485287315226, −6.00018695580266515302843014615, −4.59496019716209650841116575259, −3.70095571535773484566879797945, −2.73163938287058218220868353565,
0.843995591164748808895442507267, 2.11028221007855330747872437482, 3.64863231005412248720055935231, 5.36540137708691991979064943095, 6.64172363368839643401938891178, 7.50457621354262874570421567344, 8.456841101798046755344001517538, 8.907462080881882744299565207813, 10.48672186499619850521323950015, 11.94842326210983750310367823473