| L(s) = 1 | + (−4.93 + 0.981i)3-s + (0.275 − 0.412i)5-s + (0.527 + 0.789i)7-s + (15.0 − 6.23i)9-s + (2.09 − 10.5i)11-s + (12.3 − 12.3i)13-s + (−0.955 + 2.30i)15-s + (14.3 − 9.13i)17-s + (−5.75 − 2.38i)19-s + (−3.37 − 3.37i)21-s + (18.0 + 3.58i)23-s + (9.47 + 22.8i)25-s + (−30.4 + 20.3i)27-s + (−42.3 − 28.3i)29-s + (−3.17 − 15.9i)31-s + ⋯ |
| L(s) = 1 | + (−1.64 + 0.327i)3-s + (0.0551 − 0.0825i)5-s + (0.0753 + 0.112i)7-s + (1.67 − 0.692i)9-s + (0.190 − 0.958i)11-s + (0.947 − 0.947i)13-s + (−0.0637 + 0.153i)15-s + (0.843 − 0.537i)17-s + (−0.302 − 0.125i)19-s + (−0.160 − 0.160i)21-s + (0.782 + 0.155i)23-s + (0.378 + 0.914i)25-s + (−1.12 + 0.753i)27-s + (−1.46 − 0.976i)29-s + (−0.102 − 0.514i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 136 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.699 + 0.715i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 136 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.699 + 0.715i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.744357 - 0.313270i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.744357 - 0.313270i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 17 | \( 1 + (-14.3 + 9.13i)T \) |
| good | 3 | \( 1 + (4.93 - 0.981i)T + (8.31 - 3.44i)T^{2} \) |
| 5 | \( 1 + (-0.275 + 0.412i)T + (-9.56 - 23.0i)T^{2} \) |
| 7 | \( 1 + (-0.527 - 0.789i)T + (-18.7 + 45.2i)T^{2} \) |
| 11 | \( 1 + (-2.09 + 10.5i)T + (-111. - 46.3i)T^{2} \) |
| 13 | \( 1 + (-12.3 + 12.3i)T - 169iT^{2} \) |
| 19 | \( 1 + (5.75 + 2.38i)T + (255. + 255. i)T^{2} \) |
| 23 | \( 1 + (-18.0 - 3.58i)T + (488. + 202. i)T^{2} \) |
| 29 | \( 1 + (42.3 + 28.3i)T + (321. + 776. i)T^{2} \) |
| 31 | \( 1 + (3.17 + 15.9i)T + (-887. + 367. i)T^{2} \) |
| 37 | \( 1 + (43.3 - 8.62i)T + (1.26e3 - 523. i)T^{2} \) |
| 41 | \( 1 + (-0.149 - 0.224i)T + (-643. + 1.55e3i)T^{2} \) |
| 43 | \( 1 + (-28.5 + 11.8i)T + (1.30e3 - 1.30e3i)T^{2} \) |
| 47 | \( 1 + (-41.4 + 41.4i)T - 2.20e3iT^{2} \) |
| 53 | \( 1 + (-74.9 - 31.0i)T + (1.98e3 + 1.98e3i)T^{2} \) |
| 59 | \( 1 + (37.9 + 91.5i)T + (-2.46e3 + 2.46e3i)T^{2} \) |
| 61 | \( 1 + (-14.2 + 9.53i)T + (1.42e3 - 3.43e3i)T^{2} \) |
| 67 | \( 1 - 101. iT - 4.48e3T^{2} \) |
| 71 | \( 1 + (-65.4 + 13.0i)T + (4.65e3 - 1.92e3i)T^{2} \) |
| 73 | \( 1 + (54.0 - 80.8i)T + (-2.03e3 - 4.92e3i)T^{2} \) |
| 79 | \( 1 + (2.24 - 11.2i)T + (-5.76e3 - 2.38e3i)T^{2} \) |
| 83 | \( 1 + (-23.5 + 56.8i)T + (-4.87e3 - 4.87e3i)T^{2} \) |
| 89 | \( 1 + (111. + 111. i)T + 7.92e3iT^{2} \) |
| 97 | \( 1 + (66.6 + 44.5i)T + (3.60e3 + 8.69e3i)T^{2} \) |
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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
−12.69177054546019855365951279074, −11.54198642614648599847853653476, −11.03862077821191541880718886582, −10.06963895471269146330129787496, −8.747475943425199119858201555246, −7.20925170031638998154321168524, −5.82735784639441271410158569187, −5.37065865990565430513024443188, −3.67587578703104778492610686274, −0.76291044524416307627336179384,
1.43991759202092498031342486040, 4.18535336472469451790126968987, 5.44217457652637732910310537987, 6.49629106868297638603157115157, 7.33900609926814619372883554938, 9.038180279225778936165446105529, 10.45567157243439014477042818218, 11.03420293198250958407553510492, 12.18800881099059156692630285982, 12.65488339562988130518519371344
