| L(s) = 1 | + (−1.43 − 1.20i)2-s + (1.37 − 2.37i)3-s + (0.260 + 1.48i)4-s + (1.41 + 1.68i)5-s + (−4.82 + 1.75i)6-s + (−2.14 − 1.23i)7-s + (−0.465 + 0.805i)8-s + (−2.26 − 3.91i)9-s − 4.11i·10-s + (2.97 + 3.54i)11-s + (3.87 + 1.41i)12-s + (−0.780 − 2.14i)13-s + (1.58 + 4.36i)14-s + (5.93 − 1.04i)15-s + (4.46 − 1.62i)16-s + (−4.77 + 2.75i)17-s + ⋯ |
| L(s) = 1 | + (−1.01 − 0.850i)2-s + (0.791 − 1.37i)3-s + (0.130 + 0.740i)4-s + (0.631 + 0.752i)5-s + (−1.96 + 0.716i)6-s + (−0.811 − 0.468i)7-s + (−0.164 + 0.284i)8-s + (−0.754 − 1.30i)9-s − 1.30i·10-s + (0.896 + 1.06i)11-s + (1.11 + 0.407i)12-s + (−0.216 − 0.594i)13-s + (0.424 + 1.16i)14-s + (1.53 − 0.270i)15-s + (1.11 − 0.405i)16-s + (−1.15 + 0.669i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 73 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.368 + 0.929i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 73 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.368 + 0.929i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.411079 - 0.605344i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.411079 - 0.605344i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 73 | \( 1 + (-5.21 - 6.76i)T \) |
| good | 2 | \( 1 + (1.43 + 1.20i)T + (0.347 + 1.96i)T^{2} \) |
| 3 | \( 1 + (-1.37 + 2.37i)T + (-1.5 - 2.59i)T^{2} \) |
| 5 | \( 1 + (-1.41 - 1.68i)T + (-0.868 + 4.92i)T^{2} \) |
| 7 | \( 1 + (2.14 + 1.23i)T + (3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (-2.97 - 3.54i)T + (-1.91 + 10.8i)T^{2} \) |
| 13 | \( 1 + (0.780 + 2.14i)T + (-9.95 + 8.35i)T^{2} \) |
| 17 | \( 1 + (4.77 - 2.75i)T + (8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-3.91 - 3.28i)T + (3.29 + 18.7i)T^{2} \) |
| 23 | \( 1 + (-5.33 + 4.48i)T + (3.99 - 22.6i)T^{2} \) |
| 29 | \( 1 + (1.37 - 1.63i)T + (-5.03 - 28.5i)T^{2} \) |
| 31 | \( 1 + (4.45 + 0.784i)T + (29.1 + 10.6i)T^{2} \) |
| 37 | \( 1 + (6.26 - 5.25i)T + (6.42 - 36.4i)T^{2} \) |
| 41 | \( 1 + (2.16 + 0.789i)T + (31.4 + 26.3i)T^{2} \) |
| 43 | \( 1 + (-0.729 - 0.420i)T + (21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (0.785 - 2.15i)T + (-36.0 - 30.2i)T^{2} \) |
| 53 | \( 1 + (7.41 - 8.83i)T + (-9.20 - 52.1i)T^{2} \) |
| 59 | \( 1 + (-2.49 - 6.86i)T + (-45.1 + 37.9i)T^{2} \) |
| 61 | \( 1 + (1.49 + 0.544i)T + (46.7 + 39.2i)T^{2} \) |
| 67 | \( 1 + (-9.02 + 3.28i)T + (51.3 - 43.0i)T^{2} \) |
| 71 | \( 1 + (10.5 + 8.88i)T + (12.3 + 69.9i)T^{2} \) |
| 79 | \( 1 + (2.16 - 0.789i)T + (60.5 - 50.7i)T^{2} \) |
| 83 | \( 1 + 13.7iT - 83T^{2} \) |
| 89 | \( 1 + (3.85 - 1.40i)T + (68.1 - 57.2i)T^{2} \) |
| 97 | \( 1 + (0.286 + 0.495i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−14.14130624091523695474156718669, −12.99144518634291911250385566212, −12.20020730390299722977791043499, −10.68432840300218443022923659479, −9.742448059985497814161429582961, −8.732759408244323033885902262597, −7.32783102027994034862041590200, −6.45702907320834351905580407828, −3.01839592649874889073636594992, −1.71853879667002643790502356283,
3.44116275394134185046269460384, 5.30154194163681362827274275825, 6.84905143080480985151617605987, 8.722613527239567315414892652990, 9.257388886326033943378006343981, 9.542085132692648977834979950048, 11.26987470099938481878525604153, 13.11613637616399900125189087358, 14.18962079231498224290526971392, 15.46747251384301919583545759866
