| L(s) = 1 | + (−0.797 + 1.74i)2-s + (0.802 − 0.516i)3-s + (−1.10 − 1.27i)4-s + (−0.451 + 3.13i)5-s + (0.260 + 1.81i)6-s + (1.40 − 3.06i)7-s + (−0.584 + 0.171i)8-s + (−0.867 + 1.90i)9-s + (−5.11 − 3.28i)10-s + (0.598 − 4.16i)11-s + (−1.54 − 0.452i)12-s + (1.65 + 0.484i)13-s + (4.23 + 4.89i)14-s + (1.25 + 2.75i)15-s + (0.645 − 4.48i)16-s + (3.22 − 3.72i)17-s + ⋯ |
| L(s) = 1 | + (−0.563 + 1.23i)2-s + (0.463 − 0.297i)3-s + (−0.550 − 0.635i)4-s + (−0.201 + 1.40i)5-s + (0.106 + 0.740i)6-s + (0.529 − 1.15i)7-s + (−0.206 + 0.0606i)8-s + (−0.289 + 0.633i)9-s + (−1.61 − 1.03i)10-s + (0.180 − 1.25i)11-s + (−0.444 − 0.130i)12-s + (0.458 + 0.134i)13-s + (1.13 + 1.30i)14-s + (0.324 + 0.710i)15-s + (0.161 − 1.12i)16-s + (0.782 − 0.902i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 67 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0616 - 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 67 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0616 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.537318 + 0.571538i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.537318 + 0.571538i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 67 | \( 1 + (-1.38 + 8.06i)T \) |
| good | 2 | \( 1 + (0.797 - 1.74i)T + (-1.30 - 1.51i)T^{2} \) |
| 3 | \( 1 + (-0.802 + 0.516i)T + (1.24 - 2.72i)T^{2} \) |
| 5 | \( 1 + (0.451 - 3.13i)T + (-4.79 - 1.40i)T^{2} \) |
| 7 | \( 1 + (-1.40 + 3.06i)T + (-4.58 - 5.29i)T^{2} \) |
| 11 | \( 1 + (-0.598 + 4.16i)T + (-10.5 - 3.09i)T^{2} \) |
| 13 | \( 1 + (-1.65 - 0.484i)T + (10.9 + 7.02i)T^{2} \) |
| 17 | \( 1 + (-3.22 + 3.72i)T + (-2.41 - 16.8i)T^{2} \) |
| 19 | \( 1 + (0.0642 + 0.140i)T + (-12.4 + 14.3i)T^{2} \) |
| 23 | \( 1 + (0.215 - 0.138i)T + (9.55 - 20.9i)T^{2} \) |
| 29 | \( 1 + 9.45T + 29T^{2} \) |
| 31 | \( 1 + (6.77 - 1.98i)T + (26.0 - 16.7i)T^{2} \) |
| 37 | \( 1 + 0.210T + 37T^{2} \) |
| 41 | \( 1 + (-1.99 + 2.30i)T + (-5.83 - 40.5i)T^{2} \) |
| 43 | \( 1 + (2.71 - 3.12i)T + (-6.11 - 42.5i)T^{2} \) |
| 47 | \( 1 + (-4.08 + 2.62i)T + (19.5 - 42.7i)T^{2} \) |
| 53 | \( 1 + (-6.51 - 7.52i)T + (-7.54 + 52.4i)T^{2} \) |
| 59 | \( 1 + (12.2 - 3.61i)T + (49.6 - 31.8i)T^{2} \) |
| 61 | \( 1 + (-1.78 - 12.3i)T + (-58.5 + 17.1i)T^{2} \) |
| 71 | \( 1 + (0.0591 + 0.0683i)T + (-10.1 + 70.2i)T^{2} \) |
| 73 | \( 1 + (1.15 + 8.03i)T + (-70.0 + 20.5i)T^{2} \) |
| 79 | \( 1 + (-4.53 - 1.33i)T + (66.4 + 42.7i)T^{2} \) |
| 83 | \( 1 + (-1.11 + 7.76i)T + (-79.6 - 23.3i)T^{2} \) |
| 89 | \( 1 + (13.6 + 8.79i)T + (36.9 + 80.9i)T^{2} \) |
| 97 | \( 1 - 1.98T + 97T^{2} \) |
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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
−14.99466040051714637525151317844, −14.17435741991086381445756501782, −13.70563098585072195440502341292, −11.38161377691942488949629824145, −10.63678638563378909065662808260, −8.926969243830269663316395409729, −7.60769014594052696962501366870, −7.27940410093381041168146266172, −5.79524943413779407056364304213, −3.29184118294998280142534453394,
1.83313592674506673222455824036, 3.85789049749485178226937806643, 5.59281815900085107345857671729, 8.226347535588794446383100924072, 9.053052406918777138658867591654, 9.698761669322761598461549674390, 11.32391163735033107619535576573, 12.34512026619576493304063099421, 12.71733970429050442436528863691, 14.79632522723328434120406975661
