| L(s) = 1 | + (−0.118 + 0.822i)2-s + (−0.927 − 1.07i)3-s + (1.25 + 0.368i)4-s + (1.58 + 1.02i)5-s + (0.990 − 0.636i)6-s + (−0.0441 + 0.306i)7-s + (−1.14 + 2.50i)8-s + (0.141 − 0.982i)9-s + (−1.02 + 1.18i)10-s + (−5.16 − 3.31i)11-s + (−0.770 − 1.68i)12-s + (0.568 + 1.24i)13-s + (−0.247 − 0.0725i)14-s + (−0.380 − 2.64i)15-s + (0.281 + 0.181i)16-s + (−3.89 + 1.14i)17-s + ⋯ |
| L(s) = 1 | + (−0.0836 + 0.581i)2-s + (−0.535 − 0.618i)3-s + (0.628 + 0.184i)4-s + (0.710 + 0.456i)5-s + (0.404 − 0.259i)6-s + (−0.0166 + 0.115i)7-s + (−0.403 + 0.884i)8-s + (0.0470 − 0.327i)9-s + (−0.324 + 0.374i)10-s + (−1.55 − 1.00i)11-s + (−0.222 − 0.487i)12-s + (0.157 + 0.345i)13-s + (−0.0660 − 0.0193i)14-s + (−0.0983 − 0.683i)15-s + (0.0704 + 0.0452i)16-s + (−0.944 + 0.277i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 67 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.887 - 0.460i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 67 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.887 - 0.460i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.864538 + 0.210936i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.864538 + 0.210936i\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + (0.118 - 0.822i)T + (-1.91 - 0.563i)T^{2} \) |
| 3 | \( 1 + (0.927 + 1.07i)T + (-0.426 + 2.96i)T^{2} \) |
| 5 | \( 1 + (-1.58 - 1.02i)T + (2.07 + 4.54i)T^{2} \) |
| 7 | \( 1 + (0.0441 - 0.306i)T + (-6.71 - 1.97i)T^{2} \) |
| 11 | \( 1 + (5.16 + 3.31i)T + (4.56 + 10.0i)T^{2} \) |
| 13 | \( 1 + (-0.568 - 1.24i)T + (-8.51 + 9.82i)T^{2} \) |
| 17 | \( 1 + (3.89 - 1.14i)T + (14.3 - 9.19i)T^{2} \) |
| 19 | \( 1 + (0.632 + 4.39i)T + (-18.2 + 5.35i)T^{2} \) |
| 23 | \( 1 + (-3.85 - 4.45i)T + (-3.27 + 22.7i)T^{2} \) |
| 29 | \( 1 - 0.562T + 29T^{2} \) |
| 31 | \( 1 + (0.926 - 2.02i)T + (-20.3 - 23.4i)T^{2} \) |
| 37 | \( 1 - 6.62T + 37T^{2} \) |
| 41 | \( 1 + (-3.65 + 1.07i)T + (34.4 - 22.1i)T^{2} \) |
| 43 | \( 1 + (-6.00 + 1.76i)T + (36.1 - 23.2i)T^{2} \) |
| 47 | \( 1 + (3.38 + 3.90i)T + (-6.68 + 46.5i)T^{2} \) |
| 53 | \( 1 + (-0.820 - 0.241i)T + (44.5 + 28.6i)T^{2} \) |
| 59 | \( 1 + (0.875 - 1.91i)T + (-38.6 - 44.5i)T^{2} \) |
| 61 | \( 1 + (10.9 - 7.04i)T + (25.3 - 55.4i)T^{2} \) |
| 71 | \( 1 + (-13.2 - 3.89i)T + (59.7 + 38.3i)T^{2} \) |
| 73 | \( 1 + (11.2 - 7.21i)T + (30.3 - 66.4i)T^{2} \) |
| 79 | \( 1 + (-5.27 - 11.5i)T + (-51.7 + 59.7i)T^{2} \) |
| 83 | \( 1 + (7.74 + 4.97i)T + (34.4 + 75.4i)T^{2} \) |
| 89 | \( 1 + (-8.97 + 10.3i)T + (-12.6 - 88.0i)T^{2} \) |
| 97 | \( 1 - 4.95T + 97T^{2} \) |
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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
−15.17997477012357571300725455931, −13.72461195620829384187801786167, −12.85254152190116334035856003359, −11.42908850938774234852947207150, −10.72794042091613965384664915570, −8.929959346109533075236349457114, −7.49045908886368435482101620658, −6.44567533526347525634385089809, −5.60275975580619756849578536306, −2.62458566673581514342219196113,
2.32548765358519133328041546853, 4.72606227537667689513781724143, 5.94203733466157870163696208484, 7.63061552802360249780645416113, 9.530086000058467349924464708423, 10.41738873910643721121994798909, 11.03305593221181047839044995453, 12.54789988588975931911011857262, 13.27034887524323688196703832274, 15.02871978885054788314123010160
