| L(s) = 1 | − 1.41i·2-s + 1.73·3-s − 2.00·4-s + (−4.59 + 1.96i)5-s − 2.44i·6-s + (−6.81 + 1.57i)7-s + 2.82i·8-s + 2.99·9-s + (2.77 + 6.50i)10-s − 8.15·11-s − 3.46·12-s − 14.6·13-s + (2.23 + 9.64i)14-s + (−7.96 + 3.40i)15-s + 4.00·16-s − 5.81·17-s + ⋯ |
| L(s) = 1 | − 0.707i·2-s + 0.577·3-s − 0.500·4-s + (−0.919 + 0.393i)5-s − 0.408i·6-s + (−0.974 + 0.225i)7-s + 0.353i·8-s + 0.333·9-s + (0.277 + 0.650i)10-s − 0.741·11-s − 0.288·12-s − 1.12·13-s + (0.159 + 0.688i)14-s + (−0.530 + 0.226i)15-s + 0.250·16-s − 0.342·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.590 - 0.807i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.590 - 0.807i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.0944820 + 0.186138i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0944820 + 0.186138i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 1.41iT \) |
| 3 | \( 1 - 1.73T \) |
| 5 | \( 1 + (4.59 - 1.96i)T \) |
| 7 | \( 1 + (6.81 - 1.57i)T \) |
| good | 11 | \( 1 + 8.15T + 121T^{2} \) |
| 13 | \( 1 + 14.6T + 169T^{2} \) |
| 17 | \( 1 + 5.81T + 289T^{2} \) |
| 19 | \( 1 - 33.7iT - 361T^{2} \) |
| 23 | \( 1 + 37.2iT - 529T^{2} \) |
| 29 | \( 1 + 9.25T + 841T^{2} \) |
| 31 | \( 1 + 19.2iT - 961T^{2} \) |
| 37 | \( 1 - 63.4iT - 1.36e3T^{2} \) |
| 41 | \( 1 - 8.25iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 42.0iT - 1.84e3T^{2} \) |
| 47 | \( 1 + 23.3T + 2.20e3T^{2} \) |
| 53 | \( 1 + 71.3iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 42.9iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 34.2iT - 3.72e3T^{2} \) |
| 67 | \( 1 - 4.99iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 38.8T + 5.04e3T^{2} \) |
| 73 | \( 1 + 124.T + 5.32e3T^{2} \) |
| 79 | \( 1 + 56.1T + 6.24e3T^{2} \) |
| 83 | \( 1 - 90.3T + 6.88e3T^{2} \) |
| 89 | \( 1 - 16.2iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 82.7T + 9.40e3T^{2} \) |
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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
−12.50230415740348507486168588581, −11.64474927155534819219261443683, −10.32787091961635141159805853519, −9.849443424879381515433136045547, −8.492225982243347690460766671027, −7.70179639764691050425231310472, −6.41467593266471632013991448514, −4.67432514798781894414602439266, −3.44711580464565117195048397591, −2.48797164005557841424733456838,
0.10301537076912037164693067707, 2.94401261777563716786596863987, 4.24342656507745404748432306797, 5.39952589810268343123357338227, 7.17644087369505937265714502674, 7.43043856275233340580439810690, 8.837290404047414229822578537288, 9.457949648220596747225907056179, 10.70679551651194604327076888490, 12.04600381709029860060312341409
