L(s) = 1 | + (−1.22 − 0.707i)2-s + (1 + 1.41i)3-s + (0.999 + 1.73i)4-s + 1.41i·5-s + (−0.224 − 2.43i)6-s + (−2 + 1.73i)7-s − 2.82i·8-s + (−1.00 + 2.82i)9-s + (1.00 − 1.73i)10-s − 2.44·11-s + (−1.44 + 3.14i)12-s − 2·13-s + (3.67 − 0.707i)14-s + (−2.00 + 1.41i)15-s + (−2.00 + 3.46i)16-s + 7.34·17-s + ⋯ |
L(s) = 1 | + (−0.866 − 0.499i)2-s + (0.577 + 0.816i)3-s + (0.499 + 0.866i)4-s + 0.632i·5-s + (−0.0917 − 0.995i)6-s + (−0.755 + 0.654i)7-s − 0.999i·8-s + (−0.333 + 0.942i)9-s + (0.316 − 0.547i)10-s − 0.738·11-s + (−0.418 + 0.908i)12-s − 0.554·13-s + (0.981 − 0.188i)14-s + (−0.516 + 0.365i)15-s + (−0.500 + 0.866i)16-s + 1.78·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 168 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.239 - 0.970i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 168 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.239 - 0.970i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.643807 + 0.504426i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.643807 + 0.504426i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (1.22 + 0.707i)T \) |
| 3 | \( 1 + (-1 - 1.41i)T \) |
| 7 | \( 1 + (2 - 1.73i)T \) |
good | 5 | \( 1 - 1.41iT - 5T^{2} \) |
| 11 | \( 1 + 2.44T + 11T^{2} \) |
| 13 | \( 1 + 2T + 13T^{2} \) |
| 17 | \( 1 - 7.34T + 17T^{2} \) |
| 19 | \( 1 - 4T + 19T^{2} \) |
| 23 | \( 1 + 1.41iT - 23T^{2} \) |
| 29 | \( 1 - 4.89T + 29T^{2} \) |
| 31 | \( 1 - 6.92iT - 31T^{2} \) |
| 37 | \( 1 + 10.3iT - 37T^{2} \) |
| 41 | \( 1 + 2.44T + 41T^{2} \) |
| 43 | \( 1 + 3.46iT - 43T^{2} \) |
| 47 | \( 1 + 4.89T + 47T^{2} \) |
| 53 | \( 1 + 53T^{2} \) |
| 59 | \( 1 - 5.65iT - 59T^{2} \) |
| 61 | \( 1 - 10T + 61T^{2} \) |
| 67 | \( 1 - 3.46iT - 67T^{2} \) |
| 71 | \( 1 + 1.41iT - 71T^{2} \) |
| 73 | \( 1 - 73T^{2} \) |
| 79 | \( 1 - 8T + 79T^{2} \) |
| 83 | \( 1 + 11.3iT - 83T^{2} \) |
| 89 | \( 1 + 7.34T + 89T^{2} \) |
| 97 | \( 1 - 13.8iT - 97T^{2} \) |
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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
−12.73542278970594767168825356313, −11.88627644330103307286487497831, −10.54240166167569631707882416848, −10.06590554287868399006859203580, −9.176654591269148900012949611976, −8.110675502425022742908573129188, −7.08140929354083034200814023059, −5.37419154995384350458999495620, −3.39157678668300922816589119719, −2.67273148817510130623183120037,
1.00485030685704020045278654916, 3.01008435776676640178231297580, 5.26794249345049940889028434929, 6.56605561701785345537505526240, 7.61488125754655462796501116452, 8.205088192316876085636137458578, 9.569136053688078681341223991858, 10.05783400298158049191372772812, 11.68236104645239978344735374133, 12.66336711332198442364971692047