L(s) = 1 | + (−0.866 + 1.5i)11-s + 1.73i·17-s + i·19-s + (0.5 − 0.866i)25-s + (1.5 − 0.866i)41-s + (0.866 + 0.5i)43-s + (0.5 + 0.866i)49-s + (−0.866 − 1.5i)59-s + (−0.866 + 0.5i)67-s − 73-s + (−0.5 + 0.866i)97-s + 1.73·107-s + ⋯ |
L(s) = 1 | + (−0.866 + 1.5i)11-s + 1.73i·17-s + i·19-s + (0.5 − 0.866i)25-s + (1.5 − 0.866i)41-s + (0.866 + 0.5i)43-s + (0.5 + 0.866i)49-s + (−0.866 − 1.5i)59-s + (−0.866 + 0.5i)67-s − 73-s + (−0.5 + 0.866i)97-s + 1.73·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.422 - 0.906i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.422 - 0.906i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.000322007\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.000322007\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 7 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 11 | \( 1 + (0.866 - 1.5i)T + (-0.5 - 0.866i)T^{2} \) |
| 13 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 17 | \( 1 - 1.73iT - T^{2} \) |
| 19 | \( 1 - iT - T^{2} \) |
| 23 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 29 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 31 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 37 | \( 1 - T^{2} \) |
| 41 | \( 1 + (-1.5 + 0.866i)T + (0.5 - 0.866i)T^{2} \) |
| 43 | \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \) |
| 47 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 53 | \( 1 + T^{2} \) |
| 59 | \( 1 + (0.866 + 1.5i)T + (-0.5 + 0.866i)T^{2} \) |
| 61 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 67 | \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 + T + T^{2} \) |
| 79 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 83 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 89 | \( 1 - T^{2} \) |
| 97 | \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.768568283645917151980432812103, −8.846224988193966430622710417297, −7.938566902550447214034421477236, −7.47399823976513068375033448673, −6.37993157883344568757058476937, −5.68819124423706306263499621154, −4.60331481726555117664409483617, −3.95101750871998102850184635250, −2.61756118153172588546874078105, −1.66403554509954787578173124639,
0.78312763341026005372339263250, 2.62225438757791549003331827602, 3.17077711068374732011850422639, 4.51249009048629713882919057816, 5.33414598472279731136179283812, 6.03248201438562069025797984885, 7.15722311472711143265453724016, 7.67084296676855900867471817619, 8.781278584208087241775730348652, 9.154737724619339469272197034332