L(s) = 1 | + (0.866 + 0.5i)2-s + (0.499 + 0.866i)4-s + (0.866 + 0.5i)7-s + 0.999i·8-s + (−0.5 + 0.866i)9-s + (0.499 + 0.866i)14-s + (−0.5 + 0.866i)16-s + (−0.866 − 1.5i)17-s + (−0.866 + 0.499i)18-s + (−0.866 − 0.5i)23-s + 0.999i·28-s + (1.5 − 0.866i)31-s + (−0.866 + 0.499i)32-s − 1.73i·34-s − 0.999·36-s + ⋯ |
L(s) = 1 | + (0.866 + 0.5i)2-s + (0.499 + 0.866i)4-s + (0.866 + 0.5i)7-s + 0.999i·8-s + (−0.5 + 0.866i)9-s + (0.499 + 0.866i)14-s + (−0.5 + 0.866i)16-s + (−0.866 − 1.5i)17-s + (−0.866 + 0.499i)18-s + (−0.866 − 0.5i)23-s + 0.999i·28-s + (1.5 − 0.866i)31-s + (−0.866 + 0.499i)32-s − 1.73i·34-s − 0.999·36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.185 - 0.982i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.185 - 0.982i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.819331691\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.819331691\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.866 - 0.5i)T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (-0.866 - 0.5i)T \) |
good | 3 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 11 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 13 | \( 1 - T^{2} \) |
| 17 | \( 1 + (0.866 + 1.5i)T + (-0.5 + 0.866i)T^{2} \) |
| 19 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 23 | \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \) |
| 29 | \( 1 - T^{2} \) |
| 31 | \( 1 + (-1.5 + 0.866i)T + (0.5 - 0.866i)T^{2} \) |
| 37 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 41 | \( 1 - 1.73iT - T^{2} \) |
| 43 | \( 1 + T^{2} \) |
| 47 | \( 1 + (-0.866 + 1.5i)T + (-0.5 - 0.866i)T^{2} \) |
| 53 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 59 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 61 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 67 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 71 | \( 1 - T + T^{2} \) |
| 73 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 79 | \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 83 | \( 1 - T^{2} \) |
| 89 | \( 1 + (1.5 + 0.866i)T + (0.5 + 0.866i)T^{2} \) |
| 97 | \( 1 + 1.73T + T^{2} \) |
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
−9.958476515902195701237817851950, −8.781392494037221777613175471733, −8.193938934566448453445434076459, −7.52601253111640034678465204902, −6.55040576083291127361691162647, −5.67685385553752730178630997665, −4.88010053989908111659696060361, −4.34113853677316512842851802228, −2.81208054691695784311617190260, −2.17577986626440423989841620957,
1.30472567027342386452105907802, 2.45628824756722494273533155425, 3.74530240559608923087225185491, 4.25678018355909960867551659412, 5.34142690993631580743450149420, 6.17247791406425204569962969618, 6.86947400123642982821511951848, 8.002606754456780941009865741862, 8.802307216095779468861826973481, 9.788235466997496343501137048845