L(s) = 1 | − 3-s + 5-s + 7-s + 9-s + 11-s − 13-s − 15-s + 17-s + 19-s − 21-s + 23-s + 25-s − 27-s + 29-s − 31-s − 33-s + 35-s − 37-s + 39-s + 41-s − 43-s + 45-s − 47-s + 49-s − 51-s + 53-s + 55-s + ⋯ |
L(s) = 1 | − 3-s + 5-s + 7-s + 9-s + 11-s − 13-s − 15-s + 17-s + 19-s − 21-s + 23-s + 25-s − 27-s + 29-s − 31-s − 33-s + 35-s − 37-s + 39-s + 41-s − 43-s + 45-s − 47-s + 49-s − 51-s + 53-s + 55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1016 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1016 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.688208407\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.688208407\) |
\(L(1)\) |
\(\approx\) |
\(1.173544436\) |
\(L(1)\) |
\(\approx\) |
\(1.173544436\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 127 | \( 1 \) |
good | 3 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| 13 | \( 1 - T \) |
| 17 | \( 1 + T \) |
| 19 | \( 1 + T \) |
| 23 | \( 1 + T \) |
| 29 | \( 1 + T \) |
| 31 | \( 1 - T \) |
| 37 | \( 1 - T \) |
| 41 | \( 1 + T \) |
| 43 | \( 1 - T \) |
| 47 | \( 1 - T \) |
| 53 | \( 1 + T \) |
| 59 | \( 1 - T \) |
| 61 | \( 1 - T \) |
| 67 | \( 1 - T \) |
| 71 | \( 1 - T \) |
| 73 | \( 1 + T \) |
| 79 | \( 1 - T \) |
| 83 | \( 1 - T \) |
| 89 | \( 1 - T \) |
| 97 | \( 1 - T \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−21.4962182897574536328354580111, −21.23911600243020199140709527366, −20.156206508807639705230449631428, −19.13771455605172250945642932925, −18.136903564698083237808286180746, −17.7373131918548057664220088648, −16.86493527887887657601550977118, −16.614314664633948579393164130157, −15.22262518509573188355232737999, −14.43293945699911674533138752919, −13.83787040288952746991755559065, −12.66898400660828274757596271231, −12.004058365695495309565346538588, −11.30228958660319146815978171031, −10.36813848566123162003596065011, −9.704715900678302392868889351287, −8.85417405210986642480688940639, −7.50720110373379362648098391047, −6.88856070309343276546399082794, −5.81591213874615316812052526106, −5.18062669021066411606291335154, −4.51733326187690357048974013507, −3.106366904704409574486914549771, −1.715625821656284657629063641168, −1.12412568757777392511668734613,
1.12412568757777392511668734613, 1.715625821656284657629063641168, 3.106366904704409574486914549771, 4.51733326187690357048974013507, 5.18062669021066411606291335154, 5.81591213874615316812052526106, 6.88856070309343276546399082794, 7.50720110373379362648098391047, 8.85417405210986642480688940639, 9.704715900678302392868889351287, 10.36813848566123162003596065011, 11.30228958660319146815978171031, 12.004058365695495309565346538588, 12.66898400660828274757596271231, 13.83787040288952746991755559065, 14.43293945699911674533138752919, 15.22262518509573188355232737999, 16.614314664633948579393164130157, 16.86493527887887657601550977118, 17.7373131918548057664220088648, 18.136903564698083237808286180746, 19.13771455605172250945642932925, 20.156206508807639705230449631428, 21.23911600243020199140709527366, 21.4962182897574536328354580111