| 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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 712 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 712 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.727148308\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.727148308\) |
| \(L(1)\) |
\(\approx\) |
\(0.9418892991\) |
| \(L(1)\) |
\(\approx\) |
\(0.9418892991\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 89 | \( 1 \) |
| 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 | \( 1 - T \) |
| 97 | \( 1 + T \) |
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\(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−22.610874548054419191264499863914, −21.50738234579961549291857967606, −20.99889652466775821736655651719, −19.92793133198550039926128179847, −18.94047345144956901523257833205, −18.41810115158636974131173261087, −17.30125442201409338791967032136, −16.835286583002438858020966677012, −15.89211984219173031767042699943, −15.09010365192380803558557767636, −14.33118803614086508971339634332, −13.08821813283749402423197158930, −12.14191891713250473721189327685, −11.51309437725528390978570052866, −11.00670497936107086428368722409, −10.03353215815746373782011902884, −8.65601858807521944726946170250, −8.01368067132183598581812190009, −6.90028697971980306601640290607, −6.189021115830070679898960997174, −4.93635119287207170832428098728, −4.33595491128364856268326564091, −3.339331236122840645934133346279, −1.49893015838926699713475026924, −0.78240996382308683695777159193,
0.78240996382308683695777159193, 1.49893015838926699713475026924, 3.339331236122840645934133346279, 4.33595491128364856268326564091, 4.93635119287207170832428098728, 6.189021115830070679898960997174, 6.90028697971980306601640290607, 8.01368067132183598581812190009, 8.65601858807521944726946170250, 10.03353215815746373782011902884, 11.00670497936107086428368722409, 11.51309437725528390978570052866, 12.14191891713250473721189327685, 13.08821813283749402423197158930, 14.33118803614086508971339634332, 15.09010365192380803558557767636, 15.89211984219173031767042699943, 16.835286583002438858020966677012, 17.30125442201409338791967032136, 18.41810115158636974131173261087, 18.94047345144956901523257833205, 19.92793133198550039926128179847, 20.99889652466775821736655651719, 21.50738234579961549291857967606, 22.610874548054419191264499863914
