| L(s) = 1 | − 0.445·2-s + 1.24·3-s − 0.801·4-s − 0.554·6-s + 7-s + 0.801·8-s + 0.554·9-s − 1.80·11-s − 12-s − 0.445·13-s − 0.445·14-s + 0.445·16-s − 0.445·17-s − 0.246·18-s + 1.24·19-s + 1.24·21-s + 0.801·22-s + 1.24·23-s + 24-s + 25-s + 0.198·26-s − 0.554·27-s − 0.801·28-s + 2·31-s − 32-s − 2.24·33-s + 0.198·34-s + ⋯ |
| L(s) = 1 | − 0.445·2-s + 1.24·3-s − 0.801·4-s − 0.554·6-s + 7-s + 0.801·8-s + 0.554·9-s − 1.80·11-s − 12-s − 0.445·13-s − 0.445·14-s + 0.445·16-s − 0.445·17-s − 0.246·18-s + 1.24·19-s + 1.24·21-s + 0.801·22-s + 1.24·23-s + 24-s + 25-s + 0.198·26-s − 0.554·27-s − 0.801·28-s + 2·31-s − 32-s − 2.24·33-s + 0.198·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3143 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3143 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.320540153\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.320540153\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 - T \) |
| 449 | \( 1 - T \) |
| good | 2 | \( 1 + 0.445T + T^{2} \) |
| 3 | \( 1 - 1.24T + T^{2} \) |
| 5 | \( 1 - T^{2} \) |
| 11 | \( 1 + 1.80T + T^{2} \) |
| 13 | \( 1 + 0.445T + T^{2} \) |
| 17 | \( 1 + 0.445T + T^{2} \) |
| 19 | \( 1 - 1.24T + T^{2} \) |
| 23 | \( 1 - 1.24T + T^{2} \) |
| 29 | \( 1 - T^{2} \) |
| 31 | \( 1 - 2T + T^{2} \) |
| 37 | \( 1 - T^{2} \) |
| 41 | \( 1 - T^{2} \) |
| 43 | \( 1 - T^{2} \) |
| 47 | \( 1 - 2T + T^{2} \) |
| 53 | \( 1 + 0.445T + T^{2} \) |
| 59 | \( 1 - T^{2} \) |
| 61 | \( 1 - T^{2} \) |
| 67 | \( 1 - 2T + T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 + 1.80T + T^{2} \) |
| 79 | \( 1 - T^{2} \) |
| 83 | \( 1 + 1.80T + T^{2} \) |
| 89 | \( 1 - T^{2} \) |
| 97 | \( 1 - T^{2} \) |
| 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
−8.680965474394414601453518147995, −8.279575233329217300637187021187, −7.65366614303774600085624882311, −7.15590548875075533990267870572, −5.47649315559261804327360410747, −4.98104027321698295513497579566, −4.25750484767067329263302140432, −2.97996335570932468904214133660, −2.49941839368549164631417402884, −1.10389936718079066607815721868,
1.10389936718079066607815721868, 2.49941839368549164631417402884, 2.97996335570932468904214133660, 4.25750484767067329263302140432, 4.98104027321698295513497579566, 5.47649315559261804327360410747, 7.15590548875075533990267870572, 7.65366614303774600085624882311, 8.279575233329217300637187021187, 8.680965474394414601453518147995
