L(s) = 1 | + 2-s + 4-s − 5-s + 4·7-s + 8-s − 10-s + 4·11-s + 2·13-s + 4·14-s + 16-s − 20-s + 4·22-s − 2·23-s + 25-s + 2·26-s + 4·28-s − 2·29-s − 31-s + 32-s − 4·35-s + 2·37-s − 40-s + 4·43-s + 4·44-s − 2·46-s + 9·49-s + 50-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 1/2·4-s − 0.447·5-s + 1.51·7-s + 0.353·8-s − 0.316·10-s + 1.20·11-s + 0.554·13-s + 1.06·14-s + 1/4·16-s − 0.223·20-s + 0.852·22-s − 0.417·23-s + 1/5·25-s + 0.392·26-s + 0.755·28-s − 0.371·29-s − 0.179·31-s + 0.176·32-s − 0.676·35-s + 0.328·37-s − 0.158·40-s + 0.609·43-s + 0.603·44-s − 0.294·46-s + 9/7·49-s + 0.141·50-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2790 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2790 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.597765796\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.597765796\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 31 | \( 1 + T \) |
good | 7 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.654500372321901816842255950761, −7.970440067639789960903941990640, −7.31741015618283792753525367787, −6.41613721269874027079435062809, −5.66404730325203102454538368625, −4.72010516126955509032073982441, −4.17241278723967403560521862244, −3.37996514411777076461954948223, −2.04448547604731713691349330231, −1.19304226927182692012665951795,
1.19304226927182692012665951795, 2.04448547604731713691349330231, 3.37996514411777076461954948223, 4.17241278723967403560521862244, 4.72010516126955509032073982441, 5.66404730325203102454538368625, 6.41613721269874027079435062809, 7.31741015618283792753525367787, 7.970440067639789960903941990640, 8.654500372321901816842255950761