L(s) = 1 | − 2-s + 3-s + 4-s + 5-s − 6-s − 8-s + 9-s − 10-s − 11-s + 12-s − 13-s + 15-s + 16-s − 18-s + 3·19-s + 20-s + 22-s + 7·23-s − 24-s + 25-s + 26-s + 27-s − 8·29-s − 30-s + 2·31-s − 32-s − 33-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.447·5-s − 0.408·6-s − 0.353·8-s + 1/3·9-s − 0.316·10-s − 0.301·11-s + 0.288·12-s − 0.277·13-s + 0.258·15-s + 1/4·16-s − 0.235·18-s + 0.688·19-s + 0.223·20-s + 0.213·22-s + 1.45·23-s − 0.204·24-s + 1/5·25-s + 0.196·26-s + 0.192·27-s − 1.48·29-s − 0.182·30-s + 0.359·31-s − 0.176·32-s − 0.174·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.663286917\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.663286917\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 \) |
good | 11 | \( 1 + T + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 3 T + p T^{2} \) |
| 23 | \( 1 - 7 T + p T^{2} \) |
| 29 | \( 1 + 8 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 - 11 T + p T^{2} \) |
| 41 | \( 1 - 11 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 - 5 T + p T^{2} \) |
| 53 | \( 1 + 11 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + p T^{2} \) |
| 67 | \( 1 + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 8 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 97 | \( 1 - 16 T + p T^{2} \) |
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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
−9.217092299643704712052668053705, −9.089567991514105168634090054667, −7.68107648255631024309043088683, −7.54284655431929603006570853838, −6.35268861081301845967524360081, −5.49912118830569381810269317967, −4.39821198685569555876220275458, −3.10630202053253186340416927410, −2.33523710039910241765913893337, −1.04195073180868306135810059408,
1.04195073180868306135810059408, 2.33523710039910241765913893337, 3.10630202053253186340416927410, 4.39821198685569555876220275458, 5.49912118830569381810269317967, 6.35268861081301845967524360081, 7.54284655431929603006570853838, 7.68107648255631024309043088683, 9.089567991514105168634090054667, 9.217092299643704712052668053705