L(s) = 1 | − 5-s + 2·11-s + 3·17-s − 19-s − 3·23-s + 25-s − 4·29-s + 5·31-s − 10·37-s + 6·41-s − 6·43-s − 8·47-s − 7·49-s − 3·53-s − 2·55-s − 5·61-s − 2·67-s − 2·71-s + 6·73-s + 11·79-s + 9·83-s − 3·85-s + 10·89-s + 95-s + 8·97-s + 12·101-s + 12·103-s + ⋯ |
L(s) = 1 | − 0.447·5-s + 0.603·11-s + 0.727·17-s − 0.229·19-s − 0.625·23-s + 1/5·25-s − 0.742·29-s + 0.898·31-s − 1.64·37-s + 0.937·41-s − 0.914·43-s − 1.16·47-s − 49-s − 0.412·53-s − 0.269·55-s − 0.640·61-s − 0.244·67-s − 0.237·71-s + 0.702·73-s + 1.23·79-s + 0.987·83-s − 0.325·85-s + 1.05·89-s + 0.102·95-s + 0.812·97-s + 1.19·101-s + 1.18·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8640 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8640 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + T \) |
good | 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 + T + p T^{2} \) |
| 23 | \( 1 + 3 T + p T^{2} \) |
| 29 | \( 1 + 4 T + p T^{2} \) |
| 31 | \( 1 - 5 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 6 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 + 3 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 5 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 + 2 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 - 11 T + p T^{2} \) |
| 83 | \( 1 - 9 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 97 | \( 1 - 8 T + p T^{2} \) |
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
−7.60259235952935514720652414613, −6.61153228491734451127615031699, −6.24885591073633140195221244557, −5.24546795737060504445937875000, −4.66545048215845096549195243134, −3.71608500870595018034114829373, −3.30428403891060970294542390653, −2.13920113574481211367628389973, −1.25157934419921887410456849049, 0,
1.25157934419921887410456849049, 2.13920113574481211367628389973, 3.30428403891060970294542390653, 3.71608500870595018034114829373, 4.66545048215845096549195243134, 5.24546795737060504445937875000, 6.24885591073633140195221244557, 6.61153228491734451127615031699, 7.60259235952935514720652414613