L(s) = 1 | − 5-s − 2·7-s + 2·11-s + 2·13-s + 2·17-s + 2·19-s + 2·23-s + 25-s − 6·29-s − 4·31-s + 2·35-s − 2·37-s + 10·41-s − 8·43-s + 2·47-s − 3·49-s + 6·53-s − 2·55-s − 2·59-s + 10·61-s − 2·65-s + 8·67-s − 8·71-s − 6·73-s − 4·77-s − 16·79-s + 12·83-s + ⋯ |
L(s) = 1 | − 0.447·5-s − 0.755·7-s + 0.603·11-s + 0.554·13-s + 0.485·17-s + 0.458·19-s + 0.417·23-s + 1/5·25-s − 1.11·29-s − 0.718·31-s + 0.338·35-s − 0.328·37-s + 1.56·41-s − 1.21·43-s + 0.291·47-s − 3/7·49-s + 0.824·53-s − 0.269·55-s − 0.260·59-s + 1.28·61-s − 0.248·65-s + 0.977·67-s − 0.949·71-s − 0.702·73-s − 0.455·77-s − 1.80·79-s + 1.31·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5760 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5760 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.597138349\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.597138349\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 - 2 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 2 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 + 16 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 97 | \( 1 + 6 T + p T^{2} \) |
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
−8.100275121872087910406371987558, −7.30728481903991716257258635407, −6.81412470800861899319944862793, −5.94585849934142218074591390200, −5.37291408001919163931956668109, −4.29322876900235326517823942506, −3.61215550157467460326063588431, −3.03668573321568444849002381910, −1.78967517400994784014490693135, −0.67830229800011456252117306616,
0.67830229800011456252117306616, 1.78967517400994784014490693135, 3.03668573321568444849002381910, 3.61215550157467460326063588431, 4.29322876900235326517823942506, 5.37291408001919163931956668109, 5.94585849934142218074591390200, 6.81412470800861899319944862793, 7.30728481903991716257258635407, 8.100275121872087910406371987558