L(s) = 1 | + 3-s + 2·7-s − 2·9-s + 3·11-s + 4·13-s + 3·17-s − 5·19-s + 2·21-s + 6·23-s − 5·27-s − 2·31-s + 3·33-s − 2·37-s + 4·39-s − 3·41-s − 4·43-s + 12·47-s − 3·49-s + 3·51-s − 6·53-s − 5·57-s + 2·61-s − 4·63-s − 13·67-s + 6·69-s − 12·71-s − 11·73-s + ⋯ |
L(s) = 1 | + 0.577·3-s + 0.755·7-s − 2/3·9-s + 0.904·11-s + 1.10·13-s + 0.727·17-s − 1.14·19-s + 0.436·21-s + 1.25·23-s − 0.962·27-s − 0.359·31-s + 0.522·33-s − 0.328·37-s + 0.640·39-s − 0.468·41-s − 0.609·43-s + 1.75·47-s − 3/7·49-s + 0.420·51-s − 0.824·53-s − 0.662·57-s + 0.256·61-s − 0.503·63-s − 1.58·67-s + 0.722·69-s − 1.42·71-s − 1.28·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.786141350\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.786141350\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
good | 3 | \( 1 - T + p T^{2} \) |
| 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 + 5 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 3 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 13 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 + 11 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 + 9 T + p T^{2} \) |
| 89 | \( 1 - 15 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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
−11.25059176210607080544821585304, −10.50949322635495827498326780236, −9.085661088246745777194399285992, −8.675601200283592737413989432141, −7.75100536807499741629900854519, −6.54224267531139226528846678306, −5.51387933113612053830599215796, −4.19405944572803907683579533501, −3.10855907135943805702538312208, −1.55304465588130540352143428093,
1.55304465588130540352143428093, 3.10855907135943805702538312208, 4.19405944572803907683579533501, 5.51387933113612053830599215796, 6.54224267531139226528846678306, 7.75100536807499741629900854519, 8.675601200283592737413989432141, 9.085661088246745777194399285992, 10.50949322635495827498326780236, 11.25059176210607080544821585304