| L(s) = 1 | + 3-s + 3·7-s − 2·9-s + 5·11-s + 6·17-s + 2·19-s + 3·21-s + 6·23-s − 5·27-s − 6·29-s + 4·31-s + 5·33-s − 37-s − 9·41-s − 4·43-s + 7·47-s + 2·49-s + 6·51-s − 9·53-s + 2·57-s − 4·59-s − 8·61-s − 6·63-s + 12·67-s + 6·69-s + 3·71-s + 5·73-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1.13·7-s − 2/3·9-s + 1.50·11-s + 1.45·17-s + 0.458·19-s + 0.654·21-s + 1.25·23-s − 0.962·27-s − 1.11·29-s + 0.718·31-s + 0.870·33-s − 0.164·37-s − 1.40·41-s − 0.609·43-s + 1.02·47-s + 2/7·49-s + 0.840·51-s − 1.23·53-s + 0.264·57-s − 0.520·59-s − 1.02·61-s − 0.755·63-s + 1.46·67-s + 0.722·69-s + 0.356·71-s + 0.585·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3700 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.027630815\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.027630815\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 37 | \( 1 + T \) |
| good | 3 | \( 1 - T + p T^{2} \) |
| 7 | \( 1 - 3 T + p T^{2} \) |
| 11 | \( 1 - 5 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 + 9 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 7 T + p T^{2} \) |
| 53 | \( 1 + 9 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 - 3 T + p T^{2} \) |
| 73 | \( 1 - 5 T + p T^{2} \) |
| 79 | \( 1 - 6 T + p T^{2} \) |
| 83 | \( 1 - T + p T^{2} \) |
| 89 | \( 1 - 2 T + p T^{2} \) |
| 97 | \( 1 + p T^{2} \) |
| 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
−8.468855010390743638464396248842, −7.919131120412641908685069714009, −7.20903657466271929470629370288, −6.31266220770231430562018298260, −5.42000856842620612735134617272, −4.79744205114466790913823205832, −3.67617283202346632372250852545, −3.15724192229422619035887123295, −1.89070845616260793551241380535, −1.08982337987453724323678127858,
1.08982337987453724323678127858, 1.89070845616260793551241380535, 3.15724192229422619035887123295, 3.67617283202346632372250852545, 4.79744205114466790913823205832, 5.42000856842620612735134617272, 6.31266220770231430562018298260, 7.20903657466271929470629370288, 7.919131120412641908685069714009, 8.468855010390743638464396248842
