| L(s) = 1 | + 3-s + 9-s + 2·11-s + 2·13-s + 4·17-s + 8·23-s + 27-s + 2·31-s + 2·33-s + 8·37-s + 2·39-s − 2·41-s + 2·43-s + 10·47-s + 4·51-s − 2·53-s + 4·59-s + 10·61-s − 2·67-s + 8·69-s + 12·71-s − 10·73-s − 16·79-s + 81-s − 16·83-s + 14·89-s + 2·93-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1/3·9-s + 0.603·11-s + 0.554·13-s + 0.970·17-s + 1.66·23-s + 0.192·27-s + 0.359·31-s + 0.348·33-s + 1.31·37-s + 0.320·39-s − 0.312·41-s + 0.304·43-s + 1.45·47-s + 0.560·51-s − 0.274·53-s + 0.520·59-s + 1.28·61-s − 0.244·67-s + 0.963·69-s + 1.42·71-s − 1.17·73-s − 1.80·79-s + 1/9·81-s − 1.75·83-s + 1.48·89-s + 0.207·93-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 235200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 235200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.379069911\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.379069911\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
| good | 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 - 10 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + 16 T + p T^{2} \) |
| 83 | \( 1 + 16 T + p T^{2} \) |
| 89 | \( 1 - 14 T + p T^{2} \) |
| 97 | \( 1 + 6 T + p T^{2} \) |
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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
−12.80783673796627, −12.68149613583307, −11.93307222460168, −11.50282340835252, −11.13808807262601, −10.51218959978729, −10.11104389903063, −9.540147150296317, −9.182702340767381, −8.680033051464979, −8.302027047815861, −7.724199385712190, −7.228882145207619, −6.825026403366721, −6.269796886286753, −5.638398343993129, −5.291765990555502, −4.488090999166247, −4.107820978653783, −3.551198476333617, −2.933789617012846, −2.611930172920119, −1.728953277383248, −1.114890885942714, −0.7062712995359448,
0.7062712995359448, 1.114890885942714, 1.728953277383248, 2.611930172920119, 2.933789617012846, 3.551198476333617, 4.107820978653783, 4.488090999166247, 5.291765990555502, 5.638398343993129, 6.269796886286753, 6.825026403366721, 7.228882145207619, 7.724199385712190, 8.302027047815861, 8.680033051464979, 9.182702340767381, 9.540147150296317, 10.11104389903063, 10.51218959978729, 11.13808807262601, 11.50282340835252, 11.93307222460168, 12.68149613583307, 12.80783673796627
