| L(s) = 1 | + 3-s + 9-s − 4·11-s − 13-s − 17-s + 4·19-s + 27-s − 2·29-s − 4·33-s − 6·37-s − 39-s − 6·41-s − 4·43-s − 7·49-s − 51-s − 6·53-s + 4·57-s − 4·59-s + 6·61-s + 12·67-s + 16·71-s + 6·73-s − 8·79-s + 81-s + 12·83-s − 2·87-s + 2·89-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1/3·9-s − 1.20·11-s − 0.277·13-s − 0.242·17-s + 0.917·19-s + 0.192·27-s − 0.371·29-s − 0.696·33-s − 0.986·37-s − 0.160·39-s − 0.937·41-s − 0.609·43-s − 49-s − 0.140·51-s − 0.824·53-s + 0.529·57-s − 0.520·59-s + 0.768·61-s + 1.46·67-s + 1.89·71-s + 0.702·73-s − 0.900·79-s + 1/9·81-s + 1.31·83-s − 0.214·87-s + 0.211·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 265200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 265200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.505355146\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.505355146\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 13 | \( 1 + T \) |
| 17 | \( 1 + T \) |
| good | 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 - 16 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 2 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−12.78750079125131, −12.53640732621634, −11.82443268736616, −11.48022686477679, −10.85342583826439, −10.51634116451720, −9.884492575434164, −9.655372071744407, −9.133344384391459, −8.481218568660215, −8.150912451862988, −7.703323853971853, −7.284934637510508, −6.648380714218627, −6.336450230614898, −5.392616857581402, −5.103870008078630, −4.845576895117008, −3.840294370747658, −3.567888018492993, −2.934925926051437, −2.429162670011511, −1.883805778715982, −1.229034364566142, −0.3141640806282564,
0.3141640806282564, 1.229034364566142, 1.883805778715982, 2.429162670011511, 2.934925926051437, 3.567888018492993, 3.840294370747658, 4.845576895117008, 5.103870008078630, 5.392616857581402, 6.336450230614898, 6.648380714218627, 7.284934637510508, 7.703323853971853, 8.150912451862988, 8.481218568660215, 9.133344384391459, 9.655372071744407, 9.884492575434164, 10.51634116451720, 10.85342583826439, 11.48022686477679, 11.82443268736616, 12.53640732621634, 12.78750079125131
