L(s) = 1 | + 3-s + 7-s + 9-s − 11-s + 2·13-s + 6·17-s − 4·19-s + 21-s − 8·23-s + 27-s + 6·29-s − 4·31-s − 33-s + 2·37-s + 2·39-s − 6·41-s − 12·43-s + 49-s + 6·51-s + 2·53-s − 4·57-s + 4·59-s + 2·61-s + 63-s + 8·67-s − 8·69-s + 8·71-s + ⋯ |
L(s) = 1 | + 0.577·3-s + 0.377·7-s + 1/3·9-s − 0.301·11-s + 0.554·13-s + 1.45·17-s − 0.917·19-s + 0.218·21-s − 1.66·23-s + 0.192·27-s + 1.11·29-s − 0.718·31-s − 0.174·33-s + 0.328·37-s + 0.320·39-s − 0.937·41-s − 1.82·43-s + 1/7·49-s + 0.840·51-s + 0.274·53-s − 0.529·57-s + 0.520·59-s + 0.256·61-s + 0.125·63-s + 0.977·67-s − 0.963·69-s + 0.949·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 369600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 369600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.253314112\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.253314112\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 11 | \( 1 + T \) |
good | 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 12 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + 12 T + p T^{2} \) |
| 83 | \( 1 - 8 T + p T^{2} \) |
| 89 | \( 1 - 18 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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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
−12.52478261927379, −11.97848839402374, −11.72310158117119, −11.20287252105763, −10.44316445355699, −10.23696094358825, −9.959087420704282, −9.303390387093813, −8.726896663222409, −8.285157500875865, −8.090569254714627, −7.599808848632676, −7.041775301081632, −6.360724213657699, −6.169878333720094, −5.385404484087363, −5.051582791893604, −4.453842681225111, −3.736945135768605, −3.595287061255275, −2.919737249079896, −2.191512062056003, −1.850895255815996, −1.185314864905352, −0.4527356348880673,
0.4527356348880673, 1.185314864905352, 1.850895255815996, 2.191512062056003, 2.919737249079896, 3.595287061255275, 3.736945135768605, 4.453842681225111, 5.051582791893604, 5.385404484087363, 6.169878333720094, 6.360724213657699, 7.041775301081632, 7.599808848632676, 8.090569254714627, 8.285157500875865, 8.726896663222409, 9.303390387093813, 9.959087420704282, 10.23696094358825, 10.44316445355699, 11.20287252105763, 11.72310158117119, 11.97848839402374, 12.52478261927379