| L(s) = 1 | + 3-s − 2·7-s + 9-s + 11-s − 4·13-s − 17-s − 2·19-s − 2·21-s + 2·23-s + 27-s + 2·29-s + 4·31-s + 33-s − 6·37-s − 4·39-s + 6·41-s + 2·43-s − 3·49-s − 51-s + 12·53-s − 2·57-s + 14·59-s + 6·61-s − 2·63-s − 4·67-s + 2·69-s + 2·71-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.755·7-s + 1/3·9-s + 0.301·11-s − 1.10·13-s − 0.242·17-s − 0.458·19-s − 0.436·21-s + 0.417·23-s + 0.192·27-s + 0.371·29-s + 0.718·31-s + 0.174·33-s − 0.986·37-s − 0.640·39-s + 0.937·41-s + 0.304·43-s − 3/7·49-s − 0.140·51-s + 1.64·53-s − 0.264·57-s + 1.82·59-s + 0.768·61-s − 0.251·63-s − 0.488·67-s + 0.240·69-s + 0.237·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 224400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 224400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.325121842\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.325121842\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 11 | \( 1 - T \) |
| 17 | \( 1 + T \) |
| good | 7 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 12 T + p T^{2} \) |
| 59 | \( 1 - 14 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 2 T + p T^{2} \) |
| 73 | \( 1 - 8 T + p T^{2} \) |
| 79 | \( 1 - 2 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 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
−12.95540300014107, −12.64539936748243, −11.94175894012110, −11.79414491234127, −11.05368203391354, −10.44557678916198, −10.12679004644715, −9.671226323019815, −9.210398147893155, −8.752176134022404, −8.330180919025132, −7.721604283101046, −7.222218208089899, −6.715922511460148, −6.496887749120117, −5.683891022197473, −5.183608331029698, −4.645893905248793, −3.969792857724110, −3.699470422343200, −2.820742419648251, −2.568784230244928, −1.989553698067353, −1.110848089654933, −0.4316593065953595,
0.4316593065953595, 1.110848089654933, 1.989553698067353, 2.568784230244928, 2.820742419648251, 3.699470422343200, 3.969792857724110, 4.645893905248793, 5.183608331029698, 5.683891022197473, 6.496887749120117, 6.715922511460148, 7.222218208089899, 7.721604283101046, 8.330180919025132, 8.752176134022404, 9.210398147893155, 9.671226323019815, 10.12679004644715, 10.44557678916198, 11.05368203391354, 11.79414491234127, 11.94175894012110, 12.64539936748243, 12.95540300014107
