| L(s) = 1 | + 3-s + 7-s − 2·9-s + 3·11-s − 7·13-s + 5·17-s − 4·19-s + 21-s + 4·23-s − 5·27-s + 5·29-s − 2·31-s + 3·33-s − 7·39-s + 6·41-s + 8·43-s − 9·47-s + 49-s + 5·51-s − 4·57-s − 12·59-s + 6·61-s − 2·63-s + 16·67-s + 4·69-s − 4·71-s + 14·73-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.377·7-s − 2/3·9-s + 0.904·11-s − 1.94·13-s + 1.21·17-s − 0.917·19-s + 0.218·21-s + 0.834·23-s − 0.962·27-s + 0.928·29-s − 0.359·31-s + 0.522·33-s − 1.12·39-s + 0.937·41-s + 1.21·43-s − 1.31·47-s + 1/7·49-s + 0.700·51-s − 0.529·57-s − 1.56·59-s + 0.768·61-s − 0.251·63-s + 1.95·67-s + 0.481·69-s − 0.474·71-s + 1.63·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 11200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.349252655\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.349252655\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 - T \) |
| good | 3 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 + 7 T + p T^{2} \) |
| 17 | \( 1 - 5 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - 5 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + 9 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 - 16 T + p T^{2} \) |
| 71 | \( 1 + 4 T + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 + 9 T + p T^{2} \) |
| 83 | \( 1 + 8 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 3 T + p T^{2} \) |
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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
−16.73259219183479, −15.84079634879205, −15.06745138013482, −14.65194969675213, −14.26137501402672, −14.02070915823145, −12.86568891678044, −12.50578019497817, −11.93727756264098, −11.30236106637369, −10.73107562712502, −9.767082449638933, −9.577096667905372, −8.782639435182916, −8.225702882592320, −7.594635317692514, −7.043517540518450, −6.250865512182707, −5.470985300981855, −4.824810149789511, −4.134986070944127, −3.211833403539010, −2.611643101491491, −1.844035472779567, −0.6760871575082525,
0.6760871575082525, 1.844035472779567, 2.611643101491491, 3.211833403539010, 4.134986070944127, 4.824810149789511, 5.470985300981855, 6.250865512182707, 7.043517540518450, 7.594635317692514, 8.225702882592320, 8.782639435182916, 9.577096667905372, 9.767082449638933, 10.73107562712502, 11.30236106637369, 11.93727756264098, 12.50578019497817, 12.86568891678044, 14.02070915823145, 14.26137501402672, 14.65194969675213, 15.06745138013482, 15.84079634879205, 16.73259219183479
