| L(s) = 1 | − 3-s + 5-s − 7-s − 2·9-s + 11-s − 3·13-s − 15-s − 7·17-s + 4·19-s + 21-s + 25-s + 5·27-s + 5·29-s + 10·31-s − 33-s − 35-s + 4·37-s + 3·39-s − 10·41-s + 8·43-s − 2·45-s + 47-s + 49-s + 7·51-s + 4·53-s + 55-s − 4·57-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.447·5-s − 0.377·7-s − 2/3·9-s + 0.301·11-s − 0.832·13-s − 0.258·15-s − 1.69·17-s + 0.917·19-s + 0.218·21-s + 1/5·25-s + 0.962·27-s + 0.928·29-s + 1.79·31-s − 0.174·33-s − 0.169·35-s + 0.657·37-s + 0.480·39-s − 1.56·41-s + 1.21·43-s − 0.298·45-s + 0.145·47-s + 1/7·49-s + 0.980·51-s + 0.549·53-s + 0.134·55-s − 0.529·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.189283797\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.189283797\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| good | 3 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 - T + p T^{2} \) |
| 13 | \( 1 + 3 T + p T^{2} \) |
| 17 | \( 1 + 7 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 5 T + p T^{2} \) |
| 31 | \( 1 - 10 T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 - T + p T^{2} \) |
| 53 | \( 1 - 4 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + 11 T + p T^{2} \) |
| 83 | \( 1 + 8 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - T + p T^{2} \) |
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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
−8.993417693062439548052213697576, −8.452680031565151949530077712891, −7.30295905151208216917022291938, −6.54958122235842678005318111968, −6.01855147310902740311669436609, −5.04913615714797379840824847905, −4.42057895944109784173321806697, −3.04884428162282165924019129498, −2.29151683298808169914787877527, −0.70997319903356004338141848831,
0.70997319903356004338141848831, 2.29151683298808169914787877527, 3.04884428162282165924019129498, 4.42057895944109784173321806697, 5.04913615714797379840824847905, 6.01855147310902740311669436609, 6.54958122235842678005318111968, 7.30295905151208216917022291938, 8.452680031565151949530077712891, 8.993417693062439548052213697576
