| L(s) = 1 | − 3-s − 2·7-s − 2·9-s + 4·11-s − 4·13-s − 7·17-s + 3·19-s + 2·21-s + 5·27-s − 4·29-s − 6·31-s − 4·33-s + 2·37-s + 4·39-s + 6·41-s − 5·43-s − 10·47-s − 3·49-s + 7·51-s − 3·57-s + 5·59-s + 4·61-s + 4·63-s − 5·67-s − 14·71-s − 15·73-s − 8·77-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.755·7-s − 2/3·9-s + 1.20·11-s − 1.10·13-s − 1.69·17-s + 0.688·19-s + 0.436·21-s + 0.962·27-s − 0.742·29-s − 1.07·31-s − 0.696·33-s + 0.328·37-s + 0.640·39-s + 0.937·41-s − 0.762·43-s − 1.45·47-s − 3/7·49-s + 0.980·51-s − 0.397·57-s + 0.650·59-s + 0.512·61-s + 0.503·63-s − 0.610·67-s − 1.66·71-s − 1.75·73-s − 0.911·77-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 211600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 211600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 23 | \( 1 \) |
| good | 3 | \( 1 + T + p T^{2} \) |
| 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + 7 T + p T^{2} \) |
| 19 | \( 1 - 3 T + p T^{2} \) |
| 29 | \( 1 + 4 T + p T^{2} \) |
| 31 | \( 1 + 6 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 5 T + p T^{2} \) |
| 47 | \( 1 + 10 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 - 5 T + p T^{2} \) |
| 61 | \( 1 - 4 T + p T^{2} \) |
| 67 | \( 1 + 5 T + p T^{2} \) |
| 71 | \( 1 + 14 T + p T^{2} \) |
| 73 | \( 1 + 15 T + p T^{2} \) |
| 79 | \( 1 + 12 T + p T^{2} \) |
| 83 | \( 1 - 15 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 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
−13.33983200222700, −12.85223599228776, −12.72170817369619, −11.81891319891623, −11.52143059623121, −11.48252984299741, −10.71124920854393, −10.18873596397627, −9.700441498114003, −9.190834020699633, −8.923320484304837, −8.449737708885197, −7.505711421515517, −7.294658660261532, −6.690203801804600, −6.197380132804745, −5.968240208482457, −5.184931458728935, −4.758594943604342, −4.251966638693653, −3.529459068619054, −3.126311579112148, −2.398097390191665, −1.871608410354767, −1.057514022376802, 0, 0,
1.057514022376802, 1.871608410354767, 2.398097390191665, 3.126311579112148, 3.529459068619054, 4.251966638693653, 4.758594943604342, 5.184931458728935, 5.968240208482457, 6.197380132804745, 6.690203801804600, 7.294658660261532, 7.505711421515517, 8.449737708885197, 8.923320484304837, 9.190834020699633, 9.700441498114003, 10.18873596397627, 10.71124920854393, 11.48252984299741, 11.52143059623121, 11.81891319891623, 12.72170817369619, 12.85223599228776, 13.33983200222700
