| L(s) = 1 | + 3-s − 7-s + 9-s − 11-s − 2·13-s + 8·17-s − 2·19-s − 21-s + 23-s + 27-s + 29-s + 6·31-s − 33-s + 9·37-s − 2·39-s − 43-s + 6·47-s + 49-s + 8·51-s + 2·53-s − 2·57-s − 6·59-s + 8·61-s − 63-s − 3·67-s + 69-s + 7·71-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.377·7-s + 1/3·9-s − 0.301·11-s − 0.554·13-s + 1.94·17-s − 0.458·19-s − 0.218·21-s + 0.208·23-s + 0.192·27-s + 0.185·29-s + 1.07·31-s − 0.174·33-s + 1.47·37-s − 0.320·39-s − 0.152·43-s + 0.875·47-s + 1/7·49-s + 1.12·51-s + 0.274·53-s − 0.264·57-s − 0.781·59-s + 1.02·61-s − 0.125·63-s − 0.366·67-s + 0.120·69-s + 0.830·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.134363238\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.134363238\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| good | 11 | \( 1 + T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 8 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 - T + p T^{2} \) |
| 29 | \( 1 - T + p T^{2} \) |
| 31 | \( 1 - 6 T + p T^{2} \) |
| 37 | \( 1 - 9 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 + T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 + 3 T + p T^{2} \) |
| 71 | \( 1 - 7 T + p T^{2} \) |
| 73 | \( 1 - 16 T + p T^{2} \) |
| 79 | \( 1 - T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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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
−9.178086174923145361356295439092, −8.173616640061513612311687231369, −7.73796767919337607548327161651, −6.85752965895438882990876656904, −5.94851442054367400733493384203, −5.09165818083365444342952164291, −4.10481586394577524474348106422, −3.15803289674165168177952723150, −2.40596014358214489415565420385, −0.964817304461697686364302398834,
0.964817304461697686364302398834, 2.40596014358214489415565420385, 3.15803289674165168177952723150, 4.10481586394577524474348106422, 5.09165818083365444342952164291, 5.94851442054367400733493384203, 6.85752965895438882990876656904, 7.73796767919337607548327161651, 8.173616640061513612311687231369, 9.178086174923145361356295439092
