| L(s) = 1 | − 5-s − 7-s + 2·11-s + 4·13-s + 3·17-s + 4·19-s − 4·25-s + 8·29-s + 8·31-s + 35-s − 11·37-s − 7·41-s + 11·43-s − 11·47-s + 49-s − 2·53-s − 2·55-s − 11·59-s − 6·61-s − 4·65-s − 4·67-s + 4·71-s + 12·73-s − 2·77-s − 79-s + 3·83-s − 3·85-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 0.377·7-s + 0.603·11-s + 1.10·13-s + 0.727·17-s + 0.917·19-s − 4/5·25-s + 1.48·29-s + 1.43·31-s + 0.169·35-s − 1.80·37-s − 1.09·41-s + 1.67·43-s − 1.60·47-s + 1/7·49-s − 0.274·53-s − 0.269·55-s − 1.43·59-s − 0.768·61-s − 0.496·65-s − 0.488·67-s + 0.474·71-s + 1.40·73-s − 0.227·77-s − 0.112·79-s + 0.329·83-s − 0.325·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.973743474\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.973743474\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + T \) |
| good | 5 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 8 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 11 T + p T^{2} \) |
| 41 | \( 1 + 7 T + p T^{2} \) |
| 43 | \( 1 - 11 T + p T^{2} \) |
| 47 | \( 1 + 11 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 + 11 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 4 T + p T^{2} \) |
| 73 | \( 1 - 12 T + p T^{2} \) |
| 79 | \( 1 + T + p T^{2} \) |
| 83 | \( 1 - 3 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 97 | \( 1 - 4 T + p T^{2} \) |
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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
−8.114254772917833441441155246901, −7.42155564328025071651458700394, −6.52561632933346709600267026989, −6.13872722473593126170654432436, −5.17518619108817257099147769958, −4.39170807200882318922758601012, −3.47352292161249878138274379876, −3.10871663684848926164811214996, −1.68537973814508942743088327227, −0.77715738817083852520333411823,
0.77715738817083852520333411823, 1.68537973814508942743088327227, 3.10871663684848926164811214996, 3.47352292161249878138274379876, 4.39170807200882318922758601012, 5.17518619108817257099147769958, 6.13872722473593126170654432436, 6.52561632933346709600267026989, 7.42155564328025071651458700394, 8.114254772917833441441155246901
