| L(s) = 1 | − 3-s + 2·5-s + 9-s − 4·11-s + 2·13-s − 2·15-s − 2·17-s − 4·19-s + 8·23-s − 25-s − 27-s + 6·29-s + 8·31-s + 4·33-s + 6·37-s − 2·39-s + 6·41-s − 4·43-s + 2·45-s + 2·51-s − 2·53-s − 8·55-s + 4·57-s + 4·59-s + 2·61-s + 4·65-s + 4·67-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.894·5-s + 1/3·9-s − 1.20·11-s + 0.554·13-s − 0.516·15-s − 0.485·17-s − 0.917·19-s + 1.66·23-s − 1/5·25-s − 0.192·27-s + 1.11·29-s + 1.43·31-s + 0.696·33-s + 0.986·37-s − 0.320·39-s + 0.937·41-s − 0.609·43-s + 0.298·45-s + 0.280·51-s − 0.274·53-s − 1.07·55-s + 0.529·57-s + 0.520·59-s + 0.256·61-s + 0.496·65-s + 0.488·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.630172600\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.630172600\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−8.959311313448748386566246803110, −8.304042274266532161103040174092, −7.35977874808944660694472335106, −6.41480629060511724242965007307, −5.98894980010644060477023411018, −5.02364207902997377499660058417, −4.44992200701182918033448908949, −3.00889317555517420664706578512, −2.17837496002353596178446785744, −0.852162344962028721020593862686,
0.852162344962028721020593862686, 2.17837496002353596178446785744, 3.00889317555517420664706578512, 4.44992200701182918033448908949, 5.02364207902997377499660058417, 5.98894980010644060477023411018, 6.41480629060511724242965007307, 7.35977874808944660694472335106, 8.304042274266532161103040174092, 8.959311313448748386566246803110
