| L(s) = 1 | + 2·3-s + 5-s − 2·7-s + 9-s + 4·11-s + 6·13-s + 2·15-s + 2·17-s − 8·19-s − 4·21-s − 6·23-s + 25-s − 4·27-s + 2·29-s + 4·31-s + 8·33-s − 2·35-s − 2·37-s + 12·39-s − 10·41-s + 2·43-s + 45-s − 2·47-s − 3·49-s + 4·51-s − 2·53-s + 4·55-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 0.447·5-s − 0.755·7-s + 1/3·9-s + 1.20·11-s + 1.66·13-s + 0.516·15-s + 0.485·17-s − 1.83·19-s − 0.872·21-s − 1.25·23-s + 1/5·25-s − 0.769·27-s + 0.371·29-s + 0.718·31-s + 1.39·33-s − 0.338·35-s − 0.328·37-s + 1.92·39-s − 1.56·41-s + 0.304·43-s + 0.149·45-s − 0.291·47-s − 3/7·49-s + 0.560·51-s − 0.274·53-s + 0.539·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.927469047\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.927469047\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - T \) |
| good | 3 | \( 1 - 2 T + p T^{2} \) |
| 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 8 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 + 2 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 6 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 10 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 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
−11.68387141915772806684905453674, −10.49220259147110233894421354953, −9.613228912214399908431984206107, −8.714543115468972113059546452375, −8.239480380770897255724570167213, −6.60040674980892955704654655808, −6.06254412327818625828699807195, −4.09416326053364009314265749502, −3.28314496469795666027926296042, −1.81188448910292297689023276847,
1.81188448910292297689023276847, 3.28314496469795666027926296042, 4.09416326053364009314265749502, 6.06254412327818625828699807195, 6.60040674980892955704654655808, 8.239480380770897255724570167213, 8.714543115468972113059546452375, 9.613228912214399908431984206107, 10.49220259147110233894421354953, 11.68387141915772806684905453674
