| L(s) = 1 | + 3-s + 5-s − 2·9-s + 2·11-s − 4·13-s + 15-s − 6·19-s + 3·23-s + 25-s − 5·27-s + 3·29-s + 2·33-s + 12·37-s − 4·39-s − 7·41-s + 9·43-s − 2·45-s + 6·53-s + 2·55-s − 6·57-s + 10·59-s − 5·61-s − 4·65-s − 11·67-s + 3·69-s − 10·71-s − 8·73-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.447·5-s − 2/3·9-s + 0.603·11-s − 1.10·13-s + 0.258·15-s − 1.37·19-s + 0.625·23-s + 1/5·25-s − 0.962·27-s + 0.557·29-s + 0.348·33-s + 1.97·37-s − 0.640·39-s − 1.09·41-s + 1.37·43-s − 0.298·45-s + 0.824·53-s + 0.269·55-s − 0.794·57-s + 1.30·59-s − 0.640·61-s − 0.496·65-s − 1.34·67-s + 0.361·69-s − 1.18·71-s − 0.936·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 15680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 15680 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.361798044\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.361798044\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 - 3 T + p T^{2} \) |
| 29 | \( 1 - 3 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 12 T + p T^{2} \) |
| 41 | \( 1 + 7 T + p T^{2} \) |
| 43 | \( 1 - 9 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 10 T + p T^{2} \) |
| 61 | \( 1 + 5 T + p T^{2} \) |
| 67 | \( 1 + 11 T + p T^{2} \) |
| 71 | \( 1 + 10 T + p T^{2} \) |
| 73 | \( 1 + 8 T + p T^{2} \) |
| 79 | \( 1 - 6 T + p T^{2} \) |
| 83 | \( 1 - 3 T + p T^{2} \) |
| 89 | \( 1 - 17 T + p T^{2} \) |
| 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
−16.04016545085958, −15.11254639111323, −14.74461102158423, −14.55982894213701, −13.82708173239360, −13.19268000035299, −12.85232183047942, −11.90422004092535, −11.73633640770675, −10.76895870677485, −10.37486900844829, −9.595044793477540, −9.092370006313377, −8.690697485439547, −7.951130461022201, −7.377487730030339, −6.598900830168303, −6.083211490936693, −5.378984855627141, −4.564830463787836, −4.031945477577028, −2.985645609694286, −2.535909881004686, −1.811011855763082, −0.6231682857590702,
0.6231682857590702, 1.811011855763082, 2.535909881004686, 2.985645609694286, 4.031945477577028, 4.564830463787836, 5.378984855627141, 6.083211490936693, 6.598900830168303, 7.377487730030339, 7.951130461022201, 8.690697485439547, 9.092370006313377, 9.595044793477540, 10.37486900844829, 10.76895870677485, 11.73633640770675, 11.90422004092535, 12.85232183047942, 13.19268000035299, 13.82708173239360, 14.55982894213701, 14.74461102158423, 15.11254639111323, 16.04016545085958
