| L(s) = 1 | + 2·3-s + 5-s + 7-s + 9-s + 4·11-s + 13-s + 2·15-s − 2·17-s + 19-s + 2·21-s + 7·23-s − 4·25-s − 4·27-s − 5·29-s + 9·31-s + 8·33-s + 35-s − 2·37-s + 2·39-s + 2·41-s − 43-s + 45-s − 9·47-s + 49-s − 4·51-s + 3·53-s + 4·55-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 0.447·5-s + 0.377·7-s + 1/3·9-s + 1.20·11-s + 0.277·13-s + 0.516·15-s − 0.485·17-s + 0.229·19-s + 0.436·21-s + 1.45·23-s − 4/5·25-s − 0.769·27-s − 0.928·29-s + 1.61·31-s + 1.39·33-s + 0.169·35-s − 0.328·37-s + 0.320·39-s + 0.312·41-s − 0.152·43-s + 0.149·45-s − 1.31·47-s + 1/7·49-s − 0.560·51-s + 0.412·53-s + 0.539·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1456 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1456 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.989953146\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.989953146\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 - T \) |
| 13 | \( 1 - T \) |
| good | 3 | \( 1 - 2 T + p T^{2} \) |
| 5 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - T + p T^{2} \) |
| 23 | \( 1 - 7 T + p T^{2} \) |
| 29 | \( 1 + 5 T + p T^{2} \) |
| 31 | \( 1 - 9 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + T + p T^{2} \) |
| 47 | \( 1 + 9 T + p T^{2} \) |
| 53 | \( 1 - 3 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 14 T + p T^{2} \) |
| 67 | \( 1 + 10 T + p T^{2} \) |
| 71 | \( 1 - 14 T + p T^{2} \) |
| 73 | \( 1 - 3 T + p T^{2} \) |
| 79 | \( 1 + 5 T + p T^{2} \) |
| 83 | \( 1 + 5 T + p T^{2} \) |
| 89 | \( 1 + 9 T + p T^{2} \) |
| 97 | \( 1 + T + p T^{2} \) |
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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
−9.384639507251552777367553326751, −8.735471227383272204170095485479, −8.144388668476396997222119983247, −7.14918000366524593178728481657, −6.37684020436893069227778189405, −5.34298608831346054320252005808, −4.23622923362218866686267765866, −3.39334742454479681717934942712, −2.38361565588352713517870291579, −1.35836937709627763557977801195,
1.35836937709627763557977801195, 2.38361565588352713517870291579, 3.39334742454479681717934942712, 4.23622923362218866686267765866, 5.34298608831346054320252005808, 6.37684020436893069227778189405, 7.14918000366524593178728481657, 8.144388668476396997222119983247, 8.735471227383272204170095485479, 9.384639507251552777367553326751
