| L(s) = 1 | − 3-s − 5-s − 2·9-s − 3·11-s + 5·13-s + 15-s − 3·17-s − 2·19-s + 6·23-s + 25-s + 5·27-s − 3·29-s − 4·31-s + 3·33-s − 2·37-s − 5·39-s + 12·41-s − 10·43-s + 2·45-s + 9·47-s + 3·51-s − 12·53-s + 3·55-s + 2·57-s + 8·61-s − 5·65-s − 4·67-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s − 2/3·9-s − 0.904·11-s + 1.38·13-s + 0.258·15-s − 0.727·17-s − 0.458·19-s + 1.25·23-s + 1/5·25-s + 0.962·27-s − 0.557·29-s − 0.718·31-s + 0.522·33-s − 0.328·37-s − 0.800·39-s + 1.87·41-s − 1.52·43-s + 0.298·45-s + 1.31·47-s + 0.420·51-s − 1.64·53-s + 0.404·55-s + 0.264·57-s + 1.02·61-s − 0.620·65-s − 0.488·67-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
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 - 5 T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + 3 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 12 T + p T^{2} \) |
| 43 | \( 1 + 10 T + p T^{2} \) |
| 47 | \( 1 - 9 T + p T^{2} \) |
| 53 | \( 1 + 12 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 12 T + p T^{2} \) |
| 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
−16.27918058683733, −15.70521205619242, −15.34866029952413, −14.63538749084514, −14.08799270286763, −13.31053556615522, −12.94368303078648, −12.46612160133975, −11.50298562059002, −11.23528110807813, −10.80675915703285, −10.33180779386039, −9.261472568973643, −8.802161270444983, −8.318985739712541, −7.594941507796575, −6.944351412433016, −6.220785031076169, −5.738494598494589, −5.055805374194028, −4.414894259348672, −3.559915844925244, −2.943367486130850, −2.040492009437095, −0.9144680225949532, 0,
0.9144680225949532, 2.040492009437095, 2.943367486130850, 3.559915844925244, 4.414894259348672, 5.055805374194028, 5.738494598494589, 6.220785031076169, 6.944351412433016, 7.594941507796575, 8.318985739712541, 8.802161270444983, 9.261472568973643, 10.33180779386039, 10.80675915703285, 11.23528110807813, 11.50298562059002, 12.46612160133975, 12.94368303078648, 13.31053556615522, 14.08799270286763, 14.63538749084514, 15.34866029952413, 15.70521205619242, 16.27918058683733
