| L(s) = 1 | + 3-s + 5-s + 7-s − 2·9-s + 3·11-s + 15-s − 3·17-s + 5·19-s + 21-s − 9·23-s + 25-s − 5·27-s + 9·29-s − 8·31-s + 3·33-s + 35-s + 7·37-s + 3·41-s − 43-s − 2·45-s − 6·49-s − 3·51-s − 6·53-s + 3·55-s + 5·57-s + 9·59-s + 61-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.447·5-s + 0.377·7-s − 2/3·9-s + 0.904·11-s + 0.258·15-s − 0.727·17-s + 1.14·19-s + 0.218·21-s − 1.87·23-s + 1/5·25-s − 0.962·27-s + 1.67·29-s − 1.43·31-s + 0.522·33-s + 0.169·35-s + 1.15·37-s + 0.468·41-s − 0.152·43-s − 0.298·45-s − 6/7·49-s − 0.420·51-s − 0.824·53-s + 0.404·55-s + 0.662·57-s + 1.17·59-s + 0.128·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 54080 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 54080 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 13 | \( 1 \) |
| good | 3 | \( 1 - T + p T^{2} \) |
| 7 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 - 5 T + p T^{2} \) |
| 23 | \( 1 + 9 T + p T^{2} \) |
| 29 | \( 1 - 9 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 7 T + p T^{2} \) |
| 41 | \( 1 - 3 T + p T^{2} \) |
| 43 | \( 1 + T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 9 T + p T^{2} \) |
| 61 | \( 1 - T + p T^{2} \) |
| 67 | \( 1 - 5 T + p T^{2} \) |
| 71 | \( 1 + 9 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 - 3 T + p T^{2} \) |
| 97 | \( 1 - 17 T + p T^{2} \) |
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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
−14.59134136724242, −14.16518373483555, −13.87474060976727, −13.27557805347598, −12.74895941592321, −11.92900062089685, −11.74010958574543, −11.17836346964690, −10.58704189799138, −9.838341124194680, −9.528636192239562, −8.984411616973328, −8.466610526494904, −7.943640256759027, −7.503180854240443, −6.603936660832329, −6.295906666954218, −5.597757615390627, −5.100951742035385, −4.235392202772066, −3.853779598925407, −3.032487976211365, −2.475974582593373, −1.807577248145980, −1.126107314106583, 0,
1.126107314106583, 1.807577248145980, 2.475974582593373, 3.032487976211365, 3.853779598925407, 4.235392202772066, 5.100951742035385, 5.597757615390627, 6.295906666954218, 6.603936660832329, 7.503180854240443, 7.943640256759027, 8.466610526494904, 8.984411616973328, 9.528636192239562, 9.838341124194680, 10.58704189799138, 11.17836346964690, 11.74010958574543, 11.92900062089685, 12.74895941592321, 13.27557805347598, 13.87474060976727, 14.16518373483555, 14.59134136724242
