| L(s) = 1 | + 3-s − 5-s + 2·7-s + 9-s − 5·11-s + 5·13-s − 15-s + 5·17-s − 6·19-s + 2·21-s + 9·23-s + 25-s + 27-s − 8·29-s + 5·31-s − 5·33-s − 2·35-s − 8·37-s + 5·39-s − 7·41-s − 43-s − 45-s + 8·47-s − 3·49-s + 5·51-s + 5·53-s + 5·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.447·5-s + 0.755·7-s + 1/3·9-s − 1.50·11-s + 1.38·13-s − 0.258·15-s + 1.21·17-s − 1.37·19-s + 0.436·21-s + 1.87·23-s + 1/5·25-s + 0.192·27-s − 1.48·29-s + 0.898·31-s − 0.870·33-s − 0.338·35-s − 1.31·37-s + 0.800·39-s − 1.09·41-s − 0.152·43-s − 0.149·45-s + 1.16·47-s − 3/7·49-s + 0.700·51-s + 0.686·53-s + 0.674·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 41280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 41280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 43 | \( 1 + T \) |
| good | 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + 5 T + p T^{2} \) |
| 13 | \( 1 - 5 T + p T^{2} \) |
| 17 | \( 1 - 5 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 - 9 T + p T^{2} \) |
| 29 | \( 1 + 8 T + p T^{2} \) |
| 31 | \( 1 - 5 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 + 7 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - 5 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + p T^{2} \) |
| 67 | \( 1 - 9 T + p T^{2} \) |
| 71 | \( 1 + 10 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 + 16 T + p T^{2} \) |
| 83 | \( 1 + 9 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 3 T + p T^{2} \) |
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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
−14.94293492158673, −14.71877561797694, −13.85122549995040, −13.49535159781128, −12.96344365484659, −12.58799462203761, −11.88891376818854, −11.16338631859277, −10.85802173160336, −10.41547890138656, −9.816244240345272, −8.928002882759685, −8.489467401775184, −8.262102358167727, −7.567382944319583, −7.137662800884069, −6.436789413966375, −5.448940958887056, −5.321090442555494, −4.448807620995336, −3.852470203124127, −3.181438882627057, −2.672479602185853, −1.739456157026542, −1.144378164473301, 0,
1.144378164473301, 1.739456157026542, 2.672479602185853, 3.181438882627057, 3.852470203124127, 4.448807620995336, 5.321090442555494, 5.448940958887056, 6.436789413966375, 7.137662800884069, 7.567382944319583, 8.262102358167727, 8.489467401775184, 8.928002882759685, 9.816244240345272, 10.41547890138656, 10.85802173160336, 11.16338631859277, 11.88891376818854, 12.58799462203761, 12.96344365484659, 13.49535159781128, 13.85122549995040, 14.71877561797694, 14.94293492158673
