| L(s) = 1 | − 3-s + 9-s + 5·11-s + 2·13-s + 6·17-s − 2·19-s − 5·23-s − 27-s + 5·29-s + 4·31-s − 5·33-s + 37-s − 2·39-s + 12·41-s + 5·43-s + 2·47-s − 6·51-s + 14·53-s + 2·57-s + 2·59-s − 5·67-s + 5·69-s − 9·71-s + 10·73-s + 11·79-s + 81-s − 16·83-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1/3·9-s + 1.50·11-s + 0.554·13-s + 1.45·17-s − 0.458·19-s − 1.04·23-s − 0.192·27-s + 0.928·29-s + 0.718·31-s − 0.870·33-s + 0.164·37-s − 0.320·39-s + 1.87·41-s + 0.762·43-s + 0.291·47-s − 0.840·51-s + 1.92·53-s + 0.264·57-s + 0.260·59-s − 0.610·67-s + 0.601·69-s − 1.06·71-s + 1.17·73-s + 1.23·79-s + 1/9·81-s − 1.75·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 235200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 235200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.782278712\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.782278712\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 7 | \( 1 \) |
| good | 11 | \( 1 - 5 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + 5 T + p T^{2} \) |
| 29 | \( 1 - 5 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - T + p T^{2} \) |
| 41 | \( 1 - 12 T + p T^{2} \) |
| 43 | \( 1 - 5 T + p T^{2} \) |
| 47 | \( 1 - 2 T + p T^{2} \) |
| 53 | \( 1 - 14 T + p T^{2} \) |
| 59 | \( 1 - 2 T + p T^{2} \) |
| 61 | \( 1 + p T^{2} \) |
| 67 | \( 1 + 5 T + p T^{2} \) |
| 71 | \( 1 + 9 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 - 11 T + p T^{2} \) |
| 83 | \( 1 + 16 T + p T^{2} \) |
| 89 | \( 1 - 14 T + p T^{2} \) |
| 97 | \( 1 - 8 T + p T^{2} \) |
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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
−12.78972888831664, −12.21876490794276, −12.00803874401233, −11.71332467356180, −11.01993486916351, −10.65854950209209, −10.08703798839378, −9.755370176349972, −9.200065319839368, −8.700717492407637, −8.249033194034176, −7.618397895375559, −7.257678862884475, −6.533921114311363, −6.204505877751171, −5.815117874012902, −5.348387516652915, −4.451110560750462, −4.210281025791765, −3.715313869503393, −3.083608257600212, −2.348155084466590, −1.683666795093788, −0.8737265731051905, −0.7770877880244884,
0.7770877880244884, 0.8737265731051905, 1.683666795093788, 2.348155084466590, 3.083608257600212, 3.715313869503393, 4.210281025791765, 4.451110560750462, 5.348387516652915, 5.815117874012902, 6.204505877751171, 6.533921114311363, 7.257678862884475, 7.618397895375559, 8.249033194034176, 8.700717492407637, 9.200065319839368, 9.755370176349972, 10.08703798839378, 10.65854950209209, 11.01993486916351, 11.71332467356180, 12.00803874401233, 12.21876490794276, 12.78972888831664
