| L(s) = 1 | − 3-s − 5-s − 7-s + 9-s − 2·13-s + 15-s + 6·17-s + 4·19-s + 21-s − 23-s + 25-s − 27-s − 10·29-s + 4·31-s + 35-s + 10·37-s + 2·39-s − 2·41-s + 4·43-s − 45-s + 49-s − 6·51-s − 6·53-s − 4·57-s + 4·59-s − 6·61-s − 63-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s − 0.377·7-s + 1/3·9-s − 0.554·13-s + 0.258·15-s + 1.45·17-s + 0.917·19-s + 0.218·21-s − 0.208·23-s + 1/5·25-s − 0.192·27-s − 1.85·29-s + 0.718·31-s + 0.169·35-s + 1.64·37-s + 0.320·39-s − 0.312·41-s + 0.609·43-s − 0.149·45-s + 1/7·49-s − 0.840·51-s − 0.824·53-s − 0.529·57-s + 0.520·59-s − 0.768·61-s − 0.125·63-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 38640 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38640 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 23 | \( 1 + T \) |
| good | 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 10 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + 16 T + p T^{2} \) |
| 83 | \( 1 + 16 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 + 6 T + p T^{2} \) |
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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
−15.19541625458464, −14.45853324376123, −14.24259070443557, −13.41453064382431, −12.85555657315232, −12.50487282563986, −11.86315643374227, −11.51087123422274, −11.03361930959881, −10.25197210115568, −9.787049692815118, −9.490719839216130, −8.706816699984677, −7.899262506317822, −7.509020534829574, −7.158218747763234, −6.255945333887347, −5.757769553342376, −5.317365917457048, −4.541481484139377, −3.980177381698039, −3.237048433770309, −2.724170525731764, −1.638285750493973, −0.8945132471802122, 0,
0.8945132471802122, 1.638285750493973, 2.724170525731764, 3.237048433770309, 3.980177381698039, 4.541481484139377, 5.317365917457048, 5.757769553342376, 6.255945333887347, 7.158218747763234, 7.509020534829574, 7.899262506317822, 8.706816699984677, 9.490719839216130, 9.787049692815118, 10.25197210115568, 11.03361930959881, 11.51087123422274, 11.86315643374227, 12.50487282563986, 12.85555657315232, 13.41453064382431, 14.24259070443557, 14.45853324376123, 15.19541625458464
