L(s) = 1 | + 3-s + 2·5-s + 7-s + 9-s + 2·11-s − 4·13-s + 2·15-s + 17-s + 2·19-s + 21-s − 4·23-s − 25-s + 27-s + 2·33-s + 2·35-s + 8·37-s − 4·39-s − 2·41-s + 4·43-s + 2·45-s + 49-s + 51-s + 6·53-s + 4·55-s + 2·57-s + 10·59-s + 6·61-s + ⋯ |
L(s) = 1 | + 0.577·3-s + 0.894·5-s + 0.377·7-s + 1/3·9-s + 0.603·11-s − 1.10·13-s + 0.516·15-s + 0.242·17-s + 0.458·19-s + 0.218·21-s − 0.834·23-s − 1/5·25-s + 0.192·27-s + 0.348·33-s + 0.338·35-s + 1.31·37-s − 0.640·39-s − 0.312·41-s + 0.609·43-s + 0.298·45-s + 1/7·49-s + 0.140·51-s + 0.824·53-s + 0.539·55-s + 0.264·57-s + 1.30·59-s + 0.768·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 22848 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 22848 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.795700317\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.795700317\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 17 | \( 1 - T \) |
good | 5 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 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 | \( 1 - 10 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 + 12 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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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
−15.41257107133620, −14.67365339175757, −14.38390936573270, −14.10033751048642, −13.23044917045455, −13.06165860233018, −12.14714897215137, −11.77476775530275, −11.20469129487580, −10.15066709916453, −10.03963924097071, −9.495071955834997, −8.845116585901425, −8.339895305756977, −7.414531047113745, −7.344025492265821, −6.303277995446811, −5.833061664066512, −5.156904956626469, −4.434566848845625, −3.844713202750398, −2.930438446975555, −2.269374143114563, −1.728543206797101, −0.7624464658848059,
0.7624464658848059, 1.728543206797101, 2.269374143114563, 2.930438446975555, 3.844713202750398, 4.434566848845625, 5.156904956626469, 5.833061664066512, 6.303277995446811, 7.344025492265821, 7.414531047113745, 8.339895305756977, 8.845116585901425, 9.495071955834997, 10.03963924097071, 10.15066709916453, 11.20469129487580, 11.77476775530275, 12.14714897215137, 13.06165860233018, 13.23044917045455, 14.10033751048642, 14.38390936573270, 14.67365339175757, 15.41257107133620