| L(s) = 1 | − 5-s + 2·7-s − 4·11-s − 2·13-s − 5·17-s − 5·19-s − 23-s + 25-s + 2·29-s + 7·31-s − 2·35-s − 6·37-s + 4·43-s − 4·47-s − 3·49-s − 9·53-s + 4·55-s − 14·59-s − 11·61-s + 2·65-s + 14·67-s − 12·73-s − 8·77-s − 3·79-s + 83-s + 5·85-s − 4·91-s + ⋯ |
| L(s) = 1 | − 0.447·5-s + 0.755·7-s − 1.20·11-s − 0.554·13-s − 1.21·17-s − 1.14·19-s − 0.208·23-s + 1/5·25-s + 0.371·29-s + 1.25·31-s − 0.338·35-s − 0.986·37-s + 0.609·43-s − 0.583·47-s − 3/7·49-s − 1.23·53-s + 0.539·55-s − 1.82·59-s − 1.40·61-s + 0.248·65-s + 1.71·67-s − 1.40·73-s − 0.911·77-s − 0.337·79-s + 0.109·83-s + 0.542·85-s − 0.419·91-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1080 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1080 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 + T \) |
| good | 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 5 T + p T^{2} \) |
| 19 | \( 1 + 5 T + p T^{2} \) |
| 23 | \( 1 + T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 - 7 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 + 9 T + p T^{2} \) |
| 59 | \( 1 + 14 T + p T^{2} \) |
| 61 | \( 1 + 11 T + p T^{2} \) |
| 67 | \( 1 - 14 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 12 T + p T^{2} \) |
| 79 | \( 1 + 3 T + p T^{2} \) |
| 83 | \( 1 - T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 - 16 T + p T^{2} \) |
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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
−9.438199634414655966671221781462, −8.392487762438800568739369150245, −8.001744768198934904474847883292, −7.02981671194631499195835940912, −6.10221061334331611396169073880, −4.83906285718461367327183444990, −4.47121123035280332113441908167, −2.97136231353523248199016031137, −1.95034099206603296055540986865, 0,
1.95034099206603296055540986865, 2.97136231353523248199016031137, 4.47121123035280332113441908167, 4.83906285718461367327183444990, 6.10221061334331611396169073880, 7.02981671194631499195835940912, 8.001744768198934904474847883292, 8.392487762438800568739369150245, 9.438199634414655966671221781462
