| L(s) = 1 | − 3-s + 5-s + 9-s + 4·11-s − 2·13-s − 15-s + 2·17-s − 19-s − 4·23-s + 25-s − 27-s − 6·29-s − 4·31-s − 4·33-s + 6·37-s + 2·39-s + 10·41-s − 4·43-s + 45-s + 12·47-s − 7·49-s − 2·51-s − 6·53-s + 4·55-s + 57-s − 12·59-s + 2·61-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.447·5-s + 1/3·9-s + 1.20·11-s − 0.554·13-s − 0.258·15-s + 0.485·17-s − 0.229·19-s − 0.834·23-s + 1/5·25-s − 0.192·27-s − 1.11·29-s − 0.718·31-s − 0.696·33-s + 0.986·37-s + 0.320·39-s + 1.56·41-s − 0.609·43-s + 0.149·45-s + 1.75·47-s − 49-s − 0.280·51-s − 0.824·53-s + 0.539·55-s + 0.132·57-s − 1.56·59-s + 0.256·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 18240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 18240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 19 | \( 1 + T \) |
| good | 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−16.21630317879920, −15.56733433112491, −14.78238738038503, −14.48376114613374, −13.98583976464254, −13.22307764353756, −12.65980409831002, −12.23359876233312, −11.60954711490558, −11.12451737019382, −10.52104055371817, −9.837798398643502, −9.347754813642484, −8.978584983184948, −7.936449176787951, −7.484242359555010, −6.800527079617705, −6.035277241618493, −5.838949810580975, −4.955757571760503, −4.245081412824579, −3.714945252623037, −2.718764893085116, −1.850913936294586, −1.161491410029140, 0,
1.161491410029140, 1.850913936294586, 2.718764893085116, 3.714945252623037, 4.245081412824579, 4.955757571760503, 5.838949810580975, 6.035277241618493, 6.800527079617705, 7.484242359555010, 7.936449176787951, 8.978584983184948, 9.347754813642484, 9.837798398643502, 10.52104055371817, 11.12451737019382, 11.60954711490558, 12.23359876233312, 12.65980409831002, 13.22307764353756, 13.98583976464254, 14.48376114613374, 14.78238738038503, 15.56733433112491, 16.21630317879920
