| L(s) = 1 | + 2·7-s + 11-s + 2·13-s − 2·19-s + 23-s − 5·25-s + 10·29-s − 4·31-s + 2·37-s + 2·41-s − 2·43-s − 8·47-s − 3·49-s + 4·53-s − 12·59-s − 6·61-s − 2·67-s − 6·73-s + 2·77-s − 2·79-s + 4·83-s + 8·89-s + 4·91-s − 2·97-s + 101-s + 103-s + 107-s + ⋯ |
| L(s) = 1 | + 0.755·7-s + 0.301·11-s + 0.554·13-s − 0.458·19-s + 0.208·23-s − 25-s + 1.85·29-s − 0.718·31-s + 0.328·37-s + 0.312·41-s − 0.304·43-s − 1.16·47-s − 3/7·49-s + 0.549·53-s − 1.56·59-s − 0.768·61-s − 0.244·67-s − 0.702·73-s + 0.227·77-s − 0.225·79-s + 0.439·83-s + 0.847·89-s + 0.419·91-s − 0.203·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 36432 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 36432 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 11 | \( 1 - T \) |
| 23 | \( 1 - T \) |
| good | 5 | \( 1 + p T^{2} \) |
| 7 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 - 10 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - 4 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 + 2 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 - 8 T + p T^{2} \) |
| 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
−15.02269530279163, −14.69932833641920, −14.11757270871205, −13.59858916441521, −13.21460679063503, −12.37851551044664, −12.09859373183060, −11.32977285160052, −11.11185706029479, −10.38509257919223, −9.947705728039647, −9.178586436086536, −8.754686442720321, −8.096731314194565, −7.770075203766857, −7.003012824189060, −6.307365410695691, −5.987940319393547, −5.071602096192863, −4.640408890596989, −3.996493570751001, −3.303686991832380, −2.549173550531379, −1.706208051304225, −1.165326244160833, 0,
1.165326244160833, 1.706208051304225, 2.549173550531379, 3.303686991832380, 3.996493570751001, 4.640408890596989, 5.071602096192863, 5.987940319393547, 6.307365410695691, 7.003012824189060, 7.770075203766857, 8.096731314194565, 8.754686442720321, 9.178586436086536, 9.947705728039647, 10.38509257919223, 11.11185706029479, 11.32977285160052, 12.09859373183060, 12.37851551044664, 13.21460679063503, 13.59858916441521, 14.11757270871205, 14.69932833641920, 15.02269530279163
