| L(s) = 1 | + 5-s − 2·7-s − 4·11-s + 4·13-s − 2·17-s + 8·19-s + 8·23-s + 25-s + 4·29-s − 31-s − 2·35-s + 12·37-s − 10·41-s − 8·43-s + 4·47-s − 3·49-s + 6·53-s − 4·55-s + 2·59-s − 10·61-s + 4·65-s + 6·67-s − 6·71-s − 4·73-s + 8·77-s − 8·79-s + 4·83-s + ⋯ |
| L(s) = 1 | + 0.447·5-s − 0.755·7-s − 1.20·11-s + 1.10·13-s − 0.485·17-s + 1.83·19-s + 1.66·23-s + 1/5·25-s + 0.742·29-s − 0.179·31-s − 0.338·35-s + 1.97·37-s − 1.56·41-s − 1.21·43-s + 0.583·47-s − 3/7·49-s + 0.824·53-s − 0.539·55-s + 0.260·59-s − 1.28·61-s + 0.496·65-s + 0.733·67-s − 0.712·71-s − 0.468·73-s + 0.911·77-s − 0.900·79-s + 0.439·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 89280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 89280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 31 | \( 1 + T \) |
| good | 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - 8 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 12 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 2 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 6 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 + 18 T + p T^{2} \) |
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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
−13.81775933646144, −13.54367155216190, −13.23259310766290, −12.87455776526396, −12.17421571542734, −11.60922343682211, −11.10780510701666, −10.66281710453933, −10.12827720057629, −9.600483245103481, −9.279759534087795, −8.553332930916519, −8.198223508598944, −7.486983844608935, −6.945171533618822, −6.528925745156921, −5.866779174158678, −5.353420951276547, −4.967024420810173, −4.213376981519491, −3.389144886862094, −2.938887311155318, −2.600141953463850, −1.467776894434546, −0.9911405212153330, 0,
0.9911405212153330, 1.467776894434546, 2.600141953463850, 2.938887311155318, 3.389144886862094, 4.213376981519491, 4.967024420810173, 5.353420951276547, 5.866779174158678, 6.528925745156921, 6.945171533618822, 7.486983844608935, 8.198223508598944, 8.553332930916519, 9.279759534087795, 9.600483245103481, 10.12827720057629, 10.66281710453933, 11.10780510701666, 11.60922343682211, 12.17421571542734, 12.87455776526396, 13.23259310766290, 13.54367155216190, 13.81775933646144
