| L(s) = 1 | − 5-s + 4.47·7-s + 4.47·11-s + 4·13-s − 2·17-s − 8.94·23-s + 25-s + 6·29-s − 8.94·31-s − 4.47·35-s + 8·37-s − 8·41-s + 8.94·47-s + 13.0·49-s + 6·53-s − 4.47·55-s + 4.47·59-s + 10·61-s − 4·65-s − 8.94·67-s + 8.94·71-s + 6·73-s + 20.0·77-s + 8.94·79-s − 8.94·83-s + 2·85-s − 4·89-s + ⋯ |
| L(s) = 1 | − 0.447·5-s + 1.69·7-s + 1.34·11-s + 1.10·13-s − 0.485·17-s − 1.86·23-s + 0.200·25-s + 1.11·29-s − 1.60·31-s − 0.755·35-s + 1.31·37-s − 1.24·41-s + 1.30·47-s + 1.85·49-s + 0.824·53-s − 0.603·55-s + 0.582·59-s + 1.28·61-s − 0.496·65-s − 1.09·67-s + 1.06·71-s + 0.702·73-s + 2.27·77-s + 1.00·79-s − 0.981·83-s + 0.216·85-s − 0.423·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1440 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1440 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.094036490\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.094036490\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 4.47T + 7T^{2} \) |
| 11 | \( 1 - 4.47T + 11T^{2} \) |
| 13 | \( 1 - 4T + 13T^{2} \) |
| 17 | \( 1 + 2T + 17T^{2} \) |
| 19 | \( 1 + 19T^{2} \) |
| 23 | \( 1 + 8.94T + 23T^{2} \) |
| 29 | \( 1 - 6T + 29T^{2} \) |
| 31 | \( 1 + 8.94T + 31T^{2} \) |
| 37 | \( 1 - 8T + 37T^{2} \) |
| 41 | \( 1 + 8T + 41T^{2} \) |
| 43 | \( 1 + 43T^{2} \) |
| 47 | \( 1 - 8.94T + 47T^{2} \) |
| 53 | \( 1 - 6T + 53T^{2} \) |
| 59 | \( 1 - 4.47T + 59T^{2} \) |
| 61 | \( 1 - 10T + 61T^{2} \) |
| 67 | \( 1 + 8.94T + 67T^{2} \) |
| 71 | \( 1 - 8.94T + 71T^{2} \) |
| 73 | \( 1 - 6T + 73T^{2} \) |
| 79 | \( 1 - 8.94T + 79T^{2} \) |
| 83 | \( 1 + 8.94T + 83T^{2} \) |
| 89 | \( 1 + 4T + 89T^{2} \) |
| 97 | \( 1 - 2T + 97T^{2} \) |
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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
−9.373791719654431270600819972011, −8.495905321502352456562376203856, −8.166052791193086010050666688408, −7.16226301259578703183065798768, −6.26337537079386845543709924020, −5.34944635106809400960610371164, −4.20678341357415009135671802046, −3.86396293560103417260816712209, −2.12132454984846043350366901398, −1.16402186534327768941641869551,
1.16402186534327768941641869551, 2.12132454984846043350366901398, 3.86396293560103417260816712209, 4.20678341357415009135671802046, 5.34944635106809400960610371164, 6.26337537079386845543709924020, 7.16226301259578703183065798768, 8.166052791193086010050666688408, 8.495905321502352456562376203856, 9.373791719654431270600819972011
