| L(s) = 1 | + 3-s + 5-s + 4·7-s − 2·9-s + 11-s − 2·13-s + 15-s − 2·19-s + 4·21-s + 9·23-s − 4·25-s − 5·27-s + 4·29-s + 5·31-s + 33-s + 4·35-s − 9·37-s − 2·39-s + 2·41-s − 6·43-s − 2·45-s − 4·47-s + 9·49-s − 6·53-s + 55-s − 2·57-s − 5·59-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.447·5-s + 1.51·7-s − 2/3·9-s + 0.301·11-s − 0.554·13-s + 0.258·15-s − 0.458·19-s + 0.872·21-s + 1.87·23-s − 4/5·25-s − 0.962·27-s + 0.742·29-s + 0.898·31-s + 0.174·33-s + 0.676·35-s − 1.47·37-s − 0.320·39-s + 0.312·41-s − 0.914·43-s − 0.298·45-s − 0.583·47-s + 9/7·49-s − 0.824·53-s + 0.134·55-s − 0.264·57-s − 0.650·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 352 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.839422549\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.839422549\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 11 | \( 1 - T \) |
| good | 3 | \( 1 - T + p T^{2} \) |
| 5 | \( 1 - T + p T^{2} \) |
| 7 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 - 9 T + p T^{2} \) |
| 29 | \( 1 - 4 T + p T^{2} \) |
| 31 | \( 1 - 5 T + p T^{2} \) |
| 37 | \( 1 + 9 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + 6 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 5 T + p T^{2} \) |
| 61 | \( 1 + p T^{2} \) |
| 67 | \( 1 + 13 T + p T^{2} \) |
| 71 | \( 1 + T + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 - 14 T + p T^{2} \) |
| 89 | \( 1 + 13 T + p T^{2} \) |
| 97 | \( 1 + 19 T + p T^{2} \) |
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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
−11.43031674585287936480172767837, −10.65536531680425134672181778028, −9.477460186491048050604925394554, −8.595119251819881391518110548197, −7.956649639760417596283329475573, −6.77843662469483305850122200054, −5.43215102206894998656777956795, −4.57725485407179341808498131930, −2.97331376469435176368062512619, −1.71677882912616726095004967526,
1.71677882912616726095004967526, 2.97331376469435176368062512619, 4.57725485407179341808498131930, 5.43215102206894998656777956795, 6.77843662469483305850122200054, 7.956649639760417596283329475573, 8.595119251819881391518110548197, 9.477460186491048050604925394554, 10.65536531680425134672181778028, 11.43031674585287936480172767837
