| L(s) = 1 | + 3-s − 5-s − 7-s + 9-s + 4·11-s − 2·13-s − 15-s + 4·19-s − 21-s + 23-s + 25-s + 27-s + 8·29-s + 8·31-s + 4·33-s + 35-s + 6·37-s − 2·39-s − 8·41-s + 8·43-s − 45-s + 49-s − 12·53-s − 4·55-s + 4·57-s + 4·59-s + 10·61-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.447·5-s − 0.377·7-s + 1/3·9-s + 1.20·11-s − 0.554·13-s − 0.258·15-s + 0.917·19-s − 0.218·21-s + 0.208·23-s + 1/5·25-s + 0.192·27-s + 1.48·29-s + 1.43·31-s + 0.696·33-s + 0.169·35-s + 0.986·37-s − 0.320·39-s − 1.24·41-s + 1.21·43-s − 0.149·45-s + 1/7·49-s − 1.64·53-s − 0.539·55-s + 0.529·57-s + 0.520·59-s + 1.28·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 38640 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38640 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.227101153\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.227101153\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 23 | \( 1 - T \) |
| good | 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - 8 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 + 8 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 12 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 - 16 T + p T^{2} \) |
| 79 | \( 1 + 16 T + p T^{2} \) |
| 83 | \( 1 - 16 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 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
−14.72317179834207, −14.35908375924915, −13.82806908760277, −13.41870176903359, −12.64315666609118, −12.21249974557825, −11.76521007628641, −11.30668294870040, −10.52011195412209, −9.888986768617603, −9.564551783322336, −9.012871074986956, −8.326630020043872, −7.984885621591833, −7.218050635431973, −6.740462006193570, −6.298521459738016, −5.463097131917443, −4.614941852242101, −4.315501458524931, −3.462953309426889, −3.005520356355242, −2.342996397454182, −1.303094439518239, −0.7051620552050128,
0.7051620552050128, 1.303094439518239, 2.342996397454182, 3.005520356355242, 3.462953309426889, 4.315501458524931, 4.614941852242101, 5.463097131917443, 6.298521459738016, 6.740462006193570, 7.218050635431973, 7.984885621591833, 8.326630020043872, 9.012871074986956, 9.564551783322336, 9.888986768617603, 10.52011195412209, 11.30668294870040, 11.76521007628641, 12.21249974557825, 12.64315666609118, 13.41870176903359, 13.82806908760277, 14.35908375924915, 14.72317179834207
