| L(s) = 1 | − 5-s + 7-s + 5.65·11-s + 2·13-s − 7.65·17-s + 5.65·19-s − 5.65·23-s + 25-s + 7.65·29-s + 4·31-s − 35-s + 0.343·37-s − 2·41-s − 9.65·43-s + 13.6·47-s + 49-s − 7.65·53-s − 5.65·55-s + 4·59-s + 11.6·61-s − 2·65-s + 1.65·67-s − 15.3·71-s + 6·73-s + 5.65·77-s − 11.3·79-s + 4·83-s + ⋯ |
| L(s) = 1 | − 0.447·5-s + 0.377·7-s + 1.70·11-s + 0.554·13-s − 1.85·17-s + 1.29·19-s − 1.17·23-s + 0.200·25-s + 1.42·29-s + 0.718·31-s − 0.169·35-s + 0.0564·37-s − 0.312·41-s − 1.47·43-s + 1.99·47-s + 0.142·49-s − 1.05·53-s − 0.762·55-s + 0.520·59-s + 1.49·61-s − 0.248·65-s + 0.202·67-s − 1.81·71-s + 0.702·73-s + 0.644·77-s − 1.27·79-s + 0.439·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.931068129\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.931068129\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| good | 11 | \( 1 - 5.65T + 11T^{2} \) |
| 13 | \( 1 - 2T + 13T^{2} \) |
| 17 | \( 1 + 7.65T + 17T^{2} \) |
| 19 | \( 1 - 5.65T + 19T^{2} \) |
| 23 | \( 1 + 5.65T + 23T^{2} \) |
| 29 | \( 1 - 7.65T + 29T^{2} \) |
| 31 | \( 1 - 4T + 31T^{2} \) |
| 37 | \( 1 - 0.343T + 37T^{2} \) |
| 41 | \( 1 + 2T + 41T^{2} \) |
| 43 | \( 1 + 9.65T + 43T^{2} \) |
| 47 | \( 1 - 13.6T + 47T^{2} \) |
| 53 | \( 1 + 7.65T + 53T^{2} \) |
| 59 | \( 1 - 4T + 59T^{2} \) |
| 61 | \( 1 - 11.6T + 61T^{2} \) |
| 67 | \( 1 - 1.65T + 67T^{2} \) |
| 71 | \( 1 + 15.3T + 71T^{2} \) |
| 73 | \( 1 - 6T + 73T^{2} \) |
| 79 | \( 1 + 11.3T + 79T^{2} \) |
| 83 | \( 1 - 4T + 83T^{2} \) |
| 89 | \( 1 - 14T + 89T^{2} \) |
| 97 | \( 1 - 6T + 97T^{2} \) |
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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
−8.719705295066796590010324459653, −8.393499457682629763557493487576, −7.28416789533615624429928783557, −6.62253044099171401687101428330, −5.98738154203801305752689045656, −4.73133848626764174879982771492, −4.17068702885430910471767880845, −3.32111187748106030094283408738, −2.03132498007677724299276864886, −0.924747989442578615768519181329,
0.924747989442578615768519181329, 2.03132498007677724299276864886, 3.32111187748106030094283408738, 4.17068702885430910471767880845, 4.73133848626764174879982771492, 5.98738154203801305752689045656, 6.62253044099171401687101428330, 7.28416789533615624429928783557, 8.393499457682629763557493487576, 8.719705295066796590010324459653
