L(s) = 1 | − 3-s − 5-s + 7-s + 9-s + 2·11-s + 2·13-s + 15-s − 21-s − 4·23-s − 4·25-s − 27-s − 6·29-s + 5·31-s − 2·33-s − 35-s − 37-s − 2·39-s + 2·41-s − 45-s + 47-s − 6·49-s + 9·53-s − 2·55-s + 5·59-s − 12·61-s + 63-s − 2·65-s + ⋯ |
L(s) = 1 | − 0.577·3-s − 0.447·5-s + 0.377·7-s + 1/3·9-s + 0.603·11-s + 0.554·13-s + 0.258·15-s − 0.218·21-s − 0.834·23-s − 4/5·25-s − 0.192·27-s − 1.11·29-s + 0.898·31-s − 0.348·33-s − 0.169·35-s − 0.164·37-s − 0.320·39-s + 0.312·41-s − 0.149·45-s + 0.145·47-s − 6/7·49-s + 1.23·53-s − 0.269·55-s + 0.650·59-s − 1.53·61-s + 0.125·63-s − 0.248·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8016 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8016 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 167 | \( 1 + T \) |
good | 5 | \( 1 + T + p T^{2} \) |
| 7 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 5 T + p T^{2} \) |
| 37 | \( 1 + T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 - T + p T^{2} \) |
| 53 | \( 1 - 9 T + p T^{2} \) |
| 59 | \( 1 - 5 T + p T^{2} \) |
| 61 | \( 1 + 12 T + p T^{2} \) |
| 67 | \( 1 + 5 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 + 12 T + p T^{2} \) |
| 83 | \( 1 - 7 T + p T^{2} \) |
| 89 | \( 1 - 13 T + p T^{2} \) |
| 97 | \( 1 + 5 T + p T^{2} \) |
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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
−7.52240700993340936687374214812, −6.76810686303031874426001461319, −6.01575981240570538392680899419, −5.55996654956437593977801337256, −4.51124807618467947271451063990, −4.07141757232704287475485090768, −3.26688999561646248839658052939, −2.05164129044917110544199028408, −1.21337405114246561900628972517, 0,
1.21337405114246561900628972517, 2.05164129044917110544199028408, 3.26688999561646248839658052939, 4.07141757232704287475485090768, 4.51124807618467947271451063990, 5.55996654956437593977801337256, 6.01575981240570538392680899419, 6.76810686303031874426001461319, 7.52240700993340936687374214812