| L(s) = 1 | − 3-s + 4·7-s + 9-s + 4·11-s + 4·13-s − 4·17-s − 4·19-s − 4·21-s − 8·23-s − 27-s − 6·29-s − 4·31-s − 4·33-s + 2·37-s − 4·39-s + 10·41-s − 43-s − 4·47-s + 9·49-s + 4·51-s − 2·53-s + 4·57-s + 12·59-s + 4·63-s + 8·67-s + 8·69-s − 12·71-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1.51·7-s + 1/3·9-s + 1.20·11-s + 1.10·13-s − 0.970·17-s − 0.917·19-s − 0.872·21-s − 1.66·23-s − 0.192·27-s − 1.11·29-s − 0.718·31-s − 0.696·33-s + 0.328·37-s − 0.640·39-s + 1.56·41-s − 0.152·43-s − 0.583·47-s + 9/7·49-s + 0.560·51-s − 0.274·53-s + 0.529·57-s + 1.56·59-s + 0.503·63-s + 0.977·67-s + 0.963·69-s − 1.42·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 206400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 206400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 5 | \( 1 \) |
| 43 | \( 1 + T \) |
| good | 7 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 + 4 T + p T^{2} \) |
| 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
−13.17600521381532, −12.81031841349554, −12.24866786966359, −11.63614268498509, −11.41974574427019, −11.01217059748818, −10.72825603388559, −10.06494337022069, −9.408966403964225, −8.988741525541665, −8.558774606162002, −7.964394983334453, −7.720583670537659, −6.929050169746359, −6.385403294811194, −6.164549778130678, −5.466918820416328, −5.058957062279092, −4.267342473411551, −3.971250064214549, −3.770313504381079, −2.485901677320166, −1.945391693227217, −1.558202247001020, −0.8994490012426741, 0,
0.8994490012426741, 1.558202247001020, 1.945391693227217, 2.485901677320166, 3.770313504381079, 3.971250064214549, 4.267342473411551, 5.058957062279092, 5.466918820416328, 6.164549778130678, 6.385403294811194, 6.929050169746359, 7.720583670537659, 7.964394983334453, 8.558774606162002, 8.988741525541665, 9.408966403964225, 10.06494337022069, 10.72825603388559, 11.01217059748818, 11.41974574427019, 11.63614268498509, 12.24866786966359, 12.81031841349554, 13.17600521381532
