| L(s) = 1 | − 1.40·3-s + 3.56·5-s − 7-s − 1.02·9-s − 11-s + 13-s − 5.01·15-s + 6.72·17-s − 7.75·19-s + 1.40·21-s − 4.37·23-s + 7.72·25-s + 5.65·27-s + 5.83·29-s + 8.23·31-s + 1.40·33-s − 3.56·35-s − 11.3·37-s − 1.40·39-s + 2.41·41-s − 3.42·43-s − 3.65·45-s + 11.3·47-s + 49-s − 9.45·51-s − 5.39·53-s − 3.56·55-s + ⋯ |
| L(s) = 1 | − 0.811·3-s + 1.59·5-s − 0.377·7-s − 0.341·9-s − 0.301·11-s + 0.277·13-s − 1.29·15-s + 1.63·17-s − 1.77·19-s + 0.306·21-s − 0.912·23-s + 1.54·25-s + 1.08·27-s + 1.08·29-s + 1.47·31-s + 0.244·33-s − 0.602·35-s − 1.87·37-s − 0.224·39-s + 0.377·41-s − 0.521·43-s − 0.545·45-s + 1.64·47-s + 0.142·49-s − 1.32·51-s − 0.740·53-s − 0.480·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.759122267\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.759122267\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| 13 | \( 1 - T \) |
| good | 3 | \( 1 + 1.40T + 3T^{2} \) |
| 5 | \( 1 - 3.56T + 5T^{2} \) |
| 17 | \( 1 - 6.72T + 17T^{2} \) |
| 19 | \( 1 + 7.75T + 19T^{2} \) |
| 23 | \( 1 + 4.37T + 23T^{2} \) |
| 29 | \( 1 - 5.83T + 29T^{2} \) |
| 31 | \( 1 - 8.23T + 31T^{2} \) |
| 37 | \( 1 + 11.3T + 37T^{2} \) |
| 41 | \( 1 - 2.41T + 41T^{2} \) |
| 43 | \( 1 + 3.42T + 43T^{2} \) |
| 47 | \( 1 - 11.3T + 47T^{2} \) |
| 53 | \( 1 + 5.39T + 53T^{2} \) |
| 59 | \( 1 + 7.81T + 59T^{2} \) |
| 61 | \( 1 - 0.208T + 61T^{2} \) |
| 67 | \( 1 + 7.95T + 67T^{2} \) |
| 71 | \( 1 + 7.39T + 71T^{2} \) |
| 73 | \( 1 - 13.7T + 73T^{2} \) |
| 79 | \( 1 - 13.8T + 79T^{2} \) |
| 83 | \( 1 + 0.786T + 83T^{2} \) |
| 89 | \( 1 + 4.96T + 89T^{2} \) |
| 97 | \( 1 - 9.57T + 97T^{2} \) |
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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
−7.899959542757719525450462280665, −6.78519792273778270723328859208, −6.14591499104522067566822713210, −5.99775778366139011346882586464, −5.21671925163590351931652516950, −4.56686540017723249946100711538, −3.37622579966656992925326068329, −2.57491864740782565399512809724, −1.75748713674346219483067069202, −0.68409684466504180866080661694,
0.68409684466504180866080661694, 1.75748713674346219483067069202, 2.57491864740782565399512809724, 3.37622579966656992925326068329, 4.56686540017723249946100711538, 5.21671925163590351931652516950, 5.99775778366139011346882586464, 6.14591499104522067566822713210, 6.78519792273778270723328859208, 7.899959542757719525450462280665
