| L(s) = 1 | + 0.732·3-s − 5-s + 2.43·7-s − 2.46·9-s + 4.37·11-s + 13-s − 0.732·15-s − 4.66·17-s − 3.49·19-s + 1.78·21-s + 5.93·23-s + 25-s − 4·27-s − 7.86·29-s + 8.63·31-s + 3.20·33-s − 2.43·35-s + 6.07·37-s + 0.732·39-s + 9.54·41-s + 1.61·43-s + 2.46·45-s + 8.48·47-s − 1.04·49-s − 3.41·51-s − 4.66·53-s − 4.37·55-s + ⋯ |
| L(s) = 1 | + 0.422·3-s − 0.447·5-s + 0.921·7-s − 0.821·9-s + 1.31·11-s + 0.277·13-s − 0.189·15-s − 1.13·17-s − 0.801·19-s + 0.389·21-s + 1.23·23-s + 0.200·25-s − 0.769·27-s − 1.46·29-s + 1.55·31-s + 0.557·33-s − 0.412·35-s + 0.999·37-s + 0.117·39-s + 1.49·41-s + 0.245·43-s + 0.367·45-s + 1.23·47-s − 0.149·49-s − 0.478·51-s − 0.640·53-s − 0.589·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4160 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.271072547\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.271072547\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + T \) |
| 13 | \( 1 - T \) |
| good | 3 | \( 1 - 0.732T + 3T^{2} \) |
| 7 | \( 1 - 2.43T + 7T^{2} \) |
| 11 | \( 1 - 4.37T + 11T^{2} \) |
| 17 | \( 1 + 4.66T + 17T^{2} \) |
| 19 | \( 1 + 3.49T + 19T^{2} \) |
| 23 | \( 1 - 5.93T + 23T^{2} \) |
| 29 | \( 1 + 7.86T + 29T^{2} \) |
| 31 | \( 1 - 8.63T + 31T^{2} \) |
| 37 | \( 1 - 6.07T + 37T^{2} \) |
| 41 | \( 1 - 9.54T + 41T^{2} \) |
| 43 | \( 1 - 1.61T + 43T^{2} \) |
| 47 | \( 1 - 8.48T + 47T^{2} \) |
| 53 | \( 1 + 4.66T + 53T^{2} \) |
| 59 | \( 1 - 3.49T + 59T^{2} \) |
| 61 | \( 1 - 10.3T + 61T^{2} \) |
| 67 | \( 1 + 4.97T + 67T^{2} \) |
| 71 | \( 1 - 11.6T + 71T^{2} \) |
| 73 | \( 1 - 0.849T + 73T^{2} \) |
| 79 | \( 1 + 0.936T + 79T^{2} \) |
| 83 | \( 1 + 1.56T + 83T^{2} \) |
| 89 | \( 1 + 4.40T + 89T^{2} \) |
| 97 | \( 1 - 1.12T + 97T^{2} \) |
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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
−8.424531450373029041025012227380, −7.87387199172803283243092249552, −6.96007816140640083951268987635, −6.30174167277228664350897511483, −5.44034360387503689704672559129, −4.38166942394250923368283073272, −4.02257211848626614291833943122, −2.88173198544043189642113308803, −2.03514204322905284204635246290, −0.857625817343156607566275756888,
0.857625817343156607566275756888, 2.03514204322905284204635246290, 2.88173198544043189642113308803, 4.02257211848626614291833943122, 4.38166942394250923368283073272, 5.44034360387503689704672559129, 6.30174167277228664350897511483, 6.96007816140640083951268987635, 7.87387199172803283243092249552, 8.424531450373029041025012227380
