| L(s) = 1 | − 1.20·2-s − 0.555·4-s + 0.607·5-s + 7-s + 3.07·8-s − 0.730·10-s + 6.41·11-s − 3.74·13-s − 1.20·14-s − 2.58·16-s − 4.87·17-s − 7.90·19-s − 0.337·20-s − 7.71·22-s + 0.948·23-s − 4.63·25-s + 4.50·26-s − 0.555·28-s + 3.93·29-s − 0.0625·31-s − 3.04·32-s + 5.86·34-s + 0.607·35-s − 1.32·37-s + 9.50·38-s + 1.86·40-s − 9.12·41-s + ⋯ |
| L(s) = 1 | − 0.849·2-s − 0.277·4-s + 0.271·5-s + 0.377·7-s + 1.08·8-s − 0.231·10-s + 1.93·11-s − 1.03·13-s − 0.321·14-s − 0.645·16-s − 1.18·17-s − 1.81·19-s − 0.0754·20-s − 1.64·22-s + 0.197·23-s − 0.926·25-s + 0.883·26-s − 0.104·28-s + 0.731·29-s − 0.0112·31-s − 0.537·32-s + 1.00·34-s + 0.102·35-s − 0.217·37-s + 1.54·38-s + 0.295·40-s − 1.42·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8001 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8001 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.004613818\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.004613818\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 - T \) |
| 127 | \( 1 + T \) |
| good | 2 | \( 1 + 1.20T + 2T^{2} \) |
| 5 | \( 1 - 0.607T + 5T^{2} \) |
| 11 | \( 1 - 6.41T + 11T^{2} \) |
| 13 | \( 1 + 3.74T + 13T^{2} \) |
| 17 | \( 1 + 4.87T + 17T^{2} \) |
| 19 | \( 1 + 7.90T + 19T^{2} \) |
| 23 | \( 1 - 0.948T + 23T^{2} \) |
| 29 | \( 1 - 3.93T + 29T^{2} \) |
| 31 | \( 1 + 0.0625T + 31T^{2} \) |
| 37 | \( 1 + 1.32T + 37T^{2} \) |
| 41 | \( 1 + 9.12T + 41T^{2} \) |
| 43 | \( 1 - 9.44T + 43T^{2} \) |
| 47 | \( 1 - 3.29T + 47T^{2} \) |
| 53 | \( 1 - 2.65T + 53T^{2} \) |
| 59 | \( 1 + 12.2T + 59T^{2} \) |
| 61 | \( 1 - 5.05T + 61T^{2} \) |
| 67 | \( 1 + 2.97T + 67T^{2} \) |
| 71 | \( 1 + 4.14T + 71T^{2} \) |
| 73 | \( 1 - 4.25T + 73T^{2} \) |
| 79 | \( 1 - 3.94T + 79T^{2} \) |
| 83 | \( 1 - 8.92T + 83T^{2} \) |
| 89 | \( 1 - 4.00T + 89T^{2} \) |
| 97 | \( 1 - 19.0T + 97T^{2} \) |
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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
−7.991373731375048100497638443191, −7.17459236812521734988225564732, −6.60959466054584550959563761361, −5.96850223096142512086408617745, −4.72410747367316435908527334789, −4.44640094129912710031841522578, −3.68552928063692624619569057990, −2.21818461507758348635744181045, −1.73922744386168369017639262444, −0.57369394848680706780429421423,
0.57369394848680706780429421423, 1.73922744386168369017639262444, 2.21818461507758348635744181045, 3.68552928063692624619569057990, 4.44640094129912710031841522578, 4.72410747367316435908527334789, 5.96850223096142512086408617745, 6.60959466054584550959563761361, 7.17459236812521734988225564732, 7.991373731375048100497638443191
