L(s) = 1 | − 0.242·2-s − 1.94·4-s + 2.79·5-s − 7-s + 0.955·8-s − 0.678·10-s + 4.59·11-s − 4.92·13-s + 0.242·14-s + 3.65·16-s − 2.21·17-s + 0.299·19-s − 5.42·20-s − 1.11·22-s − 2.16·23-s + 2.82·25-s + 1.19·26-s + 1.94·28-s − 5.57·29-s + 6.00·31-s − 2.79·32-s + 0.536·34-s − 2.79·35-s + 8.81·37-s − 0.0726·38-s + 2.67·40-s − 10.5·41-s + ⋯ |
L(s) = 1 | − 0.171·2-s − 0.970·4-s + 1.25·5-s − 0.377·7-s + 0.337·8-s − 0.214·10-s + 1.38·11-s − 1.36·13-s + 0.0648·14-s + 0.912·16-s − 0.536·17-s + 0.0687·19-s − 1.21·20-s − 0.237·22-s − 0.452·23-s + 0.564·25-s + 0.234·26-s + 0.366·28-s − 1.03·29-s + 1.07·31-s − 0.494·32-s + 0.0920·34-s − 0.472·35-s + 1.44·37-s − 0.0117·38-s + 0.422·40-s − 1.64·41-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
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 + T \) |
| 127 | \( 1 + T \) |
good | 2 | \( 1 + 0.242T + 2T^{2} \) |
| 5 | \( 1 - 2.79T + 5T^{2} \) |
| 11 | \( 1 - 4.59T + 11T^{2} \) |
| 13 | \( 1 + 4.92T + 13T^{2} \) |
| 17 | \( 1 + 2.21T + 17T^{2} \) |
| 19 | \( 1 - 0.299T + 19T^{2} \) |
| 23 | \( 1 + 2.16T + 23T^{2} \) |
| 29 | \( 1 + 5.57T + 29T^{2} \) |
| 31 | \( 1 - 6.00T + 31T^{2} \) |
| 37 | \( 1 - 8.81T + 37T^{2} \) |
| 41 | \( 1 + 10.5T + 41T^{2} \) |
| 43 | \( 1 + 6.69T + 43T^{2} \) |
| 47 | \( 1 - 7.58T + 47T^{2} \) |
| 53 | \( 1 + 1.04T + 53T^{2} \) |
| 59 | \( 1 + 5.65T + 59T^{2} \) |
| 61 | \( 1 - 0.161T + 61T^{2} \) |
| 67 | \( 1 - 2.60T + 67T^{2} \) |
| 71 | \( 1 + 2.49T + 71T^{2} \) |
| 73 | \( 1 + 5.94T + 73T^{2} \) |
| 79 | \( 1 + 13.1T + 79T^{2} \) |
| 83 | \( 1 + 9.40T + 83T^{2} \) |
| 89 | \( 1 - 9.33T + 89T^{2} \) |
| 97 | \( 1 + 11.8T + 97T^{2} \) |
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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
−7.45841985777973452252176519845, −6.72119657941708862332902582751, −6.07892488386335115460581879221, −5.43950626925740967005999908772, −4.63469782870010935962206301964, −4.07360946456560379465743366615, −3.06643282773240484468104034297, −2.10321870705675055431920552204, −1.27927944899865137883141531782, 0,
1.27927944899865137883141531782, 2.10321870705675055431920552204, 3.06643282773240484468104034297, 4.07360946456560379465743366615, 4.63469782870010935962206301964, 5.43950626925740967005999908772, 6.07892488386335115460581879221, 6.72119657941708862332902582751, 7.45841985777973452252176519845