| L(s) = 1 | + 2-s − 4-s − 4·5-s − 7-s − 3·8-s − 4·10-s + 2·13-s − 14-s − 16-s − 6·17-s + 2·19-s + 4·20-s + 11·25-s + 2·26-s + 28-s + 10·29-s + 6·31-s + 5·32-s − 6·34-s + 4·35-s + 10·37-s + 2·38-s + 12·40-s − 10·41-s − 4·43-s + 6·47-s + 49-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1/2·4-s − 1.78·5-s − 0.377·7-s − 1.06·8-s − 1.26·10-s + 0.554·13-s − 0.267·14-s − 1/4·16-s − 1.45·17-s + 0.458·19-s + 0.894·20-s + 11/5·25-s + 0.392·26-s + 0.188·28-s + 1.85·29-s + 1.07·31-s + 0.883·32-s − 1.02·34-s + 0.676·35-s + 1.64·37-s + 0.324·38-s + 1.89·40-s − 1.56·41-s − 0.609·43-s + 0.875·47-s + 1/7·49-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 | \( 1 - T + p T^{2} \) |
| 5 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 10 T + p T^{2} \) |
| 31 | \( 1 - 6 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 + 12 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 - 8 T + p T^{2} \) |
| 97 | \( 1 + 4 T + p T^{2} \) |
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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
−7.51607636876563303393019456353, −6.61640193413984728017469398231, −6.25018209417494606018337637666, −5.07849319527270821904284624409, −4.43323680106416345156296795500, −4.13764045774431626902829747774, −3.23503801963960853838561576057, −2.75520987414909956653954415494, −0.945663678476916805004674382623, 0,
0.945663678476916805004674382623, 2.75520987414909956653954415494, 3.23503801963960853838561576057, 4.13764045774431626902829747774, 4.43323680106416345156296795500, 5.07849319527270821904284624409, 6.25018209417494606018337637666, 6.61640193413984728017469398231, 7.51607636876563303393019456353
