| L(s) = 1 | − 0.942·2-s − 1.11·4-s − 2.02·5-s − 7-s + 2.93·8-s + 1.91·10-s − 0.393·11-s − 1.52·13-s + 0.942·14-s − 0.541·16-s + 7.22·17-s + 5.33·19-s + 2.25·20-s + 0.371·22-s − 6.14·23-s − 0.889·25-s + 1.43·26-s + 1.11·28-s − 10.3·29-s − 5.51·31-s − 5.35·32-s − 6.81·34-s + 2.02·35-s + 1.11·37-s − 5.02·38-s − 5.94·40-s + 12.0·41-s + ⋯ |
| L(s) = 1 | − 0.666·2-s − 0.555·4-s − 0.906·5-s − 0.377·7-s + 1.03·8-s + 0.604·10-s − 0.118·11-s − 0.422·13-s + 0.251·14-s − 0.135·16-s + 1.75·17-s + 1.22·19-s + 0.503·20-s + 0.0791·22-s − 1.28·23-s − 0.177·25-s + 0.281·26-s + 0.210·28-s − 1.92·29-s − 0.989·31-s − 0.946·32-s − 1.16·34-s + 0.342·35-s + 0.183·37-s − 0.815·38-s − 0.940·40-s + 1.87·41-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 | \( 1 + 0.942T + 2T^{2} \) |
| 5 | \( 1 + 2.02T + 5T^{2} \) |
| 11 | \( 1 + 0.393T + 11T^{2} \) |
| 13 | \( 1 + 1.52T + 13T^{2} \) |
| 17 | \( 1 - 7.22T + 17T^{2} \) |
| 19 | \( 1 - 5.33T + 19T^{2} \) |
| 23 | \( 1 + 6.14T + 23T^{2} \) |
| 29 | \( 1 + 10.3T + 29T^{2} \) |
| 31 | \( 1 + 5.51T + 31T^{2} \) |
| 37 | \( 1 - 1.11T + 37T^{2} \) |
| 41 | \( 1 - 12.0T + 41T^{2} \) |
| 43 | \( 1 + 0.215T + 43T^{2} \) |
| 47 | \( 1 - 4.05T + 47T^{2} \) |
| 53 | \( 1 + 6.05T + 53T^{2} \) |
| 59 | \( 1 + 3.92T + 59T^{2} \) |
| 61 | \( 1 + 3.48T + 61T^{2} \) |
| 67 | \( 1 - 9.31T + 67T^{2} \) |
| 71 | \( 1 - 6.78T + 71T^{2} \) |
| 73 | \( 1 + 11.2T + 73T^{2} \) |
| 79 | \( 1 + 5.16T + 79T^{2} \) |
| 83 | \( 1 - 6.24T + 83T^{2} \) |
| 89 | \( 1 - 13.9T + 89T^{2} \) |
| 97 | \( 1 - 10.5T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.69550738629568877945902799957, −7.34727960172734246976701947838, −5.93818601525968416137659031722, −5.49243617745770121548720204579, −4.59745136463506928446947157663, −3.71798142349099162615492223861, −3.41202575014836422948288930491, −2.04112130921024492776858718847, −0.933043255089307877706993884493, 0,
0.933043255089307877706993884493, 2.04112130921024492776858718847, 3.41202575014836422948288930491, 3.71798142349099162615492223861, 4.59745136463506928446947157663, 5.49243617745770121548720204579, 5.93818601525968416137659031722, 7.34727960172734246976701947838, 7.69550738629568877945902799957
