| L(s) = 1 | − 0.735·2-s − 1.45·4-s − 0.430·5-s − 7-s + 2.54·8-s + 0.316·10-s − 1.57·11-s + 6.79·13-s + 0.735·14-s + 1.04·16-s + 0.272·17-s − 1.03·19-s + 0.628·20-s + 1.16·22-s + 2.36·23-s − 4.81·25-s − 4.99·26-s + 1.45·28-s − 3.64·29-s − 4.13·31-s − 5.85·32-s − 0.200·34-s + 0.430·35-s + 1.67·37-s + 0.763·38-s − 1.09·40-s + 3.63·41-s + ⋯ |
| L(s) = 1 | − 0.520·2-s − 0.729·4-s − 0.192·5-s − 0.377·7-s + 0.899·8-s + 0.100·10-s − 0.476·11-s + 1.88·13-s + 0.196·14-s + 0.261·16-s + 0.0660·17-s − 0.238·19-s + 0.140·20-s + 0.247·22-s + 0.492·23-s − 0.962·25-s − 0.980·26-s + 0.275·28-s − 0.676·29-s − 0.743·31-s − 1.03·32-s − 0.0343·34-s + 0.0727·35-s + 0.274·37-s + 0.123·38-s − 0.173·40-s + 0.567·41-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 | \( 1 + 0.735T + 2T^{2} \) |
| 5 | \( 1 + 0.430T + 5T^{2} \) |
| 11 | \( 1 + 1.57T + 11T^{2} \) |
| 13 | \( 1 - 6.79T + 13T^{2} \) |
| 17 | \( 1 - 0.272T + 17T^{2} \) |
| 19 | \( 1 + 1.03T + 19T^{2} \) |
| 23 | \( 1 - 2.36T + 23T^{2} \) |
| 29 | \( 1 + 3.64T + 29T^{2} \) |
| 31 | \( 1 + 4.13T + 31T^{2} \) |
| 37 | \( 1 - 1.67T + 37T^{2} \) |
| 41 | \( 1 - 3.63T + 41T^{2} \) |
| 43 | \( 1 + 4.66T + 43T^{2} \) |
| 47 | \( 1 + 10.9T + 47T^{2} \) |
| 53 | \( 1 - 2.62T + 53T^{2} \) |
| 59 | \( 1 - 7.52T + 59T^{2} \) |
| 61 | \( 1 + 15.2T + 61T^{2} \) |
| 67 | \( 1 - 13.9T + 67T^{2} \) |
| 71 | \( 1 - 1.92T + 71T^{2} \) |
| 73 | \( 1 + 1.72T + 73T^{2} \) |
| 79 | \( 1 - 0.416T + 79T^{2} \) |
| 83 | \( 1 - 10.0T + 83T^{2} \) |
| 89 | \( 1 - 10.7T + 89T^{2} \) |
| 97 | \( 1 + 5.98T + 97T^{2} \) |
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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
−7.76507527259538840648732090459, −6.87176556834102946516633322532, −6.07761055131799641994744514989, −5.46114192763490408174244260283, −4.62477929734924652225649224109, −3.74949363067402543539763373519, −3.38907271960473314387611183322, −2.00600429797379613045303747232, −1.07343164570583139769589551979, 0,
1.07343164570583139769589551979, 2.00600429797379613045303747232, 3.38907271960473314387611183322, 3.74949363067402543539763373519, 4.62477929734924652225649224109, 5.46114192763490408174244260283, 6.07761055131799641994744514989, 6.87176556834102946516633322532, 7.76507527259538840648732090459
