| L(s) = 1 | + 0.904·2-s − 1.18·4-s + 0.974·5-s − 7-s − 2.87·8-s + 0.881·10-s − 1.96·11-s + 2.74·13-s − 0.904·14-s − 0.238·16-s + 1.83·17-s + 2.02·19-s − 1.15·20-s − 1.77·22-s + 4.95·23-s − 4.04·25-s + 2.47·26-s + 1.18·28-s − 5.22·29-s − 1.18·31-s + 5.53·32-s + 1.66·34-s − 0.974·35-s − 0.458·37-s + 1.83·38-s − 2.80·40-s − 4.63·41-s + ⋯ |
| L(s) = 1 | + 0.639·2-s − 0.591·4-s + 0.435·5-s − 0.377·7-s − 1.01·8-s + 0.278·10-s − 0.592·11-s + 0.760·13-s − 0.241·14-s − 0.0596·16-s + 0.445·17-s + 0.464·19-s − 0.257·20-s − 0.378·22-s + 1.03·23-s − 0.809·25-s + 0.486·26-s + 0.223·28-s − 0.970·29-s − 0.212·31-s + 0.979·32-s + 0.284·34-s − 0.164·35-s − 0.0754·37-s + 0.297·38-s − 0.443·40-s − 0.723·41-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 | \( 1 - 0.904T + 2T^{2} \) |
| 5 | \( 1 - 0.974T + 5T^{2} \) |
| 11 | \( 1 + 1.96T + 11T^{2} \) |
| 13 | \( 1 - 2.74T + 13T^{2} \) |
| 17 | \( 1 - 1.83T + 17T^{2} \) |
| 19 | \( 1 - 2.02T + 19T^{2} \) |
| 23 | \( 1 - 4.95T + 23T^{2} \) |
| 29 | \( 1 + 5.22T + 29T^{2} \) |
| 31 | \( 1 + 1.18T + 31T^{2} \) |
| 37 | \( 1 + 0.458T + 37T^{2} \) |
| 41 | \( 1 + 4.63T + 41T^{2} \) |
| 43 | \( 1 + 9.80T + 43T^{2} \) |
| 47 | \( 1 - 1.05T + 47T^{2} \) |
| 53 | \( 1 + 0.354T + 53T^{2} \) |
| 59 | \( 1 + 2.99T + 59T^{2} \) |
| 61 | \( 1 - 10.7T + 61T^{2} \) |
| 67 | \( 1 + 11.3T + 67T^{2} \) |
| 71 | \( 1 - 11.1T + 71T^{2} \) |
| 73 | \( 1 - 15.2T + 73T^{2} \) |
| 79 | \( 1 + 10.8T + 79T^{2} \) |
| 83 | \( 1 + 2.78T + 83T^{2} \) |
| 89 | \( 1 + 5.79T + 89T^{2} \) |
| 97 | \( 1 - 14.0T + 97T^{2} \) |
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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
−7.45051075654835943411736798786, −6.61301815591242841418813655275, −5.91179190001975171793888773343, −5.34293211466721014680395007250, −4.84292688688511349418999052995, −3.71361444267324113004485580437, −3.40700029651677134645107823047, −2.42577127890360378353694994138, −1.26125960064843551516439781170, 0,
1.26125960064843551516439781170, 2.42577127890360378353694994138, 3.40700029651677134645107823047, 3.71361444267324113004485580437, 4.84292688688511349418999052995, 5.34293211466721014680395007250, 5.91179190001975171793888773343, 6.61301815591242841418813655275, 7.45051075654835943411736798786
