L(s) = 1 | − 1.63·2-s + 0.672·4-s + 0.107·5-s − 7-s + 2.16·8-s − 0.176·10-s − 1.23·11-s − 2.67·13-s + 1.63·14-s − 4.89·16-s − 6.14·17-s + 1.57·19-s + 0.0725·20-s + 2.01·22-s − 5.97·23-s − 4.98·25-s + 4.37·26-s − 0.672·28-s + 9.60·29-s + 1.81·31-s + 3.65·32-s + 10.0·34-s − 0.107·35-s + 7.63·37-s − 2.57·38-s + 0.233·40-s + 4.73·41-s + ⋯ |
L(s) = 1 | − 1.15·2-s + 0.336·4-s + 0.0482·5-s − 0.377·7-s + 0.767·8-s − 0.0557·10-s − 0.372·11-s − 0.742·13-s + 0.436·14-s − 1.22·16-s − 1.48·17-s + 0.361·19-s + 0.0162·20-s + 0.430·22-s − 1.24·23-s − 0.997·25-s + 0.858·26-s − 0.127·28-s + 1.78·29-s + 0.325·31-s + 0.646·32-s + 1.72·34-s − 0.0182·35-s + 1.25·37-s − 0.418·38-s + 0.0369·40-s + 0.739·41-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 | \( 1 + 1.63T + 2T^{2} \) |
| 5 | \( 1 - 0.107T + 5T^{2} \) |
| 11 | \( 1 + 1.23T + 11T^{2} \) |
| 13 | \( 1 + 2.67T + 13T^{2} \) |
| 17 | \( 1 + 6.14T + 17T^{2} \) |
| 19 | \( 1 - 1.57T + 19T^{2} \) |
| 23 | \( 1 + 5.97T + 23T^{2} \) |
| 29 | \( 1 - 9.60T + 29T^{2} \) |
| 31 | \( 1 - 1.81T + 31T^{2} \) |
| 37 | \( 1 - 7.63T + 37T^{2} \) |
| 41 | \( 1 - 4.73T + 41T^{2} \) |
| 43 | \( 1 - 8.65T + 43T^{2} \) |
| 47 | \( 1 - 11.0T + 47T^{2} \) |
| 53 | \( 1 - 13.7T + 53T^{2} \) |
| 59 | \( 1 + 5.68T + 59T^{2} \) |
| 61 | \( 1 + 8.36T + 61T^{2} \) |
| 67 | \( 1 + 4.40T + 67T^{2} \) |
| 71 | \( 1 - 13.9T + 71T^{2} \) |
| 73 | \( 1 + 8.13T + 73T^{2} \) |
| 79 | \( 1 - 10.6T + 79T^{2} \) |
| 83 | \( 1 + 10.3T + 83T^{2} \) |
| 89 | \( 1 + 15.7T + 89T^{2} \) |
| 97 | \( 1 + 12.2T + 97T^{2} \) |
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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
−7.65772498415484632211544554456, −7.03930685283610596795824713396, −6.26324576139180944218132118451, −5.54307586849786280729439995123, −4.36839892091370915708217976244, −4.22808986878229730635879436886, −2.65226315046539572374974831300, −2.23587235333006437902012892666, −0.946001048035913700547777939165, 0,
0.946001048035913700547777939165, 2.23587235333006437902012892666, 2.65226315046539572374974831300, 4.22808986878229730635879436886, 4.36839892091370915708217976244, 5.54307586849786280729439995123, 6.26324576139180944218132118451, 7.03930685283610596795824713396, 7.65772498415484632211544554456