| L(s) = 1 | − 2.29·2-s + 3.25·4-s + 0.132·5-s + 7-s − 2.88·8-s − 0.304·10-s + 1.38·11-s + 3.00·13-s − 2.29·14-s + 0.0952·16-s − 4.60·17-s − 1.78·19-s + 0.432·20-s − 3.16·22-s − 6.02·23-s − 4.98·25-s − 6.87·26-s + 3.25·28-s − 2.00·29-s + 9.54·31-s + 5.54·32-s + 10.5·34-s + 0.132·35-s + 2.84·37-s + 4.09·38-s − 0.382·40-s − 2.18·41-s + ⋯ |
| L(s) = 1 | − 1.62·2-s + 1.62·4-s + 0.0593·5-s + 0.377·7-s − 1.01·8-s − 0.0962·10-s + 0.416·11-s + 0.832·13-s − 0.612·14-s + 0.0238·16-s − 1.11·17-s − 0.409·19-s + 0.0966·20-s − 0.674·22-s − 1.25·23-s − 0.996·25-s − 1.34·26-s + 0.615·28-s − 0.371·29-s + 1.71·31-s + 0.980·32-s + 1.81·34-s + 0.0224·35-s + 0.467·37-s + 0.663·38-s − 0.0604·40-s − 0.340·41-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 | \( 1 + 2.29T + 2T^{2} \) |
| 5 | \( 1 - 0.132T + 5T^{2} \) |
| 11 | \( 1 - 1.38T + 11T^{2} \) |
| 13 | \( 1 - 3.00T + 13T^{2} \) |
| 17 | \( 1 + 4.60T + 17T^{2} \) |
| 19 | \( 1 + 1.78T + 19T^{2} \) |
| 23 | \( 1 + 6.02T + 23T^{2} \) |
| 29 | \( 1 + 2.00T + 29T^{2} \) |
| 31 | \( 1 - 9.54T + 31T^{2} \) |
| 37 | \( 1 - 2.84T + 37T^{2} \) |
| 41 | \( 1 + 2.18T + 41T^{2} \) |
| 43 | \( 1 - 9.51T + 43T^{2} \) |
| 47 | \( 1 + 2.52T + 47T^{2} \) |
| 53 | \( 1 - 8.38T + 53T^{2} \) |
| 59 | \( 1 + 7.78T + 59T^{2} \) |
| 61 | \( 1 - 3.77T + 61T^{2} \) |
| 67 | \( 1 + 10.1T + 67T^{2} \) |
| 71 | \( 1 - 2.19T + 71T^{2} \) |
| 73 | \( 1 - 1.19T + 73T^{2} \) |
| 79 | \( 1 + 10.3T + 79T^{2} \) |
| 83 | \( 1 + 1.81T + 83T^{2} \) |
| 89 | \( 1 - 7.25T + 89T^{2} \) |
| 97 | \( 1 - 2.14T + 97T^{2} \) |
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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
−7.83815393269048691500439813851, −6.94293383190815849502436797429, −6.33090298287910018678363721017, −5.77907882344571277012594350869, −4.46079105354476977829002301685, −3.99463912841808748066866421825, −2.64763469278711099860502659015, −1.93757126535635117660649444368, −1.13595834172652689410151164670, 0,
1.13595834172652689410151164670, 1.93757126535635117660649444368, 2.64763469278711099860502659015, 3.99463912841808748066866421825, 4.46079105354476977829002301685, 5.77907882344571277012594350869, 6.33090298287910018678363721017, 6.94293383190815849502436797429, 7.83815393269048691500439813851
