| L(s) = 1 | + 2.66·2-s + 5.12·4-s + 2.72·5-s − 7-s + 8.32·8-s + 7.28·10-s − 1.99·11-s + 3.01·13-s − 2.66·14-s + 11.9·16-s + 1.40·17-s − 3.11·19-s + 13.9·20-s − 5.33·22-s + 3.27·23-s + 2.44·25-s + 8.03·26-s − 5.12·28-s + 6.12·29-s + 6.08·31-s + 15.3·32-s + 3.73·34-s − 2.72·35-s − 5.42·37-s − 8.30·38-s + 22.7·40-s − 11.6·41-s + ⋯ |
| L(s) = 1 | + 1.88·2-s + 2.56·4-s + 1.22·5-s − 0.377·7-s + 2.94·8-s + 2.30·10-s − 0.602·11-s + 0.835·13-s − 0.713·14-s + 2.99·16-s + 0.339·17-s − 0.714·19-s + 3.12·20-s − 1.13·22-s + 0.682·23-s + 0.489·25-s + 1.57·26-s − 0.967·28-s + 1.13·29-s + 1.09·31-s + 2.70·32-s + 0.641·34-s − 0.461·35-s − 0.892·37-s − 1.34·38-s + 3.59·40-s − 1.81·41-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
| \(L(1)\) |
\(\approx\) |
\(9.570987939\) |
| \(L(\frac12)\) |
\(\approx\) |
\(9.570987939\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 2.66T + 2T^{2} \) |
| 5 | \( 1 - 2.72T + 5T^{2} \) |
| 11 | \( 1 + 1.99T + 11T^{2} \) |
| 13 | \( 1 - 3.01T + 13T^{2} \) |
| 17 | \( 1 - 1.40T + 17T^{2} \) |
| 19 | \( 1 + 3.11T + 19T^{2} \) |
| 23 | \( 1 - 3.27T + 23T^{2} \) |
| 29 | \( 1 - 6.12T + 29T^{2} \) |
| 31 | \( 1 - 6.08T + 31T^{2} \) |
| 37 | \( 1 + 5.42T + 37T^{2} \) |
| 41 | \( 1 + 11.6T + 41T^{2} \) |
| 43 | \( 1 - 9.33T + 43T^{2} \) |
| 47 | \( 1 - 8.98T + 47T^{2} \) |
| 53 | \( 1 + 4.15T + 53T^{2} \) |
| 59 | \( 1 + 7.85T + 59T^{2} \) |
| 61 | \( 1 + 3.02T + 61T^{2} \) |
| 67 | \( 1 + 7.97T + 67T^{2} \) |
| 71 | \( 1 + 4.60T + 71T^{2} \) |
| 73 | \( 1 - 6.75T + 73T^{2} \) |
| 79 | \( 1 - 0.469T + 79T^{2} \) |
| 83 | \( 1 - 11.7T + 83T^{2} \) |
| 89 | \( 1 + 1.25T + 89T^{2} \) |
| 97 | \( 1 - 5.52T + 97T^{2} \) |
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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
−7.48920002208046550371637927297, −6.65823866814827434591495199167, −6.24563971352269801388629041839, −5.72814257055676062291835610664, −5.04350316324343036218311010178, −4.44462642924977917231181544784, −3.49257865932576193038680863443, −2.85348681797864622630321984804, −2.18341881722817586717088311934, −1.26828698600340932266300099102,
1.26828698600340932266300099102, 2.18341881722817586717088311934, 2.85348681797864622630321984804, 3.49257865932576193038680863443, 4.44462642924977917231181544784, 5.04350316324343036218311010178, 5.72814257055676062291835610664, 6.24563971352269801388629041839, 6.65823866814827434591495199167, 7.48920002208046550371637927297
