| L(s) = 1 | − 0.662·2-s − 1.56·4-s − 3.26·5-s − 7-s + 2.35·8-s + 2.16·10-s + 1.50·11-s − 4.27·13-s + 0.662·14-s + 1.55·16-s − 2.42·17-s − 3.39·19-s + 5.10·20-s − 0.994·22-s − 4.67·23-s + 5.69·25-s + 2.82·26-s + 1.56·28-s − 2.49·29-s + 5.16·31-s − 5.75·32-s + 1.60·34-s + 3.26·35-s − 9.93·37-s + 2.24·38-s − 7.71·40-s − 1.42·41-s + ⋯ |
| L(s) = 1 | − 0.468·2-s − 0.780·4-s − 1.46·5-s − 0.377·7-s + 0.834·8-s + 0.684·10-s + 0.452·11-s − 1.18·13-s + 0.177·14-s + 0.389·16-s − 0.587·17-s − 0.777·19-s + 1.14·20-s − 0.211·22-s − 0.975·23-s + 1.13·25-s + 0.554·26-s + 0.295·28-s − 0.462·29-s + 0.927·31-s − 1.01·32-s + 0.275·34-s + 0.552·35-s − 1.63·37-s + 0.364·38-s − 1.21·40-s − 0.223·41-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
| \(L(1)\) |
\(\approx\) |
\(0.03216212150\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.03216212150\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 0.662T + 2T^{2} \) |
| 5 | \( 1 + 3.26T + 5T^{2} \) |
| 11 | \( 1 - 1.50T + 11T^{2} \) |
| 13 | \( 1 + 4.27T + 13T^{2} \) |
| 17 | \( 1 + 2.42T + 17T^{2} \) |
| 19 | \( 1 + 3.39T + 19T^{2} \) |
| 23 | \( 1 + 4.67T + 23T^{2} \) |
| 29 | \( 1 + 2.49T + 29T^{2} \) |
| 31 | \( 1 - 5.16T + 31T^{2} \) |
| 37 | \( 1 + 9.93T + 37T^{2} \) |
| 41 | \( 1 + 1.42T + 41T^{2} \) |
| 43 | \( 1 - 0.721T + 43T^{2} \) |
| 47 | \( 1 - 2.25T + 47T^{2} \) |
| 53 | \( 1 + 1.84T + 53T^{2} \) |
| 59 | \( 1 + 3.44T + 59T^{2} \) |
| 61 | \( 1 + 1.15T + 61T^{2} \) |
| 67 | \( 1 + 14.4T + 67T^{2} \) |
| 71 | \( 1 + 11.7T + 71T^{2} \) |
| 73 | \( 1 - 3.97T + 73T^{2} \) |
| 79 | \( 1 + 3.64T + 79T^{2} \) |
| 83 | \( 1 + 12.5T + 83T^{2} \) |
| 89 | \( 1 + 13.9T + 89T^{2} \) |
| 97 | \( 1 + 9.82T + 97T^{2} \) |
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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
−7.88633523791510712315991100230, −7.31638522900554141900125374665, −6.72633546759783994111269853704, −5.72521909142346110282537143685, −4.73738051135302977833082533100, −4.28061180067442650317938539047, −3.72788739755826879348248063253, −2.76154585892820230268585780630, −1.55133854446952637822307110621, −0.096705303407124762013691215494,
0.096705303407124762013691215494, 1.55133854446952637822307110621, 2.76154585892820230268585780630, 3.72788739755826879348248063253, 4.28061180067442650317938539047, 4.73738051135302977833082533100, 5.72521909142346110282537143685, 6.72633546759783994111269853704, 7.31638522900554141900125374665, 7.88633523791510712315991100230
