| L(s) = 1 | − 0.856·2-s − 1.26·4-s − 5-s − 0.580·7-s + 2.79·8-s + 0.856·10-s + 4.74·11-s + 6.25·13-s + 0.497·14-s + 0.134·16-s + 5.87·17-s − 5.41·19-s + 1.26·20-s − 4.06·22-s + 8.56·23-s + 25-s − 5.35·26-s + 0.734·28-s + 3.96·29-s + 5.68·31-s − 5.71·32-s − 5.03·34-s + 0.580·35-s + 0.495·37-s + 4.63·38-s − 2.79·40-s − 11.8·41-s + ⋯ |
| L(s) = 1 | − 0.605·2-s − 0.632·4-s − 0.447·5-s − 0.219·7-s + 0.989·8-s + 0.270·10-s + 1.43·11-s + 1.73·13-s + 0.132·14-s + 0.0335·16-s + 1.42·17-s − 1.24·19-s + 0.283·20-s − 0.866·22-s + 1.78·23-s + 0.200·25-s − 1.05·26-s + 0.138·28-s + 0.736·29-s + 1.02·31-s − 1.00·32-s − 0.863·34-s + 0.0981·35-s + 0.0814·37-s + 0.752·38-s − 0.442·40-s − 1.85·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4005 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4005 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.372741451\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.372741451\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 89 | \( 1 + T \) |
| good | 2 | \( 1 + 0.856T + 2T^{2} \) |
| 7 | \( 1 + 0.580T + 7T^{2} \) |
| 11 | \( 1 - 4.74T + 11T^{2} \) |
| 13 | \( 1 - 6.25T + 13T^{2} \) |
| 17 | \( 1 - 5.87T + 17T^{2} \) |
| 19 | \( 1 + 5.41T + 19T^{2} \) |
| 23 | \( 1 - 8.56T + 23T^{2} \) |
| 29 | \( 1 - 3.96T + 29T^{2} \) |
| 31 | \( 1 - 5.68T + 31T^{2} \) |
| 37 | \( 1 - 0.495T + 37T^{2} \) |
| 41 | \( 1 + 11.8T + 41T^{2} \) |
| 43 | \( 1 + 4.78T + 43T^{2} \) |
| 47 | \( 1 - 3.78T + 47T^{2} \) |
| 53 | \( 1 - 7.91T + 53T^{2} \) |
| 59 | \( 1 + 10.6T + 59T^{2} \) |
| 61 | \( 1 - 9.77T + 61T^{2} \) |
| 67 | \( 1 - 1.35T + 67T^{2} \) |
| 71 | \( 1 - 10.7T + 71T^{2} \) |
| 73 | \( 1 - 1.31T + 73T^{2} \) |
| 79 | \( 1 - 0.492T + 79T^{2} \) |
| 83 | \( 1 + 1.62T + 83T^{2} \) |
| 97 | \( 1 + 9.61T + 97T^{2} \) |
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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
−8.433677710684544692770616738721, −8.131625004842566767324989221591, −6.87136408118121086622544446208, −6.52397758349260924235140919391, −5.45725369090682746502757070631, −4.54743888118147683986058055792, −3.79212293051712988607555376318, −3.22515363721044854592135700249, −1.45617055868804096323483676381, −0.861202382816179963795175573148,
0.861202382816179963795175573148, 1.45617055868804096323483676381, 3.22515363721044854592135700249, 3.79212293051712988607555376318, 4.54743888118147683986058055792, 5.45725369090682746502757070631, 6.52397758349260924235140919391, 6.87136408118121086622544446208, 8.131625004842566767324989221591, 8.433677710684544692770616738721
