| L(s) = 1 | − 2.89·3-s − 5-s + 2.68·7-s + 5.40·9-s + 5.20·11-s − 0.720·13-s + 2.89·15-s − 6.59·17-s + 5.39·19-s − 7.77·21-s − 2.07·23-s + 25-s − 6.97·27-s + 3.46·29-s + 6.64·31-s − 15.0·33-s − 2.68·35-s + 2.32·37-s + 2.08·39-s + 6.40·41-s + 4.43·43-s − 5.40·45-s + 3.76·47-s + 0.191·49-s + 19.1·51-s + 11.1·53-s − 5.20·55-s + ⋯ |
| L(s) = 1 | − 1.67·3-s − 0.447·5-s + 1.01·7-s + 1.80·9-s + 1.56·11-s − 0.199·13-s + 0.748·15-s − 1.59·17-s + 1.23·19-s − 1.69·21-s − 0.433·23-s + 0.200·25-s − 1.34·27-s + 0.644·29-s + 1.19·31-s − 2.62·33-s − 0.453·35-s + 0.382·37-s + 0.334·39-s + 1.00·41-s + 0.676·43-s − 0.805·45-s + 0.548·47-s + 0.0274·49-s + 2.67·51-s + 1.52·53-s − 0.701·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8020 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8020 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.343798583\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.343798583\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + T \) |
| 401 | \( 1 + T \) |
| good | 3 | \( 1 + 2.89T + 3T^{2} \) |
| 7 | \( 1 - 2.68T + 7T^{2} \) |
| 11 | \( 1 - 5.20T + 11T^{2} \) |
| 13 | \( 1 + 0.720T + 13T^{2} \) |
| 17 | \( 1 + 6.59T + 17T^{2} \) |
| 19 | \( 1 - 5.39T + 19T^{2} \) |
| 23 | \( 1 + 2.07T + 23T^{2} \) |
| 29 | \( 1 - 3.46T + 29T^{2} \) |
| 31 | \( 1 - 6.64T + 31T^{2} \) |
| 37 | \( 1 - 2.32T + 37T^{2} \) |
| 41 | \( 1 - 6.40T + 41T^{2} \) |
| 43 | \( 1 - 4.43T + 43T^{2} \) |
| 47 | \( 1 - 3.76T + 47T^{2} \) |
| 53 | \( 1 - 11.1T + 53T^{2} \) |
| 59 | \( 1 - 11.8T + 59T^{2} \) |
| 61 | \( 1 + 1.92T + 61T^{2} \) |
| 67 | \( 1 + 9.76T + 67T^{2} \) |
| 71 | \( 1 - 11.6T + 71T^{2} \) |
| 73 | \( 1 - 0.405T + 73T^{2} \) |
| 79 | \( 1 - 7.64T + 79T^{2} \) |
| 83 | \( 1 + 7.18T + 83T^{2} \) |
| 89 | \( 1 + 3.82T + 89T^{2} \) |
| 97 | \( 1 - 2.72T + 97T^{2} \) |
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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
−7.66627263428738646159344460375, −6.87378293658453508694270872464, −6.53785447787100586313276355736, −5.72044074824729848369958749822, −5.05841742703475104427084726638, −4.27824954140839788268850477746, −4.07716547802548291379637836430, −2.52445636912054633697406875612, −1.35044253067666553168069818889, −0.72104655121705753004368362322,
0.72104655121705753004368362322, 1.35044253067666553168069818889, 2.52445636912054633697406875612, 4.07716547802548291379637836430, 4.27824954140839788268850477746, 5.05841742703475104427084726638, 5.72044074824729848369958749822, 6.53785447787100586313276355736, 6.87378293658453508694270872464, 7.66627263428738646159344460375
