| L(s) = 1 | + 0.560·2-s − 7.68·4-s − 18.3·5-s + 31.3·7-s − 8.78·8-s − 10.2·10-s − 53.2·11-s + 28.9·13-s + 17.5·14-s + 56.5·16-s + 17.8·17-s − 7.34·19-s + 141.·20-s − 29.8·22-s − 201.·23-s + 212.·25-s + 16.1·26-s − 241.·28-s + 245.·29-s − 173.·31-s + 101.·32-s + 10.0·34-s − 576.·35-s − 53.5·37-s − 4.11·38-s + 161.·40-s + 27.2·41-s + ⋯ |
| L(s) = 1 | + 0.198·2-s − 0.960·4-s − 1.64·5-s + 1.69·7-s − 0.388·8-s − 0.325·10-s − 1.46·11-s + 0.616·13-s + 0.335·14-s + 0.883·16-s + 0.254·17-s − 0.0886·19-s + 1.57·20-s − 0.289·22-s − 1.82·23-s + 1.69·25-s + 0.122·26-s − 1.62·28-s + 1.57·29-s − 1.00·31-s + 0.563·32-s + 0.0504·34-s − 2.78·35-s − 0.237·37-s − 0.0175·38-s + 0.638·40-s + 0.103·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2151 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2151 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 239 | \( 1 + 239T \) |
| good | 2 | \( 1 - 0.560T + 8T^{2} \) |
| 5 | \( 1 + 18.3T + 125T^{2} \) |
| 7 | \( 1 - 31.3T + 343T^{2} \) |
| 11 | \( 1 + 53.2T + 1.33e3T^{2} \) |
| 13 | \( 1 - 28.9T + 2.19e3T^{2} \) |
| 17 | \( 1 - 17.8T + 4.91e3T^{2} \) |
| 19 | \( 1 + 7.34T + 6.85e3T^{2} \) |
| 23 | \( 1 + 201.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 245.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 173.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 53.5T + 5.06e4T^{2} \) |
| 41 | \( 1 - 27.2T + 6.89e4T^{2} \) |
| 43 | \( 1 + 367.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 341.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 767.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 838.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 429.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 508.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 819.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 435.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 792.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 293.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.32e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 96.5T + 9.12e5T^{2} \) |
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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
−8.252510990627621938008005299663, −7.971091613362713944198987594115, −7.05113984787251271013081061393, −5.50716988433621595169373151722, −5.09865633061184372830317801075, −4.13957574197131827644450340218, −3.79882521882415142660918840670, −2.43143977246957883251885617732, −0.977446190485588125618450700428, 0,
0.977446190485588125618450700428, 2.43143977246957883251885617732, 3.79882521882415142660918840670, 4.13957574197131827644450340218, 5.09865633061184372830317801075, 5.50716988433621595169373151722, 7.05113984787251271013081061393, 7.971091613362713944198987594115, 8.252510990627621938008005299663
