| L(s) = 1 | + 3·3-s − 2.85·5-s − 7-s + 6·9-s − 5.47·11-s + 3.61·13-s − 8.56·15-s + 5.23·17-s + 5·19-s − 3·21-s − 1.14·23-s + 3.14·25-s + 9·27-s − 0.527·29-s + 9.47·31-s − 16.4·33-s + 2.85·35-s + 5.32·37-s + 10.8·39-s + 2.85·41-s − 3.38·43-s − 17.1·45-s + 0.0901·47-s + 49-s + 15.7·51-s − 11.8·53-s + 15.6·55-s + ⋯ |
| L(s) = 1 | + 1.73·3-s − 1.27·5-s − 0.377·7-s + 2·9-s − 1.64·11-s + 1.00·13-s − 2.21·15-s + 1.26·17-s + 1.14·19-s − 0.654·21-s − 0.238·23-s + 0.629·25-s + 1.73·27-s − 0.0980·29-s + 1.70·31-s − 2.85·33-s + 0.482·35-s + 0.875·37-s + 1.73·39-s + 0.445·41-s − 0.515·43-s − 2.55·45-s + 0.0131·47-s + 0.142·49-s + 2.19·51-s − 1.62·53-s + 2.10·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2828 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2828 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.691587911\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.691587911\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 + T \) |
| 101 | \( 1 + T \) |
| good | 3 | \( 1 - 3T + 3T^{2} \) |
| 5 | \( 1 + 2.85T + 5T^{2} \) |
| 11 | \( 1 + 5.47T + 11T^{2} \) |
| 13 | \( 1 - 3.61T + 13T^{2} \) |
| 17 | \( 1 - 5.23T + 17T^{2} \) |
| 19 | \( 1 - 5T + 19T^{2} \) |
| 23 | \( 1 + 1.14T + 23T^{2} \) |
| 29 | \( 1 + 0.527T + 29T^{2} \) |
| 31 | \( 1 - 9.47T + 31T^{2} \) |
| 37 | \( 1 - 5.32T + 37T^{2} \) |
| 41 | \( 1 - 2.85T + 41T^{2} \) |
| 43 | \( 1 + 3.38T + 43T^{2} \) |
| 47 | \( 1 - 0.0901T + 47T^{2} \) |
| 53 | \( 1 + 11.8T + 53T^{2} \) |
| 59 | \( 1 - 15.1T + 59T^{2} \) |
| 61 | \( 1 - 3.32T + 61T^{2} \) |
| 67 | \( 1 - 5.32T + 67T^{2} \) |
| 71 | \( 1 - 2.38T + 71T^{2} \) |
| 73 | \( 1 - 10.4T + 73T^{2} \) |
| 79 | \( 1 + 6.70T + 79T^{2} \) |
| 83 | \( 1 - 16.6T + 83T^{2} \) |
| 89 | \( 1 + 17.9T + 89T^{2} \) |
| 97 | \( 1 - 9.76T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.374053627017834449220228123895, −8.085470614989398479085416164432, −7.72271416297050762702832728062, −6.88976143317582866780104570825, −5.64362150723481537048676463496, −4.61109436331917321299931749762, −3.64684601896498955381034767415, −3.22767077383827333849922121583, −2.47957564554632759801447027669, −0.956484171915648091248351948341,
0.956484171915648091248351948341, 2.47957564554632759801447027669, 3.22767077383827333849922121583, 3.64684601896498955381034767415, 4.61109436331917321299931749762, 5.64362150723481537048676463496, 6.88976143317582866780104570825, 7.72271416297050762702832728062, 8.085470614989398479085416164432, 8.374053627017834449220228123895
