| L(s) = 1 | + 0.583·2-s + 3-s − 1.65·4-s + 0.583·6-s + 4.38·7-s − 2.13·8-s + 9-s − 1.65·12-s − 1.72·13-s + 2.56·14-s + 2.07·16-s + 3.73·17-s + 0.583·18-s − 2.79·19-s + 4.38·21-s − 6.45·23-s − 2.13·24-s − 1.00·26-s + 27-s − 7.28·28-s − 7.88·29-s + 3.05·31-s + 5.48·32-s + 2.17·34-s − 1.65·36-s + 9.21·37-s − 1.62·38-s + ⋯ |
| L(s) = 1 | + 0.412·2-s + 0.577·3-s − 0.829·4-s + 0.238·6-s + 1.65·7-s − 0.755·8-s + 0.333·9-s − 0.478·12-s − 0.477·13-s + 0.684·14-s + 0.517·16-s + 0.904·17-s + 0.137·18-s − 0.640·19-s + 0.957·21-s − 1.34·23-s − 0.436·24-s − 0.196·26-s + 0.192·27-s − 1.37·28-s − 1.46·29-s + 0.548·31-s + 0.969·32-s + 0.373·34-s − 0.276·36-s + 1.51·37-s − 0.264·38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9075 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9075 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.050579326\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.050579326\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 5 | \( 1 \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 - 0.583T + 2T^{2} \) |
| 7 | \( 1 - 4.38T + 7T^{2} \) |
| 13 | \( 1 + 1.72T + 13T^{2} \) |
| 17 | \( 1 - 3.73T + 17T^{2} \) |
| 19 | \( 1 + 2.79T + 19T^{2} \) |
| 23 | \( 1 + 6.45T + 23T^{2} \) |
| 29 | \( 1 + 7.88T + 29T^{2} \) |
| 31 | \( 1 - 3.05T + 31T^{2} \) |
| 37 | \( 1 - 9.21T + 37T^{2} \) |
| 41 | \( 1 - 9.20T + 41T^{2} \) |
| 43 | \( 1 - 8.88T + 43T^{2} \) |
| 47 | \( 1 + 6.29T + 47T^{2} \) |
| 53 | \( 1 - 6.32T + 53T^{2} \) |
| 59 | \( 1 + 0.0655T + 59T^{2} \) |
| 61 | \( 1 - 5.21T + 61T^{2} \) |
| 67 | \( 1 + 6.67T + 67T^{2} \) |
| 71 | \( 1 + 15.9T + 71T^{2} \) |
| 73 | \( 1 - 3.40T + 73T^{2} \) |
| 79 | \( 1 - 3.56T + 79T^{2} \) |
| 83 | \( 1 - 7.22T + 83T^{2} \) |
| 89 | \( 1 - 9.20T + 89T^{2} \) |
| 97 | \( 1 - 3.33T + 97T^{2} \) |
| 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
−7.79742715829588259762340178469, −7.42474304693590116639523777175, −6.02588876884379020957899334489, −5.65025017353797526260525663591, −4.71752702204095993305886312055, −4.32527794785154893215265953667, −3.69238426530140799910855120295, −2.60585517363630114074351433850, −1.86268721798963396922812445623, −0.794037486301738826348798077294,
0.794037486301738826348798077294, 1.86268721798963396922812445623, 2.60585517363630114074351433850, 3.69238426530140799910855120295, 4.32527794785154893215265953667, 4.71752702204095993305886312055, 5.65025017353797526260525663591, 6.02588876884379020957899334489, 7.42474304693590116639523777175, 7.79742715829588259762340178469
