L(s) = 1 | − 0.246·2-s − 1.93·4-s − 1.24·7-s + 0.972·8-s − 2.56·11-s − 4.68·13-s + 0.307·14-s + 3.63·16-s − 5.83·17-s + 4.16·19-s + 0.634·22-s − 1.60·23-s + 1.15·26-s + 2.41·28-s + 3.21·29-s − 9.19·31-s − 2.84·32-s + 1.44·34-s − 1.27·37-s − 1.02·38-s − 8.69·41-s − 3.88·43-s + 4.98·44-s + 0.396·46-s + 3.20·47-s − 5.44·49-s + 9.07·52-s + ⋯ |
L(s) = 1 | − 0.174·2-s − 0.969·4-s − 0.471·7-s + 0.343·8-s − 0.774·11-s − 1.29·13-s + 0.0822·14-s + 0.909·16-s − 1.41·17-s + 0.956·19-s + 0.135·22-s − 0.334·23-s + 0.226·26-s + 0.456·28-s + 0.597·29-s − 1.65·31-s − 0.502·32-s + 0.247·34-s − 0.210·37-s − 0.166·38-s − 1.35·41-s − 0.591·43-s + 0.751·44-s + 0.0584·46-s + 0.468·47-s − 0.777·49-s + 1.25·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5625 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5625 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.4664405130\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4664405130\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
good | 2 | \( 1 + 0.246T + 2T^{2} \) |
| 7 | \( 1 + 1.24T + 7T^{2} \) |
| 11 | \( 1 + 2.56T + 11T^{2} \) |
| 13 | \( 1 + 4.68T + 13T^{2} \) |
| 17 | \( 1 + 5.83T + 17T^{2} \) |
| 19 | \( 1 - 4.16T + 19T^{2} \) |
| 23 | \( 1 + 1.60T + 23T^{2} \) |
| 29 | \( 1 - 3.21T + 29T^{2} \) |
| 31 | \( 1 + 9.19T + 31T^{2} \) |
| 37 | \( 1 + 1.27T + 37T^{2} \) |
| 41 | \( 1 + 8.69T + 41T^{2} \) |
| 43 | \( 1 + 3.88T + 43T^{2} \) |
| 47 | \( 1 - 3.20T + 47T^{2} \) |
| 53 | \( 1 - 13.3T + 53T^{2} \) |
| 59 | \( 1 - 6.39T + 59T^{2} \) |
| 61 | \( 1 - 3.82T + 61T^{2} \) |
| 67 | \( 1 + 10.1T + 67T^{2} \) |
| 71 | \( 1 + 13.8T + 71T^{2} \) |
| 73 | \( 1 + 11.3T + 73T^{2} \) |
| 79 | \( 1 - 11.4T + 79T^{2} \) |
| 83 | \( 1 - 2.83T + 83T^{2} \) |
| 89 | \( 1 - 1.38T + 89T^{2} \) |
| 97 | \( 1 - 0.0305T + 97T^{2} \) |
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
−8.230871937434648291252062786485, −7.37787945554749536776342719534, −6.96376470095587228264092634269, −5.82310125382323565916426903800, −5.14470687218138336572438520658, −4.61304142820689229425628321706, −3.70627129060984736909146602527, −2.84718463169421640630631421969, −1.87317647987066047457216607159, −0.36046574005908201012212072555,
0.36046574005908201012212072555, 1.87317647987066047457216607159, 2.84718463169421640630631421969, 3.70627129060984736909146602527, 4.61304142820689229425628321706, 5.14470687218138336572438520658, 5.82310125382323565916426903800, 6.96376470095587228264092634269, 7.37787945554749536776342719534, 8.230871937434648291252062786485