L(s) = 1 | + 0.0631·2-s + 0.00106·3-s − 1.99·4-s + 5-s + 6.72e−5·6-s + 3.26·7-s − 0.252·8-s − 2.99·9-s + 0.0631·10-s − 11-s − 0.00212·12-s + 0.254·13-s + 0.206·14-s + 0.00106·15-s + 3.97·16-s − 5.08·17-s − 0.189·18-s − 0.0662·19-s − 1.99·20-s + 0.00347·21-s − 0.0631·22-s + 3.74·23-s − 0.000268·24-s + 25-s + 0.0160·26-s − 0.00638·27-s − 6.51·28-s + ⋯ |
L(s) = 1 | + 0.0446·2-s + 0.000614·3-s − 0.998·4-s + 0.447·5-s + 2.74e−5·6-s + 1.23·7-s − 0.0892·8-s − 0.999·9-s + 0.0199·10-s − 0.301·11-s − 0.000613·12-s + 0.0707·13-s + 0.0550·14-s + 0.000274·15-s + 0.994·16-s − 1.23·17-s − 0.0446·18-s − 0.0152·19-s − 0.446·20-s + 0.000757·21-s − 0.0134·22-s + 0.781·23-s − 5.48e − 5·24-s + 0.200·25-s + 0.00315·26-s − 0.00122·27-s − 1.23·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4015 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.561515697\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.561515697\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| 73 | \( 1 + T \) |
good | 2 | \( 1 - 0.0631T + 2T^{2} \) |
| 3 | \( 1 - 0.00106T + 3T^{2} \) |
| 7 | \( 1 - 3.26T + 7T^{2} \) |
| 13 | \( 1 - 0.254T + 13T^{2} \) |
| 17 | \( 1 + 5.08T + 17T^{2} \) |
| 19 | \( 1 + 0.0662T + 19T^{2} \) |
| 23 | \( 1 - 3.74T + 23T^{2} \) |
| 29 | \( 1 - 3.06T + 29T^{2} \) |
| 31 | \( 1 - 0.0370T + 31T^{2} \) |
| 37 | \( 1 - 1.75T + 37T^{2} \) |
| 41 | \( 1 + 4.05T + 41T^{2} \) |
| 43 | \( 1 - 5.35T + 43T^{2} \) |
| 47 | \( 1 + 6.11T + 47T^{2} \) |
| 53 | \( 1 - 7.61T + 53T^{2} \) |
| 59 | \( 1 - 7.27T + 59T^{2} \) |
| 61 | \( 1 + 4.70T + 61T^{2} \) |
| 67 | \( 1 - 9.97T + 67T^{2} \) |
| 71 | \( 1 - 10.3T + 71T^{2} \) |
| 79 | \( 1 + 4.96T + 79T^{2} \) |
| 83 | \( 1 - 17.7T + 83T^{2} \) |
| 89 | \( 1 + 17.6T + 89T^{2} \) |
| 97 | \( 1 + 9.81T + 97T^{2} \) |
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
−8.458101891831094831071300229514, −8.048616745416659883992213431177, −6.98817579967869260654458097447, −6.06997638405097095050749223871, −5.23274262149616894671306879021, −4.87241906308266238771863565011, −4.01492432293780717914399198423, −2.91051467284936564615218854107, −1.98001539563908026201202655664, −0.71845919690758723132680107319,
0.71845919690758723132680107319, 1.98001539563908026201202655664, 2.91051467284936564615218854107, 4.01492432293780717914399198423, 4.87241906308266238771863565011, 5.23274262149616894671306879021, 6.06997638405097095050749223871, 6.98817579967869260654458097447, 8.048616745416659883992213431177, 8.458101891831094831071300229514