L(s) = 1 | − 1.56·2-s + 0.438·4-s − 5-s − 7-s + 2.43·8-s + 1.56·10-s − 2.56·11-s + 4.56·13-s + 1.56·14-s − 4.68·16-s + 4.56·17-s + 1.12·19-s − 0.438·20-s + 4·22-s + 5.12·23-s + 25-s − 7.12·26-s − 0.438·28-s + 5.68·29-s + 2.43·32-s − 7.12·34-s + 35-s + 6·37-s − 1.75·38-s − 2.43·40-s + 3.12·41-s + 9.12·43-s + ⋯ |
L(s) = 1 | − 1.10·2-s + 0.219·4-s − 0.447·5-s − 0.377·7-s + 0.862·8-s + 0.493·10-s − 0.772·11-s + 1.26·13-s + 0.417·14-s − 1.17·16-s + 1.10·17-s + 0.257·19-s − 0.0980·20-s + 0.852·22-s + 1.06·23-s + 0.200·25-s − 1.39·26-s − 0.0828·28-s + 1.05·29-s + 0.431·32-s − 1.22·34-s + 0.169·35-s + 0.986·37-s − 0.284·38-s − 0.385·40-s + 0.487·41-s + 1.39·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.6392995351\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6392995351\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 + T \) |
good | 2 | \( 1 + 1.56T + 2T^{2} \) |
| 11 | \( 1 + 2.56T + 11T^{2} \) |
| 13 | \( 1 - 4.56T + 13T^{2} \) |
| 17 | \( 1 - 4.56T + 17T^{2} \) |
| 19 | \( 1 - 1.12T + 19T^{2} \) |
| 23 | \( 1 - 5.12T + 23T^{2} \) |
| 29 | \( 1 - 5.68T + 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 - 6T + 37T^{2} \) |
| 41 | \( 1 - 3.12T + 41T^{2} \) |
| 43 | \( 1 - 9.12T + 43T^{2} \) |
| 47 | \( 1 + 3.68T + 47T^{2} \) |
| 53 | \( 1 + 3.12T + 53T^{2} \) |
| 59 | \( 1 - 4T + 59T^{2} \) |
| 61 | \( 1 + 9.36T + 61T^{2} \) |
| 67 | \( 1 + 6.24T + 67T^{2} \) |
| 71 | \( 1 + 8T + 71T^{2} \) |
| 73 | \( 1 - 4.24T + 73T^{2} \) |
| 79 | \( 1 + 6.56T + 79T^{2} \) |
| 83 | \( 1 + 4T + 83T^{2} \) |
| 89 | \( 1 + 7.12T + 89T^{2} \) |
| 97 | \( 1 + 14.8T + 97T^{2} \) |
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
−11.29128324216579441412299488111, −10.60273298920605758454148199278, −9.735326714318220717641094488874, −8.800940529534860652287928406157, −8.018914520710302787255664195411, −7.22010964104278774045158709760, −5.87175408394933631486803658423, −4.50078523425898222775851695087, −3.08681891969205900823800104935, −1.00722398457166771928851375344,
1.00722398457166771928851375344, 3.08681891969205900823800104935, 4.50078523425898222775851695087, 5.87175408394933631486803658423, 7.22010964104278774045158709760, 8.018914520710302787255664195411, 8.800940529534860652287928406157, 9.735326714318220717641094488874, 10.60273298920605758454148199278, 11.29128324216579441412299488111