L(s) = 1 | − 7-s + 4·11-s − 2·13-s + 2·17-s + 4·19-s − 8·23-s + 6·29-s − 8·31-s − 2·37-s − 2·41-s − 12·43-s − 8·47-s + 49-s − 6·53-s + 4·59-s + 2·61-s + 12·67-s − 8·71-s + 14·73-s − 4·77-s − 12·83-s − 2·89-s + 2·91-s − 10·97-s + 101-s + 103-s + 107-s + ⋯ |
L(s) = 1 | − 0.377·7-s + 1.20·11-s − 0.554·13-s + 0.485·17-s + 0.917·19-s − 1.66·23-s + 1.11·29-s − 1.43·31-s − 0.328·37-s − 0.312·41-s − 1.82·43-s − 1.16·47-s + 1/7·49-s − 0.824·53-s + 0.520·59-s + 0.256·61-s + 1.46·67-s − 0.949·71-s + 1.63·73-s − 0.455·77-s − 1.31·83-s − 0.211·89-s + 0.209·91-s − 1.01·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 100800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 100800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.596055111\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.596055111\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + T \) |
good | 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + 12 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 2 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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
−13.93970755217475, −13.23918417865836, −12.71910887225671, −12.16491067149236, −11.88341784731987, −11.44000228687970, −10.85082958498688, −10.04087615596593, −9.789916447217702, −9.540826952462369, −8.695193694816893, −8.332655774273482, −7.738533637423961, −7.142633510894681, −6.635535428100408, −6.266423709709159, −5.523601150043488, −5.092964641321978, −4.425829562034707, −3.642074015381062, −3.489074535328219, −2.661194235170348, −1.830187129796914, −1.380192478163516, −0.3957825969788781,
0.3957825969788781, 1.380192478163516, 1.830187129796914, 2.661194235170348, 3.489074535328219, 3.642074015381062, 4.425829562034707, 5.092964641321978, 5.523601150043488, 6.266423709709159, 6.635535428100408, 7.142633510894681, 7.738533637423961, 8.332655774273482, 8.695193694816893, 9.540826952462369, 9.789916447217702, 10.04087615596593, 10.85082958498688, 11.44000228687970, 11.88341784731987, 12.16491067149236, 12.71910887225671, 13.23918417865836, 13.93970755217475