L(s) = 1 | − 2-s + 4-s − 5-s + 4·7-s − 8-s + 10-s + 4·11-s − 2·13-s − 4·14-s + 16-s + 2·17-s − 19-s − 20-s − 4·22-s + 8·23-s + 25-s + 2·26-s + 4·28-s − 6·29-s + 4·31-s − 32-s − 2·34-s − 4·35-s − 10·37-s + 38-s + 40-s + 2·41-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.447·5-s + 1.51·7-s − 0.353·8-s + 0.316·10-s + 1.20·11-s − 0.554·13-s − 1.06·14-s + 1/4·16-s + 0.485·17-s − 0.229·19-s − 0.223·20-s − 0.852·22-s + 1.66·23-s + 1/5·25-s + 0.392·26-s + 0.755·28-s − 1.11·29-s + 0.718·31-s − 0.176·32-s − 0.342·34-s − 0.676·35-s − 1.64·37-s + 0.162·38-s + 0.158·40-s + 0.312·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1710 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1710 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.484840411\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.484840411\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 19 | \( 1 + T \) |
good | 7 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 - 12 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 12 T + p T^{2} \) |
| 83 | \( 1 - 8 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 18 T + p T^{2} \) |
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
−9.007501573361219672358859249069, −8.762357662077711478049716277630, −7.61167775805836824156465733476, −7.36268023391797474422359816854, −6.27098962407567852504120158218, −5.16613482743050477908409057827, −4.41523514966053667957387110649, −3.30401299683327559417441826167, −1.96082161662423778223422893983, −1.00778413777270536949067485726,
1.00778413777270536949067485726, 1.96082161662423778223422893983, 3.30401299683327559417441826167, 4.41523514966053667957387110649, 5.16613482743050477908409057827, 6.27098962407567852504120158218, 7.36268023391797474422359816854, 7.61167775805836824156465733476, 8.762357662077711478049716277630, 9.007501573361219672358859249069