| L(s) = 1 | − 5-s − 7-s + 4·11-s − 2·13-s − 2·17-s − 4·19-s + 25-s + 10·29-s + 35-s + 6·37-s + 6·41-s − 4·43-s + 8·47-s + 49-s − 6·53-s − 4·55-s + 4·59-s − 10·61-s + 2·65-s + 4·67-s + 16·71-s − 14·73-s − 4·77-s + 8·79-s + 4·83-s + 2·85-s − 10·89-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 0.377·7-s + 1.20·11-s − 0.554·13-s − 0.485·17-s − 0.917·19-s + 1/5·25-s + 1.85·29-s + 0.169·35-s + 0.986·37-s + 0.937·41-s − 0.609·43-s + 1.16·47-s + 1/7·49-s − 0.824·53-s − 0.539·55-s + 0.520·59-s − 1.28·61-s + 0.248·65-s + 0.488·67-s + 1.89·71-s − 1.63·73-s − 0.455·77-s + 0.900·79-s + 0.439·83-s + 0.216·85-s − 1.05·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.514966982\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.514966982\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 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 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 10 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 16 T + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 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
−8.903900080530951245031265181701, −8.229840493552864703836273827418, −7.31550633365076634919505584164, −6.56685109351084816736151587899, −6.04937146457284202575157728923, −4.70885712136720502272020682329, −4.21702561516116135390460587584, −3.20677409250645666329963957162, −2.19117175954656438879220051942, −0.77915345755938942605491958714,
0.77915345755938942605491958714, 2.19117175954656438879220051942, 3.20677409250645666329963957162, 4.21702561516116135390460587584, 4.70885712136720502272020682329, 6.04937146457284202575157728923, 6.56685109351084816736151587899, 7.31550633365076634919505584164, 8.229840493552864703836273827418, 8.903900080530951245031265181701
