| L(s) = 1 | − 0.913·5-s + 7-s + 2.64·11-s + 4·13-s − 3.46·17-s − 5.58·19-s − 3.46·23-s − 4.16·25-s + 8.75·29-s + 9.16·31-s − 0.913·35-s + 3·37-s + 0.913·41-s − 0.582·43-s − 13.1·47-s + 49-s + 8.66·53-s − 2.41·55-s + 3.46·59-s + 11.5·61-s − 3.65·65-s − 8.58·67-s − 4.47·71-s + 15.1·73-s + 2.64·77-s − 0.582·79-s − 9.66·83-s + ⋯ |
| L(s) = 1 | − 0.408·5-s + 0.377·7-s + 0.797·11-s + 1.10·13-s − 0.840·17-s − 1.28·19-s − 0.722·23-s − 0.833·25-s + 1.62·29-s + 1.64·31-s − 0.154·35-s + 0.493·37-s + 0.142·41-s − 0.0888·43-s − 1.91·47-s + 0.142·49-s + 1.18·53-s − 0.325·55-s + 0.450·59-s + 1.48·61-s − 0.453·65-s − 1.04·67-s − 0.530·71-s + 1.77·73-s + 0.301·77-s − 0.0655·79-s − 1.06·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9072 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9072 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.019919125\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.019919125\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 - T \) |
| good | 5 | \( 1 + 0.913T + 5T^{2} \) |
| 11 | \( 1 - 2.64T + 11T^{2} \) |
| 13 | \( 1 - 4T + 13T^{2} \) |
| 17 | \( 1 + 3.46T + 17T^{2} \) |
| 19 | \( 1 + 5.58T + 19T^{2} \) |
| 23 | \( 1 + 3.46T + 23T^{2} \) |
| 29 | \( 1 - 8.75T + 29T^{2} \) |
| 31 | \( 1 - 9.16T + 31T^{2} \) |
| 37 | \( 1 - 3T + 37T^{2} \) |
| 41 | \( 1 - 0.913T + 41T^{2} \) |
| 43 | \( 1 + 0.582T + 43T^{2} \) |
| 47 | \( 1 + 13.1T + 47T^{2} \) |
| 53 | \( 1 - 8.66T + 53T^{2} \) |
| 59 | \( 1 - 3.46T + 59T^{2} \) |
| 61 | \( 1 - 11.5T + 61T^{2} \) |
| 67 | \( 1 + 8.58T + 67T^{2} \) |
| 71 | \( 1 + 4.47T + 71T^{2} \) |
| 73 | \( 1 - 15.1T + 73T^{2} \) |
| 79 | \( 1 + 0.582T + 79T^{2} \) |
| 83 | \( 1 + 9.66T + 83T^{2} \) |
| 89 | \( 1 - 1.82T + 89T^{2} \) |
| 97 | \( 1 + 1.58T + 97T^{2} \) |
| 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
−7.966223978060829431219549348954, −6.83783733919967337812298516246, −6.43740824476057519923581741128, −5.87513068612276177319378975258, −4.70040139466614427460421285155, −4.25749985231424359529323490756, −3.63593654923881207787072078041, −2.57278663510629328545056338043, −1.71717537576295633903030801809, −0.70223894163811122285835032618,
0.70223894163811122285835032618, 1.71717537576295633903030801809, 2.57278663510629328545056338043, 3.63593654923881207787072078041, 4.25749985231424359529323490756, 4.70040139466614427460421285155, 5.87513068612276177319378975258, 6.43740824476057519923581741128, 6.83783733919967337812298516246, 7.966223978060829431219549348954
