| L(s) = 1 | + 3-s + 0.732·5-s − 7-s + 9-s + 4.19·11-s + 0.732·15-s − 1.26·17-s − 7.46·19-s − 21-s + 8.73·23-s − 4.46·25-s + 27-s + 3.46·29-s − 4.92·31-s + 4.19·33-s − 0.732·35-s + 10·37-s − 1.26·41-s + 0.928·43-s + 0.732·45-s + 6.92·47-s + 49-s − 1.26·51-s + 3.46·53-s + 3.07·55-s − 7.46·57-s + 10.9·59-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.327·5-s − 0.377·7-s + 0.333·9-s + 1.26·11-s + 0.189·15-s − 0.307·17-s − 1.71·19-s − 0.218·21-s + 1.82·23-s − 0.892·25-s + 0.192·27-s + 0.643·29-s − 0.885·31-s + 0.730·33-s − 0.123·35-s + 1.64·37-s − 0.198·41-s + 0.141·43-s + 0.109·45-s + 1.01·47-s + 0.142·49-s − 0.177·51-s + 0.475·53-s + 0.414·55-s − 0.988·57-s + 1.42·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5376 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5376 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.721548906\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.721548906\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| good | 5 | \( 1 - 0.732T + 5T^{2} \) |
| 11 | \( 1 - 4.19T + 11T^{2} \) |
| 13 | \( 1 + 13T^{2} \) |
| 17 | \( 1 + 1.26T + 17T^{2} \) |
| 19 | \( 1 + 7.46T + 19T^{2} \) |
| 23 | \( 1 - 8.73T + 23T^{2} \) |
| 29 | \( 1 - 3.46T + 29T^{2} \) |
| 31 | \( 1 + 4.92T + 31T^{2} \) |
| 37 | \( 1 - 10T + 37T^{2} \) |
| 41 | \( 1 + 1.26T + 41T^{2} \) |
| 43 | \( 1 - 0.928T + 43T^{2} \) |
| 47 | \( 1 - 6.92T + 47T^{2} \) |
| 53 | \( 1 - 3.46T + 53T^{2} \) |
| 59 | \( 1 - 10.9T + 59T^{2} \) |
| 61 | \( 1 - 1.46T + 61T^{2} \) |
| 67 | \( 1 - 7.46T + 67T^{2} \) |
| 71 | \( 1 - 14.1T + 71T^{2} \) |
| 73 | \( 1 + 10.3T + 73T^{2} \) |
| 79 | \( 1 - 9.46T + 79T^{2} \) |
| 83 | \( 1 + 8.39T + 83T^{2} \) |
| 89 | \( 1 - 1.26T + 89T^{2} \) |
| 97 | \( 1 - 6.39T + 97T^{2} \) |
| 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
−8.376240015135890807307314059851, −7.38561352211010917423906586039, −6.69213780448247960070151225424, −6.25179625482481576241879389631, −5.28200897054093502751754310643, −4.24488603290357350864706653370, −3.82152617405076016790866812527, −2.73613962062443313741311136127, −2.00821155542493407239049129768, −0.882213172867133979406920798167,
0.882213172867133979406920798167, 2.00821155542493407239049129768, 2.73613962062443313741311136127, 3.82152617405076016790866812527, 4.24488603290357350864706653370, 5.28200897054093502751754310643, 6.25179625482481576241879389631, 6.69213780448247960070151225424, 7.38561352211010917423906586039, 8.376240015135890807307314059851
