L(s) = 1 | + 2-s + 4-s − 0.267·5-s − 7-s + 8-s − 0.267·10-s − 6.19·11-s − 6.46·13-s − 14-s + 16-s − 7·17-s + 0.732·19-s − 0.267·20-s − 6.19·22-s + 4.19·23-s − 4.92·25-s − 6.46·26-s − 28-s − 1.53·29-s + 8.19·31-s + 32-s − 7·34-s + 0.267·35-s + 10.6·37-s + 0.732·38-s − 0.267·40-s − 2.53·41-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.5·4-s − 0.119·5-s − 0.377·7-s + 0.353·8-s − 0.0847·10-s − 1.86·11-s − 1.79·13-s − 0.267·14-s + 0.250·16-s − 1.69·17-s + 0.167·19-s − 0.0599·20-s − 1.32·22-s + 0.874·23-s − 0.985·25-s − 1.26·26-s − 0.188·28-s − 0.285·29-s + 1.47·31-s + 0.176·32-s − 1.20·34-s + 0.0452·35-s + 1.75·37-s + 0.118·38-s − 0.0423·40-s − 0.396·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1134 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1134 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 7 | \( 1 + T \) |
good | 5 | \( 1 + 0.267T + 5T^{2} \) |
| 11 | \( 1 + 6.19T + 11T^{2} \) |
| 13 | \( 1 + 6.46T + 13T^{2} \) |
| 17 | \( 1 + 7T + 17T^{2} \) |
| 19 | \( 1 - 0.732T + 19T^{2} \) |
| 23 | \( 1 - 4.19T + 23T^{2} \) |
| 29 | \( 1 + 1.53T + 29T^{2} \) |
| 31 | \( 1 - 8.19T + 31T^{2} \) |
| 37 | \( 1 - 10.6T + 37T^{2} \) |
| 41 | \( 1 + 2.53T + 41T^{2} \) |
| 43 | \( 1 + 1.46T + 43T^{2} \) |
| 47 | \( 1 + 4.73T + 47T^{2} \) |
| 53 | \( 1 - 9.46T + 53T^{2} \) |
| 59 | \( 1 + 4.19T + 59T^{2} \) |
| 61 | \( 1 - 3.92T + 61T^{2} \) |
| 67 | \( 1 + 6.73T + 67T^{2} \) |
| 71 | \( 1 + 6.53T + 71T^{2} \) |
| 73 | \( 1 - 8.26T + 73T^{2} \) |
| 79 | \( 1 + 9.12T + 79T^{2} \) |
| 83 | \( 1 + 16.5T + 83T^{2} \) |
| 89 | \( 1 + 9.92T + 89T^{2} \) |
| 97 | \( 1 - 10.9T + 97T^{2} \) |
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
−9.664074941591599720742791732481, −8.428341657674312088421045214990, −7.56379236408772484960119959418, −6.94693463030764101693840896249, −5.87373824601335053405090479509, −4.95132725704934400632122717422, −4.38862553849927697361249010289, −2.83636973829316666374829135119, −2.38273707489870961758895142314, 0,
2.38273707489870961758895142314, 2.83636973829316666374829135119, 4.38862553849927697361249010289, 4.95132725704934400632122717422, 5.87373824601335053405090479509, 6.94693463030764101693840896249, 7.56379236408772484960119959418, 8.428341657674312088421045214990, 9.664074941591599720742791732481