L(s) = 1 | − 2-s + 4-s + 1.23·5-s − 8-s − 1.23·10-s + 11-s + 16-s − 5.23·17-s + 7.70·19-s + 1.23·20-s − 22-s + 2.47·23-s − 3.47·25-s − 4.47·29-s + 2.76·31-s − 32-s + 5.23·34-s − 10.9·37-s − 7.70·38-s − 1.23·40-s − 5.23·41-s + 6.47·43-s + 44-s − 2.47·46-s + 3.70·47-s + 3.47·50-s + 6·53-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.5·4-s + 0.552·5-s − 0.353·8-s − 0.390·10-s + 0.301·11-s + 0.250·16-s − 1.26·17-s + 1.76·19-s + 0.276·20-s − 0.213·22-s + 0.515·23-s − 0.694·25-s − 0.830·29-s + 0.496·31-s − 0.176·32-s + 0.897·34-s − 1.79·37-s − 1.25·38-s − 0.195·40-s − 0.817·41-s + 0.986·43-s + 0.150·44-s − 0.364·46-s + 0.540·47-s + 0.491·50-s + 0.824·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9702 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9702 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.578322129\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.578322129\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 11 | \( 1 - T \) |
good | 5 | \( 1 - 1.23T + 5T^{2} \) |
| 13 | \( 1 + 13T^{2} \) |
| 17 | \( 1 + 5.23T + 17T^{2} \) |
| 19 | \( 1 - 7.70T + 19T^{2} \) |
| 23 | \( 1 - 2.47T + 23T^{2} \) |
| 29 | \( 1 + 4.47T + 29T^{2} \) |
| 31 | \( 1 - 2.76T + 31T^{2} \) |
| 37 | \( 1 + 10.9T + 37T^{2} \) |
| 41 | \( 1 + 5.23T + 41T^{2} \) |
| 43 | \( 1 - 6.47T + 43T^{2} \) |
| 47 | \( 1 - 3.70T + 47T^{2} \) |
| 53 | \( 1 - 6T + 53T^{2} \) |
| 59 | \( 1 + 1.52T + 59T^{2} \) |
| 61 | \( 1 + 61T^{2} \) |
| 67 | \( 1 - 15.4T + 67T^{2} \) |
| 71 | \( 1 - 2.47T + 71T^{2} \) |
| 73 | \( 1 - 15.7T + 73T^{2} \) |
| 79 | \( 1 - 11.4T + 79T^{2} \) |
| 83 | \( 1 - 1.23T + 83T^{2} \) |
| 89 | \( 1 + 6.47T + 89T^{2} \) |
| 97 | \( 1 + 97T^{2} \) |
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
−7.71992016479634089860880991159, −6.90951080462553017174952159381, −6.62506979527121883220720456246, −5.54100819292567442174291173581, −5.23341744295641218995391737590, −4.07505504720581808236900605838, −3.32548423912170175310326166844, −2.37519692820945728285819048018, −1.69048504214662511733193828091, −0.67258874766466695522463288606,
0.67258874766466695522463288606, 1.69048504214662511733193828091, 2.37519692820945728285819048018, 3.32548423912170175310326166844, 4.07505504720581808236900605838, 5.23341744295641218995391737590, 5.54100819292567442174291173581, 6.62506979527121883220720456246, 6.90951080462553017174952159381, 7.71992016479634089860880991159