L(s) = 1 | + 3·3-s + 5-s + 6·9-s + 11-s − 2·13-s + 3·15-s − 3·17-s + 5·19-s + 3·23-s − 4·25-s + 9·27-s − 6·29-s − 31-s + 3·33-s − 5·37-s − 6·39-s + 10·41-s + 4·43-s + 6·45-s + 47-s − 9·51-s − 9·53-s + 55-s + 15·57-s + 3·59-s − 3·61-s − 2·65-s + ⋯ |
L(s) = 1 | + 1.73·3-s + 0.447·5-s + 2·9-s + 0.301·11-s − 0.554·13-s + 0.774·15-s − 0.727·17-s + 1.14·19-s + 0.625·23-s − 4/5·25-s + 1.73·27-s − 1.11·29-s − 0.179·31-s + 0.522·33-s − 0.821·37-s − 0.960·39-s + 1.56·41-s + 0.609·43-s + 0.894·45-s + 0.145·47-s − 1.26·51-s − 1.23·53-s + 0.134·55-s + 1.98·57-s + 0.390·59-s − 0.384·61-s − 0.248·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.938805148\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.938805148\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
good | 3 | \( 1 - p T + p T^{2} \) |
| 5 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 - T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 - 5 T + p T^{2} \) |
| 23 | \( 1 - 3 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + T + p T^{2} \) |
| 37 | \( 1 + 5 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - T + p T^{2} \) |
| 53 | \( 1 + 9 T + p T^{2} \) |
| 59 | \( 1 - 3 T + p T^{2} \) |
| 61 | \( 1 + 3 T + p T^{2} \) |
| 67 | \( 1 + 11 T + p T^{2} \) |
| 71 | \( 1 + 16 T + p T^{2} \) |
| 73 | \( 1 + 7 T + p T^{2} \) |
| 79 | \( 1 - 11 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 - 9 T + p T^{2} \) |
| 97 | \( 1 + 6 T + p T^{2} \) |
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
−9.909934185067456468681392565173, −9.287262093819373491188835101633, −8.834897152421966708934214741373, −7.64209225546588621077680033121, −7.26679637140838061473697520358, −5.93137139427462572617317343004, −4.64642458837335514011656102062, −3.60825384437311163852849219399, −2.66681820396833997806059796550, −1.67920345541699519630742216151,
1.67920345541699519630742216151, 2.66681820396833997806059796550, 3.60825384437311163852849219399, 4.64642458837335514011656102062, 5.93137139427462572617317343004, 7.26679637140838061473697520358, 7.64209225546588621077680033121, 8.834897152421966708934214741373, 9.287262093819373491188835101633, 9.909934185067456468681392565173