L(s) = 1 | + 3·3-s + 5-s + 6·9-s − 2·11-s + 6·13-s + 3·15-s − 2·17-s − 9·23-s + 25-s + 9·27-s + 3·29-s − 2·31-s − 6·33-s + 8·37-s + 18·39-s − 5·41-s + 43-s + 6·45-s − 8·47-s − 6·51-s + 4·53-s − 2·55-s + 8·59-s − 7·61-s + 6·65-s − 3·67-s − 27·69-s + ⋯ |
L(s) = 1 | + 1.73·3-s + 0.447·5-s + 2·9-s − 0.603·11-s + 1.66·13-s + 0.774·15-s − 0.485·17-s − 1.87·23-s + 1/5·25-s + 1.73·27-s + 0.557·29-s − 0.359·31-s − 1.04·33-s + 1.31·37-s + 2.88·39-s − 0.780·41-s + 0.152·43-s + 0.894·45-s − 1.16·47-s − 0.840·51-s + 0.549·53-s − 0.269·55-s + 1.04·59-s − 0.896·61-s + 0.744·65-s − 0.366·67-s − 3.25·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.133303692\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.133303692\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 \) |
good | 3 | \( 1 - p T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + 9 T + p T^{2} \) |
| 29 | \( 1 - 3 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 + 5 T + p T^{2} \) |
| 43 | \( 1 - T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - 4 T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 + 7 T + p T^{2} \) |
| 67 | \( 1 + 3 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 - T + p T^{2} \) |
| 89 | \( 1 + 13 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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
−9.861341895886835928479556830898, −9.035811175689496960229121791675, −8.323416847465527228736051049638, −7.88368303014610380901450609320, −6.70298540851638126059873615067, −5.80575855175278089882628697130, −4.36419093915488757070038212385, −3.56036060499126781046326462835, −2.56199961749627251052556531480, −1.60608326000150852234991263675,
1.60608326000150852234991263675, 2.56199961749627251052556531480, 3.56036060499126781046326462835, 4.36419093915488757070038212385, 5.80575855175278089882628697130, 6.70298540851638126059873615067, 7.88368303014610380901450609320, 8.323416847465527228736051049638, 9.035811175689496960229121791675, 9.861341895886835928479556830898