L(s) = 1 | − 2·5-s − 7-s + 4·11-s + 2·13-s + 6·17-s + 8·19-s − 25-s − 6·29-s + 8·31-s + 2·35-s − 2·37-s − 2·41-s − 4·43-s + 8·47-s + 49-s − 6·53-s − 8·55-s − 6·61-s − 4·65-s − 4·67-s + 8·71-s + 10·73-s − 4·77-s + 16·79-s − 8·83-s − 12·85-s + 6·89-s + ⋯ |
L(s) = 1 | − 0.894·5-s − 0.377·7-s + 1.20·11-s + 0.554·13-s + 1.45·17-s + 1.83·19-s − 1/5·25-s − 1.11·29-s + 1.43·31-s + 0.338·35-s − 0.328·37-s − 0.312·41-s − 0.609·43-s + 1.16·47-s + 1/7·49-s − 0.824·53-s − 1.07·55-s − 0.768·61-s − 0.496·65-s − 0.488·67-s + 0.949·71-s + 1.17·73-s − 0.455·77-s + 1.80·79-s − 0.878·83-s − 1.30·85-s + 0.635·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.325718303\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.325718303\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + T \) |
good | 5 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 - 8 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 - 16 T + p T^{2} \) |
| 83 | \( 1 + 8 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 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
−11.09014316837144741766145173789, −9.886707849578587066687733658491, −9.256691510852288615675806117776, −8.093537689649372229665039418958, −7.40397429472305265033368100557, −6.35373904523042773985731435784, −5.29340219049011126125230323142, −3.89550776474805329328185747620, −3.26735667825841356907489548421, −1.15037892508990574535308118020,
1.15037892508990574535308118020, 3.26735667825841356907489548421, 3.89550776474805329328185747620, 5.29340219049011126125230323142, 6.35373904523042773985731435784, 7.40397429472305265033368100557, 8.093537689649372229665039418958, 9.256691510852288615675806117776, 9.886707849578587066687733658491, 11.09014316837144741766145173789