| L(s) = 1 | − 5-s − 2·11-s − 4·13-s − 2·17-s − 2·23-s + 25-s − 6·29-s + 4·31-s − 10·37-s + 2·41-s + 4·43-s + 8·47-s − 10·53-s + 2·55-s − 4·59-s + 8·61-s + 4·65-s + 8·67-s − 6·71-s + 4·73-s + 4·79-s − 4·83-s + 2·85-s + 10·89-s − 12·97-s + 14·101-s + 4·103-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 0.603·11-s − 1.10·13-s − 0.485·17-s − 0.417·23-s + 1/5·25-s − 1.11·29-s + 0.718·31-s − 1.64·37-s + 0.312·41-s + 0.609·43-s + 1.16·47-s − 1.37·53-s + 0.269·55-s − 0.520·59-s + 1.02·61-s + 0.496·65-s + 0.977·67-s − 0.712·71-s + 0.468·73-s + 0.450·79-s − 0.439·83-s + 0.216·85-s + 1.05·89-s − 1.21·97-s + 1.39·101-s + 0.394·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8820 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8820 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.010845866\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.010845866\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 \) |
| good | 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 97 | \( 1 + 12 T + p T^{2} \) |
| 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
−7.66780372491090711027681137181, −7.21580775238727646329921950032, −6.46515309037437206072332038071, −5.59024121611366573302519449979, −4.97190723580801965570863065554, −4.28189959975158716079165406976, −3.47972888140025473559491046033, −2.60168717473018233058317410716, −1.88240848180737697141864203578, −0.46518926125573767576091126958,
0.46518926125573767576091126958, 1.88240848180737697141864203578, 2.60168717473018233058317410716, 3.47972888140025473559491046033, 4.28189959975158716079165406976, 4.97190723580801965570863065554, 5.59024121611366573302519449979, 6.46515309037437206072332038071, 7.21580775238727646329921950032, 7.66780372491090711027681137181
