| L(s) = 1 | − 1.13·3-s − 1.24·5-s + 4.57·7-s − 1.70·9-s + 0.731·11-s + 1.94·13-s + 1.41·15-s + 4.32·17-s − 0.162·19-s − 5.20·21-s − 9.50·23-s − 3.44·25-s + 5.34·27-s + 8.98·29-s − 4.60·31-s − 0.830·33-s − 5.70·35-s − 1.75·37-s − 2.20·39-s − 0.0325·41-s + 8.64·43-s + 2.13·45-s + 11.4·47-s + 13.9·49-s − 4.91·51-s + 6.31·53-s − 0.911·55-s + ⋯ |
| L(s) = 1 | − 0.655·3-s − 0.557·5-s + 1.73·7-s − 0.569·9-s + 0.220·11-s + 0.539·13-s + 0.365·15-s + 1.04·17-s − 0.0372·19-s − 1.13·21-s − 1.98·23-s − 0.689·25-s + 1.02·27-s + 1.66·29-s − 0.827·31-s − 0.144·33-s − 0.964·35-s − 0.288·37-s − 0.353·39-s − 0.00508·41-s + 1.31·43-s + 0.317·45-s + 1.67·47-s + 1.99·49-s − 0.688·51-s + 0.867·53-s − 0.122·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.683890735\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.683890735\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 251 | \( 1 + T \) |
| good | 3 | \( 1 + 1.13T + 3T^{2} \) |
| 5 | \( 1 + 1.24T + 5T^{2} \) |
| 7 | \( 1 - 4.57T + 7T^{2} \) |
| 11 | \( 1 - 0.731T + 11T^{2} \) |
| 13 | \( 1 - 1.94T + 13T^{2} \) |
| 17 | \( 1 - 4.32T + 17T^{2} \) |
| 19 | \( 1 + 0.162T + 19T^{2} \) |
| 23 | \( 1 + 9.50T + 23T^{2} \) |
| 29 | \( 1 - 8.98T + 29T^{2} \) |
| 31 | \( 1 + 4.60T + 31T^{2} \) |
| 37 | \( 1 + 1.75T + 37T^{2} \) |
| 41 | \( 1 + 0.0325T + 41T^{2} \) |
| 43 | \( 1 - 8.64T + 43T^{2} \) |
| 47 | \( 1 - 11.4T + 47T^{2} \) |
| 53 | \( 1 - 6.31T + 53T^{2} \) |
| 59 | \( 1 + 3.32T + 59T^{2} \) |
| 61 | \( 1 + 4.48T + 61T^{2} \) |
| 67 | \( 1 - 1.39T + 67T^{2} \) |
| 71 | \( 1 - 0.425T + 71T^{2} \) |
| 73 | \( 1 + 12.0T + 73T^{2} \) |
| 79 | \( 1 - 8.01T + 79T^{2} \) |
| 83 | \( 1 + 3.79T + 83T^{2} \) |
| 89 | \( 1 + 9.80T + 89T^{2} \) |
| 97 | \( 1 + 8.07T + 97T^{2} \) |
| 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
−7.72955601581380208139571667111, −7.44098311644989807632398982921, −6.16342107454970038874688468467, −5.79338928086147930212701746068, −5.08920314884605449508785856364, −4.29682033895125357320420522243, −3.77437864260454009047588692722, −2.57645851099815640228932364398, −1.61608424162009976714839821049, −0.69767630253297720928873370273,
0.69767630253297720928873370273, 1.61608424162009976714839821049, 2.57645851099815640228932364398, 3.77437864260454009047588692722, 4.29682033895125357320420522243, 5.08920314884605449508785856364, 5.79338928086147930212701746068, 6.16342107454970038874688468467, 7.44098311644989807632398982921, 7.72955601581380208139571667111
