L(s) = 1 | + 1.60·2-s + 3-s + 0.568·4-s − 4.39·5-s + 1.60·6-s + 7-s − 2.29·8-s + 9-s − 7.04·10-s + 3.13·11-s + 0.568·12-s + 0.710·13-s + 1.60·14-s − 4.39·15-s − 4.81·16-s − 3.80·17-s + 1.60·18-s + 4.84·19-s − 2.49·20-s + 21-s + 5.02·22-s − 2.77·23-s − 2.29·24-s + 14.3·25-s + 1.13·26-s + 27-s + 0.568·28-s + ⋯ |
L(s) = 1 | + 1.13·2-s + 0.577·3-s + 0.284·4-s − 1.96·5-s + 0.654·6-s + 0.377·7-s − 0.811·8-s + 0.333·9-s − 2.22·10-s + 0.945·11-s + 0.164·12-s + 0.197·13-s + 0.428·14-s − 1.13·15-s − 1.20·16-s − 0.923·17-s + 0.377·18-s + 1.11·19-s − 0.558·20-s + 0.218·21-s + 1.07·22-s − 0.579·23-s − 0.468·24-s + 2.86·25-s + 0.223·26-s + 0.192·27-s + 0.107·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8043 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8043 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 383 | \( 1 + T \) |
good | 2 | \( 1 - 1.60T + 2T^{2} \) |
| 5 | \( 1 + 4.39T + 5T^{2} \) |
| 11 | \( 1 - 3.13T + 11T^{2} \) |
| 13 | \( 1 - 0.710T + 13T^{2} \) |
| 17 | \( 1 + 3.80T + 17T^{2} \) |
| 19 | \( 1 - 4.84T + 19T^{2} \) |
| 23 | \( 1 + 2.77T + 23T^{2} \) |
| 29 | \( 1 - 1.87T + 29T^{2} \) |
| 31 | \( 1 - 8.79T + 31T^{2} \) |
| 37 | \( 1 + 11.5T + 37T^{2} \) |
| 41 | \( 1 + 3.26T + 41T^{2} \) |
| 43 | \( 1 - 3.29T + 43T^{2} \) |
| 47 | \( 1 + 6.85T + 47T^{2} \) |
| 53 | \( 1 + 10.3T + 53T^{2} \) |
| 59 | \( 1 - 1.04T + 59T^{2} \) |
| 61 | \( 1 - 11.5T + 61T^{2} \) |
| 67 | \( 1 + 11.2T + 67T^{2} \) |
| 71 | \( 1 + 12.9T + 71T^{2} \) |
| 73 | \( 1 + 12.8T + 73T^{2} \) |
| 79 | \( 1 - 10.2T + 79T^{2} \) |
| 83 | \( 1 + 1.27T + 83T^{2} \) |
| 89 | \( 1 + 8.72T + 89T^{2} \) |
| 97 | \( 1 - 16.8T + 97T^{2} \) |
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
−7.36559925126403021090090655003, −6.85746137042598671849918913575, −6.12065041374371294669275957419, −4.88584273196312754962909350782, −4.60771160252362425050625266362, −3.86165182181422559681805604026, −3.43745062864305136879989945809, −2.73683035613487864615648341456, −1.31433367030452505031348541882, 0,
1.31433367030452505031348541882, 2.73683035613487864615648341456, 3.43745062864305136879989945809, 3.86165182181422559681805604026, 4.60771160252362425050625266362, 4.88584273196312754962909350782, 6.12065041374371294669275957419, 6.85746137042598671849918913575, 7.36559925126403021090090655003